Approximate Guarding of Monotone and Rectilinear Polygons

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Approximate Guarding of Monotone and Rectilinear Polygons"

Transcription

1 Aroximate Guarding of Monotone and Rectilinear Polygons Erik Krohn Bengt J. Nilsson Abstract We show that vertex guarding a monotone olygon is NP-hard and construct a constant factor aroximation algorithm for interior guarding monotone olygons. Using this algorithm we obtain an aroximation algorithm for interior guarding rectilinear olygons that has an aroximation factor indeendent of the number of vertices of the olygon. If the size of the smallest interior guard cover is OPT for a rectilinear olygon, our algorithm roduces a guard set of size O(OPT 2 ). Comutational geometry Art gallery roblems Monotone olygons Rectilinear olygons Aroximation algorithms 1 Introduction The art gallery roblem is erhas the best known roblem in comutational geometry. It asks for the minimum number of guards to guard a sace having obstacles. Originally, the obstacles were considered to be walls mutually connected to form a closed Jordan curve, hence, a simle olygon. Tight bounds for the number of guards necessary and sufficient were found by Chvátal [7] and Fisk [17]. Subsequently, other obstacle saces, both more general and more restricted than simle olygons have also been considered for guarding roblems, most notably, olygons with holes and simle rectilinear olygons [21, 32]. Art gallery roblems are motivated by alications such as line-of-sight transmission networks in terrains, such as, signal communications and broadcasting, cellular telehony systems and other telecommunication technologies as well as lacement of motion detectors and security cameras. We distinguish between two tyes of guarding roblems in simle olygons. Vertex guarding considers only guards ositioned at vertices of the olygon, whereas interior guarding allows the guards to be laced anywhere in the interior of the olygon. The comutational comlexity question of guarding simle olygons was settled by Aggarwal [1] and Lee and Lin [26] indeendently when they showed that the roblem is NP-hard for both vertex guards and interior guards. Further results have shown that already for very restricted subclasses of olygons the roblem is still NP-hard [2, 30]. Chen et al. [5] claim that vertex guarding a monotone olygon is NP-hard, however the details of their roof are omitted and still to be verified. We resent a new roof that vertex guarding a monotone olygon is NP-hard. The aroximation comlexity of guarding olygons has been studied by Eidenbenz and others. Eidenbenz [14] shows that olygons with holes cannot be efficiently guarded by fewer than Ω(log n) times the otimal number of interior or vertex guards, unless P=NP, where n is the number of vertices of the olygon. Brodén et al. and Eidenbenz [2, 13] indeendently rove that interior guarding simle olygons is APX-hard. Deartment of Comuter Science, University of Wisconsin Oshkosh, Oshkosh, WI, 54901, USA. Deartment of Comuter Science, Malmö University, SE Malmö, Sweden. 1

2 s t monotone rectilinear Figure 1: Illustrating the olygon classes. Any olygon (with or without holes) can be efficiently vertex guarded with logarithmic aroximation factor in n, the number of vertices of the olygon. The algorithm is a simle reduction to SET COVER and goes as follows [19]: comute the arrangement roduced by the visibility olygons of the vertices. Next, let each vertex v corresond to a set in the set cover instance consisting of elements corresonding to the faces of the arrangement that lie in the visibility olygon of v. The greedy algorithm for SET COVER will then roduce a guard cover having logarithmic aroximation factor. The above result can be imroved for simle olygons using randomization, giving an algorithm with exected running time O(nOPT 2 v log4 n) that roduces a vertex guard cover with aroximation factor O(log OPT v ) with high robability, where OPT v is the smallest vertex guard cover for the olygon [12]. Taking the same aroach one ste further, Deshande et al. [11] resent a seudo-olynomial randomized algorithm for finding a guard cover (without any restriction on lacement) with aroximation factor O(log OPT). We rove olynomial time deterministic aroximation algorithms for interior guarding of monotone and rectilinear olygons. As we have already mentioned, vertex guarding of monotone olygons is NP-hard, and furthermore, otimally guarding rectilinear olygons is also NP-hard [23]. This rovides the basis for our interest in aroximation algorithms for these roblems. The art gallery roblem concerns itself with covering olygons using star shaed ieces, the visibility olygons of the guards. Covering olygons with other tye of objects, e.g., convex olygons, etc., remains NP-hard in general; [8, 9, 16, 20, 29, 32, 33, 34]. The next section contains some useful definitions. Section 3 contains our NP-hardness roof for monotone olygons and in Sections 4 and 5 we describe the aroximation algorithms for guarding monotone and rectilinear olygons resectively. 2 Definitions A olygon P is l-monotone if there is a line of monotonicity l such that any line orthogonal to l has a simly connected intersection with P. When we talk about monotone olygons, we will henceforth assume that they are x-monotone, i.e., the x-axis is the line of monotonicity for the olygons we consider; see Figure 1. The boundary of a monotone olygon P can be subdivided into two chains, the uer chain U and the lower chain D. Let s and t be the leftmost and rightmost vertices of P resectively. The chain U consists of the boundary ath followed from s to t in clockwise direction, whereas D is the boundary ath followed from s to t in counterclockwise direction. A olygon P is rectilinear if the boundary of P consists of axis arallel line segments. Hence, at each vertex, the interior angle between the two connecting boundary edges is either 90 or 270 degrees; see Figure 1. 2

3 q r r r t SP(,t) r Figure 2: Illustrating the roof of Lemma 2.1. Let VP() denote the visibility olygon of P from the oint, i.e, the set of oints in P that can be connected with a line segment to without intersecting the outside of P. Consider a artial set of guard oints g 1,..., g m inpand the union of their visibility olygons m i=1 VP(g i), the set P \ m i=1 VP(g i) is the region of P not seen by the oints g 1,..., g m. This region consists of a set of simly connected olygonal regions called ockets bounded by either the olygon boundary or the edges of the visibility olygons. The following definitions are useful for monotone olygons. Since the x-axis is the line of monotonicity it makes sense to say that an object A in the olygon is to the left or to the right of some other object B if there is vertical line that searates the two objects. We will occasionally use A B (A B) to denote that A is to the right (to the left) of B. Let q be a oint in VP(). We denote by VP R (, q) the art of VP() that lies to the right of q. Similarly, VP L (, q) is the art of VP() to the left of q. Hence, VP() = VP L (, q) VP R (, q) for all oints q P. We also denote VP R () = VP R (, ) and VP L () = VP L (, ). In the sequel, we will also let SP(, q) denote the shortest (Euclidean) ath between oints and q inside P. LEMMA 2.1 If q is a oint on SP(, t) inside a monotone olygon P, then VP R (, q) VP R (q). PROOF: Let r be a oint to the right of q in P that is visible from. To rove that r is seen from q consider the vertical line through r and its intersection oint r with SP(, t). The three oints, r, and r define a olygon in P having three convex vertices and ossibly some reflex vertices on the ath SP(, r ). Since r sees both and r, r sees all of the ath SP(, r ) and hence also the oint q; see Figure 2. 3 NP-Hardness of Vertex Guarding Monotone Polygons In this section, we will show that vertex guarding a monotone olygon is NP-hard. The reduction is from Monotone 3SAT (M3SAT) [18, age 259 (roblem L02)]. An M3SAT instance (X, C) consists of a air of sets, a set of Boolean variables, X = {x 1, x 2,..., x n } and a set of clauses, C = {c 1, c 2,..., c m }. Each clause contains three literals, c i = x j x k x l, a ositive clause, or c i = x j x k x l, a negative clause, for 1 j, k, l n. An M3SAT instance is satisfiable, if a satisfying truth assignment for C exists such that all clauses c i are true. An ordinary 3SAT instance can easily be transformed to an M3SAT instance by taking each non-monotone clause and relacing it by three monotone ones as follows. c i = x j x k x l ( z i1 z i2 x l ) (z i1 x j x k ) (z i2 x j x k ) c i = x j x k x l (z i1 z i2 x l ) ( z i1 x j x k ) ( z i2 x j x k ) 3

4 x x d( x) x x x x d(x) b(x) Starting attern b(x) d(x) d( x) b(x) Negative Positive Variable atterns Figure 3: The different tyes of variable atterns. where z i1 and z i2 are new variables used only in these three clauses. It is easy to verify that a truth assignment makes clause c i true if and only if the truth assignment makes all the three monotone relacement clauses true as well. By aroriately dulicating clauses, we can assume that the instance has m clauses where m is odd and the instance has (m + 1)/2 ositive clauses and (m 1)/2 negative clauses. Also, let K = n (m + 1). We show that any M3SAT instance is olynomially transformable to an instance of vertex guarding a monotone olygon. We construct a monotone olygon P from the M3SAT instance such that P is guardable by K or fewer guards if and only if the M3SAT instance is satisfiable. We first resent some basic gadgets to show how the olygon is constructed. We then connect these gadgets together to create a olygon. Starting Pattern: The lower boundary of the olygon is divided into two arts, the left and the right sides. The first gadgets on the left side are the starting atterns. The starting atterns are shown to the left in Figure 3. In each attern, the bottom of the downward sike b(x) is the distinguished vertex of the attern. This area is only seen by vertices x and x and must be guarded by one of these two vertices. This attern aears along the left side of the lower boundary of the monotone olygon a total of n times, one corresonding to each variable. Variable Pattern: On the left and the right side of the lower boundary we have variable atterns that verify the assigned truth value of each variable. This attern is shown to the right in Figure 3. Once again, the bottom of the sike at b(x) must be guarded by either x or x. The attern has additional distinguished vertices that we call ledges d(x) and d( x) that must both be seen and this is what forces the choice of guard lacement at either x or x. Figure 4 shows how the starting atterns are connected to variable atterns. If we choose x j in the starting attern, we are forced to continuing to choose x j in each of the subsequent variable atterns. If we at some variable attern would choose x j instead of x j, the ledge d( x j ) is not seen. Similarly, if we in the starting attern choose x j, we are, by the same argument, forced to continuing to choose x j in each of subsequent variable atterns. Clauses: For each clause c in the boolean formula, there is a sequence of variable atterns x 1,...,x n along either the left or the right side of the lower boundary and a clause attern along the uer boundary of the olygon. On the left side of the lower boundary, the variable attern sequence corresonds to negative clauses, on the right side, to ositive clauses. The clause attern on the uer boundary consists of three vertices in an uward sike such that the to vertex of the sike is only seen by the variable atterns corresonding to the literals in the clause; see 4

5 x j x j x k x k. x j x j x j Starting atterns x j x k x k x k x k Positive variable atterns Negative variable atterns Figure 4: Variable atterns transferring logical values. Figure 5. We denote the to vertex of the sike by c to corresond to the clause. We choose our truth value for each variable in the starting variable atterns. The truth values are then mirrored in turn between variable atterns on the right side, corresonding to ositive clauses, and variable atterns on the left side, corresonding to negative clauses, of the lower boundary. Truth values do not change in the mirroring rocess since a variable x j in clause c i only sees the ledge d(x j ) in the next variable attern and none of the other ledges. Similarly x j only sees ledge d( x j ) in the next variable attern; see Figure 4. In the examle of Figure 5 the M3SAT clause corresonds to c = x 1 x 3 x 5. Hence, a vertex guard lacement that corresonds to a truth assignment that makes c true, will have at least one guard on x 1, x 3 or x 5 and can therefore see vertex c without additional guards. We still have variables x 2 and x 4 in the clause, however none of them or their negations see the vertex c. They are there simly to transfer their truth values in case these variables are needed in later clauses. The monotone olygon we construct consists of 4n +(6n + 4)m + 2 vertices. Each starting variable attern having four vertices, each variable attern six vertices, the clause sike consists of three vertices lus one blocking vertex at the start of each clause sequence on the lower boundary and the two leftmost and rightmost oints of the olygon. Consider an M3SAT instance (x 1 x 2 x 3 ) ( x 1 x 3 x 5 ) (x 3 x 4 x 5 ). Figure 6 shows how this instance is transformed into a monotone olygon and a lacement of guards corresonding to the satisfying truth assignment x 1 = x 2 = x 4 = x 5 = false, x 3 = true. Exactly K = n(m + 1) guards are required to guard the olygon since there are K bottom vertices b(x j ) at downward sikes and no vertex in the olygon can see more than one such b(x j ) vertex. If the M3SAT instance is satisfiable, then we lace guards at vertices in accordance to whether the variable is true or false in each of the sequences of variable atterns. Each clause vertex is seen since one of the literals in the associated clause is true and the corresonding vertex has a guard. 5

6 clause c x 1 x 3 x 1 x 2 x 3 x 4 x 5 x 5 Figure 5: A variable attern sequence with its clause sike. Figure 6: Examle reduction of (x 1 x 2 x 3) ( x 1 x 3 x 5) (x 3 x 4 x 5). Points with white centers mark the guards. 6

7 ke(r) R Figure 7: Illustrating the concet of a kernel exansion. The darker shaded areas are the comonents of R and the lighter shaded area is the kernel exansion of R. Suose we have a vertex guard cover of size exactly K. Since each bottom sike b(x j ) is guarded there is a guard at one of x j, x j, or b(x j ) itself. They together make u K guards so there can be no other guards. Since each clause vertex c i is also seen, we can establish which of the guards see this vertex and deduce a satisfying truth assignment from this guard lacement. We have roved the following theorem. THEOREM 1 Finding the smallest vertex guard cover for a monotone olygon is NP-hard. Note that our roof does not immediately generalize to interior guards. In the next section, we show how to aroximate the minimum number of interior guards in a monotone olygon. 4 Interior Guarding Monotone Polygons 4.1 The Guarding Algorithm Our algorithm for guarding a monotone olygon P incrementally guards P starting from the left, moving right. Hence, we are interested in the structure of the ockets that occur when guarding is done in this way. We define the main region that we will be interested in, the sear, that will guide the lacement of our guards. Let R be a, ossibly disconnected, set of oints in P and let q be a oint in P. We denote by R L (q) the oints of R to the left of q. Let v R be a leftmost oint of R, hence, if q is to the left of v R, then R L (q) =. DEFINITION 4.1 The kernel exansion of R, denoted ke(r), is formally defined as the set of oints ke(r) def = {q v R R L (q) VP(q)}, i.e., all the oints q in P to the right of v R that see everything in R L (q); see Figure 7. kernel exansion of a region consisting of two connected comonents. In Section we describe how to comute kernel exansions efficiently. Figure 7 shows the LEMMA 4.1 A kernel exansion ke(r) is a monotone olygon. 7

8 (a) uer ocket (b) uer ocket lower ocket boundary ocket (c) (d) Figure 8: Illustrating ockets. Points with white centers are guards. The shaded areas are the visibility olygons. PROOF: Let q and q be two oints in ke(r) having the same x-coordinate, hence, R L (q) = R L (q ) and assume that R L (q). Let be any oint in R L (q) seen by both q and q. Since the three oints, q, and q form a (ossibly degenerate) triangle in P, any oint between q and q will also see. This means that any vertical line has a simly connected intersection with ke(r) so the region is monotone. Assume that we have a artial guard cover in P with the roerty that all guards are to the left of any ockets remaining in P. Consider such a ocket. We say that is a boundary ocket if it is adjacent to both the uer and lower boundaries U and D of P, an uer ocket if it is adjacent only to the uer boundary U, a lower ocket if it is adjacent only to the lower boundary D and a middle ocket if it is adjacent to neither U nor D; see Figures 8(a) (c). We show in Section that our incremental guarding algorithm never roduces any middle ockets, so we can disregard them for now. Since we assume that the guards all lie to the left of all ockets, it is easy to see that the cover can only generate one boundary ocket. Let be such a boundary ocket. In Lemma 4.7 of Section we show that the kernel exansion of a olygonal region is comletely determined by the vertices of the region. We subdivide the vertices of into three sets, V M, V U and V D, where V M are the vertices interior to P, V U are the vertices coinciding with the uer boundary U of P and V D are the vertices coinciding with the lower boundary D of P. We let P U () be the straight line ath visiting the vertices in V M V U in order from left to right, i.e., all vertices of V U are visited along the uer boundary U. Similarly, we let P D () be the straight line ath visiting the vertices in V M V D in order from left to right. Consider the uer ockets resulting from a artial guard cover and enumerate them U 1, U 2,... from left to right. Similarly enumerate the lower ockets D 1,D 2,... from left to right and denote the boundary ocket B. Define two sets as follows: Q U Q D def = U 1 U 2 P U ( B ), def = D 1 D 2 P D( B ), with which we can establish the main regions of interest in the resentation. 8

9 uer sear s U U s D t uer sear ti u U Figure 9: The dark shaded area is the uer sear. DEFINITION 4.2 For X being one of U or D, let the kernel exansion of Q X be the sear with resect to X, i.e., def s X = ke(q X ). Hence, we have the two sear tyes s U and s D corresonding to the sequences of ockets we are currently considering; see Figure 9 for an examle of the uer sear s U. Since the intersection of monotone olygons is also monotone, a sear can be comuted by a lane swee algorithm going from left to right, maintaining the uer and lower boundaries of the kernel exansions; see Section The rightmost intersection oint between the uer and lower boundary of a sear is called the sear ti and we denote sear tis by u U and u D, corresonding to the two tyes of sears; see Figure 9. By the definition of these oints, u X has the roerty of being the rightmost oint that sees all the oints of the ockets of tye X to the left of it. The sears are deendent on the lacement of the reviously laced guards so we will henceforth refer to them as s U (G ) and s D (G ) given the artial guard set G. For each sear s X (G ), we similarly arameterize the sear ti u X (G ). If G =, the uer and lower sears s U ( ) and s D ( ) together with the uer and lower sear tis u U ( ) and u D ( ) are well defined since all of P is considered a boundary ocket. We can now give the details of our guarding algorithm, dislayed in Figure 10. Each iteration of the algorithm begins by comuting the sears and the sear tis, which we show how to do efficiently in Section Ste 4 selects the leftmost of u U (G) and u D (G), lacing a guard g at this oint in Ste 5. Ste 6 results in the addition of g to the guard set only if s X(G) actually intersects l g, where X denotes the remaining ocket tye different from X. We show how to erform Ste 7 efficiently in Section Since all uer and lower ockets are guarded after the algorithm has concluded, we know that the comlete boundary of P is seen by the guards laced. In fact, we rove in Lemma 4.3 that also the interior of P is seen. We claim the following theorem and dedicate Sections 4.2 and 4.3 to roving it. THEOREM 2 The algorithm GUARD-MONOTONE-POLYGON comutes a guard cover of size at most 30OPT for a monotone olygon P in olynomial time, where OPT is the size of the smallest guard cover for P. 9

10 Algorithm Inut: GUARD-MONOTONE-POLYGON A monotone olygon P Outut: A guard cover for P 1 G := while not all ockets are guarded do 2 Comute s U (G) and u U(G) if there are uer ockets 3 Comute s D (G) and u D(G) if there are lower ockets 4 if u U(G) is to the left of u D(G) then X := U, X := D else X := D, X := U endif 5 Place a guard g at u X(G) and let l g be the vertical line segment through g 6 Place a guard g at an intersection oint of l g and s X(G), if they intersect 7 Place a guard ĝ on l g so that u X(G {g, g, ĝ}) lies as far to the right as ossible 8 G := G {g,g, ĝ} endwhile return G End GUARD-MONOTONE-POLYGON Figure 10: The algorithm for guarding monotone olygons. To hel the reader, we rovide a table of the notation we introduce. Symbol Name Exlanation VP() the visibility olygon of the oint VP L (, q), the art of VP() to the left of q VP R (, q) the art of VP() to the right of q VP B (, q, r) the art of VP() between q and r R L (q) the set of oints of R to the left of q ke(r) kernel exansion the set of oints that see all of R L () Q X the union of tye X ockets s X sear the kernel exansion of Q X, ke(q X ) u X sear ti the rightmost oint of a sear, s X l the line or segment through the oint v X base v X lies on the boundary of VP(u X ) and u X lies on the boundary of VP(u X ) 4.2 Correctness and Aroximation Factor Correctness We know from the construction of algorithm GUARD-MONOTONE-POLYGON that it will guard the boundary of the olygon. However, we need to rove that it will also guard the interior of the olygon. To rove this, it is sufficient to show that our algorithm never roduces a middle ocket. We do this in two stes. The first ste is to show that all guards cannot be on one side of a middle ocket, i.e., a middle ocket can never be generated to the right of the guards as they are laced by the algorithm. The second ste is to show that when the algorithm 10

11 g u q l gd r e u l q q u q q d e d r g d Figure 11: Illustrating the roof of Lemma 4.2. laces new guards, a middle ocket to the left of these guards can never be generated. LEMMA 4.2 Consider a middle ocket of a artial guard set in a monotone olygon. Let r be the leftmost oint in. Not all guards of the artial guard set can be to the left of r. PROOF: Assume for a contradiction, that all the guards are to the left of r. Let r be the rightmost oint of and let e u and e d be the uer and lower edges resectively of that are also adjacent to r. Neither e u nor e d can be vertical since this would immediately give a contradiction, requiring a guard to the right of r. Let q be a oint interior to, below e u and above e d, let l q be the vertical line through q and let q u and q d be the intersection oints of l q with e u and e d resectively. Since all guards are to the left of r, no single guard can see both oints q u and q d, so there are at least two different guards g u and g d that see these oints. Furthermore, the lines of sight from g u to q u and from g d to q d cannot cross, otherwise, either g u or g d will see oints inside the middle ocket giving us a contradiction. Assume without loss of generality that g u is further to the left than g d, otherwise we reverse the roles of g u and g d in the following argument. Let l gd be the vertical line through g d. It intersects the line of sight from g u to q u at q. The four oints g d, q d, q u and q form a convex olygon with the guard g d in one corner. Hence, q is seen by g d, contradicting our assumtion that q is in a middle ocket; see Figure 11. Lemma 4.2 shows that as algorithm GUARD-MONOTONE-POLYGON laces guards incrementally in the olygon, it can never generate a middle ocket to the right of the rightmost guard laced so far. Next, we show that the algorithm will not generate a middle olygon to the left of this guard either, thus giving us the following lemma. LEMMA 4.3 The algorithm GUARD-MONOTONE-POLYGON never introduces middle ockets, and hence, roduces a comlete guard cover. PROOF: We make a roof by contradiction and assume that in iteration i of the algorithm, as guards g i, g i and ĝ i are ositioned in the olygon, a middle ocket i is generated between guard triles g j, g j, ĝ j and g j+1, g j+1, ĝ j+1, where j < i. We assume furthermore that i is the first iteration index that generates a middle ocket and hence that i is generated in this iteration. Let G i denote the set of guards laced by the algorithm from iteration 1 until iteration i has comleted. 11

12 Let r be a oint in i and consider the situation just after iteration i 1. The oint r belongs to a ocket i 1 that is either an uer, a lower or a boundary ocket after this iteration. Consider the situation as the algorithm laces guards g i, g i and ĝ i during iteration i. By Lemma 4.2, r lies to the left of g i since there are no other guards to the right of g i. Without loss of generality, we can assume that g i is laced at u U (G i ) in Ste 5 of the algorithm, as the argument when g i is laced at u D (G i ) is comletely analogous. We make a case analysis on whether i 1 is an uer, a lower or a boundary ocket. Assume first that i 1 is an uer ocket, then we have an immediate contradiction since g i = u U (G i 1 ) and u U (G i 1 ), by definition, sees all oints in the uer ockets to the left of itself, and hence, also the oint r. Assume next that i 1 is a lower ocket, then either the vertical line l gi through g i intersects s D (G i 1 ), in which case g i sees r, giving us a contradiction, or l g i does not intersect s D (G i 1 ). In this case, u D (G i 1 ) is to the left of u U (G i 1 ), contradicting the selection in Ste 4. If we assume that i 1 is a boundary ocket, it lies to the right of g i 1. Assume for a contradiction that the oint r, to the left of g i, is not seen by g i or g i. This means that some art of the olygon boundary hides r from g i and g i. Assume first that this is U, i.e., the shortest ath SP(r, g i) touches U at some vertex v. This means that the vertex v on U to the left of v is not seen, contradicting that g i = u U (G i 1 ), since v P U ( i 1 ); see the definition of Q U in Section 4.1. On the other hand, if SP(r, g i ) touches D at some vertex v, then the vertex v on D to the left of v is not seen, giving us that u D (G i 1 ) is to the left of u U (G i 1 ), since v P D ( i 1 ), contradicting the selection in Ste 4. Therefore, algorithm GUARD-MONOTONE-POLYGON roduces a guard cover that sees all the boundary of the olygon and it never generates a middle ocket. Hence it roduces a comlete guard cover for the monotone olygon Bases and Shadows We continue with a discussion that becomes fairly technical. We associate a secific region, a shadow, to the right of a sear and show that if two sears of the same tye, i.e., uer or lower sears, generated by a artial guard set that obeys certain conditions, then the associated shadows do not intersect. We use this information in the next section to bound the number of guards that our algorithm will roduce. We begin by defining two concets. Fix a artial guard cover G and let s X be the sears with resect to G, for the ocket tye X being U or D. To each sear s X we associate a oint called a base of the sear, denoted v X, with X being U or D. Let l ux be the vertical line through the sear ti u X of the sear s X and let Q X be the region, the set of ockets, such that s X = ke(q X ); see Definition 4.2. DEFINITION 4.3 A base of s X is the rightmost oint v X in Q X and on the boundary X of P such that the sear ti u X lies on the boundary of VP(v X ) and v X lies on the boundary of VP(u X ). A oint v U is called an uer base and a oint v D is called a lower base. Note that a base can lie on a boundary edge infinitely close to a vertex without being on the vertex. See Figure 12 for an examle of an uer base. The second concet that we define is that of a shadow. DEFINITION 4.4 For a sear s X with X being one of U or D, define the shadow of s X, denotedshd X, to be the art of the visibility olygon of the base v X strictly to the right of u X. Hence, shd X = VP R (v X, u X )\l ux, where l ux is the vertical line through u X ; see Figure

13 base v U shadow shd U Figure 12: Examle of a base and shadow for an uer sear. We arameterize the shadows, in the same way as the sears, to be deendent on the lacement of the reviously laced guards and refer to them as shd U (G ) and shd D (G ) given the artial guard set G. We rove a technical lemma that will be useful to bound the number of guards roduced by our algorithm. LEMMA 4.4 If G and G + are two artial guard covers of P such that G G + and u X (G ) G +, for X being one of U or D, then shd X (G ) shd X (G + ) =. PROOF: Since the sear tye is fixed in each case, we can simlify our notation and let u = u X (G ), u + = u X (G + ), v = v X (G ), v + = v X (G + ), shd = shd X (G ), and shd + = shd X (G + ). We denote by X the oosite boundary of X in P, i.e., if X is U then X is D and vice versa. Let Q and Q + be the ocket regions of the two guard sets with resect to X. If u + lies on the boundary X (excet for the degenerate case when u + lies on a reflex vertex of X), then we immediately have that shd + = roving the lemma. Assume now that u + does not lie on X (or that the degenerate case has occurred), then the sear ti u + is adjacent to two boundary edges e and e of the sear, where e is art of the line segment [v +, u + ]. Consider the extension of e, the other edge, from u + towards the left until it reaches the exterior of P at. We claim that the line segment [, u + ] must touch the boundary X at some oint. Assume that it does not, then there is a oint on the extension of [v +, u + ] towards the right that sees as much of Q + as u + does, thus contradicting that u + is a sear ti. This also shows that [, u + ] touches the boundary of some ocket in Q +. Let q denote the leftmost oint on X that intersects the segment [, u + ]. The line segment [, q] artitions P into two subolygons, P + and P, where P + contains u +. This gives us two cases; see Figure 13. v lies in P. If the extension of [v, u ] towards the right does not cross [, q], then all of shd also lies in P and cannot intersect shd + which lies in P +. If the extension of [v, u ] towards the right does cross [, q], then extend [q, u + ] towards the right until it reaches the exterior of P at. In this case, all of shd in P + lies on the same side of [q, ] as v +, 13

14 v + P + u shd q u + P + P v u shd q v + u + v P (a) shd + (b) shd + u v + v P + P q u + g shd (c) shd + Figure 13: Illustrating the roof of Lemma 4.4 with X = U. since v + lies on X and q lies on X. Hence, shd + lies on the oosite side of [q, ] so shd and shd + cannot intersect; see Figure 13(a). v lies in P +. If u lies in P, then all of shd lies in P and the two shadows cannot intersect. Assume now that u lies in P + and that v sees oints in shd +. We have two subcases. If u also sees oints in shd +, we have an immediate contradiction since then u sees oints in some ocket of Q + or oints of the ath P X () of Q +, if is a boundary ocket; see Figure 13(b). If u does not see oints in shd +, there is a art of the boundary X blocking vision between u and shd +. Note that u cannot see q since otherwise it would also see oints of. Therefore, there is second guard g in G + seing q. The guard g must lie in P otherwise it sees oints in. Furthermore, visibility from g into must be blocked by the boundary X, which must then cross the line segment [, q], a contradiction; see Figure 13(c). This concludes the roof Serial Guard Covers We define a secial tye of guard set that will hel us rove the aroximation factor of our algorithm. DEFINITION 4.5 We define restricted guards as follows: For a region R, a guard g is R-restricted, if we only consider the restricted visibility olygon of g to be VP(g) def = VP(g) R. A guard g is a left (or right) guard, if g is VP L (g)-restricted (or VP R (g)-restricted). Next, we define a stri subdivision of P. 14

15 G y-axis Ḡ P P Figure 14: The reflected olygon P. DEFINITION 4.6 A stri subdivision of a monotone olygon P is a subdivision of the olygon by the introduction of vertical segments connecting the uer and lower boundary. Each stri is a subolygon of P bounded by a left vertical edge (ossibly degenerating to the left end oint s), a ortion of the uer boundary U, a right vertical edge (ossibly degenerating to the right end oint t) and a ortion of the lower boundary D. We are now in a osition to define serial guard sets. DEFINITION 4.7 A guard set is serial, if P is subdivided into K stris s 0,...,s K 1 ordered from left to right so that the left edge of stri s 0 has zero or one s 0 -restricted guard, the left edge of every other stri s i, 0 < i < K, has exactly one s i -restricted guard and the right edge of each stri s i, 0 i < K, contains zero or more left guards. No other guards are laced in the olygon. We say that a serial guard set is an uer serial guard cover, if the s i -restricted guard on the left edge of each stri s i, together with the left guards on the right edges of the stris, sees the uer boundary of s i. Similarly, a guard set is a lower serial guard cover, if the s i -restricted guard on the left edge of each stri s i, together with the left guards on the right edges of the stris, sees the lower boundary of s i. LEMMA 4.5 A monotone olygon has an uer serial guard cover with at most 2OPT s i -restricted guards and at most 3OPT left guards. PROOF: Let G be a guard cover for P. Reflect P along the y-axis to get the reversed olygon P having the guard cover Ḡ, the set G reflected along the y-axis; see Figure 14. Our roof is constructive and iteratively laces restricted guards in the olygon P. Do a lane swee from left to right on P. Initially, let H 0 be the emty guard set and let l 0 be the vertical line through the leftmost oint of P. Iteratively, given the artial guard set H j and the line l j, we construct the next artial guard set H j+1 and the next vertical line l j+1 as follows: Obtain the uer sear ti u = u U (H j ) and let l j+1 be the maximal vertical line segment through u interior to P. Let H j+1 include the guards in H j and let s j be the stri in P bounded by l j and l j+1. We lace the following additional guards on l j+1, 1. an s j -restricted guard ḡ j at u, 2. a right guard at u, 3. each guard from Ḡ in s j is moved along its shortest ath to s, the rightmost oint of P, until it reaches l j+1. At this oints lace a right guard. Each of the guards thus laced is added to H j+1 ; see Figure 15. According to Lemma 2.1, any right guard ḡ R on l j+1 will see at least as much to the right of l j+1 in P as the original guard ḡ in Ḡ. Hence, we have the 15

16 Ḡ l j+1 VP(H j) ḡ j VP(H j+1) ḡ j 1 s j VP(ḡ j) l j Figure 15: Illustrating the roof of Lemma 4.5. invariant that the guard set H j+1 sees at least as much of the uer boundary of P as the guards to the left of u in ḠṪhe rocess terminates after K iterations when the lane swee reaches the rightmost oint of P. In the last iteration we have two ossibilities. Either, the right guards in H K 1 together see the uer boundary of s K 1, in which case we do not have to add any s K 1 -restricted guard at the right edge of s K 1, or s K 1 must by necessity contain guards from Ḡ. We differentiate between these cases when we count the number of guards laced. The s j -restricted guard ḡ j laced at u U (H j ) will see the uer boundary of s j together with the right guards in H j. Hence, the restricted guard set H K sees the uer boundary of P. We count the number of guards laced according to their tye, 1, 2, or 3, above. The number of Tye 3 right guards in H K is Ḡ = G since each of these right guards corresonds to a guard in G. The number of Tye 2 right guards is the same as the number of Tye 1 s j -restricted guards since both tyes are laced at uer sear tis u. It remains to count the Tye 1 s j -restricted guards. If s j contains guards from Ḡ, we can associate ḡ j to such a guard ḡ in Ḡ. In articular, if the rocess laces guard ḡ K 1 in the last iteration, there is a guard from Ḡ in s K 1 and we can associate ḡ K 1 to this guard. On the other hand, if s j contains no guards from Ḡ, then an uer base v U(H j ) of s U (H j ) is not seen by the guards in Ḡ to the left of l j, i.e., in s 0,..., s j 1, since no right guard in H j sees the base. Therefore, the base v U (H j ) must be seen by a guard ḡ in Ḡ lying in the uer shadow shd U(H j ). Since there is a Tye 2 right guard at the osition of ḡ j, the rerequisites of Lemma 4.4 are fulfilled and we know that no two uer shadows shd U (H j ) and shd U (H j ) intersect for j j, and hence, ḡ can only see one base. We can therefore associate ḡ j to a guard ḡ from Ḡ in the uer shadow shd U(H j ). Note that, if s K 1 contains no guard from Ḡ, then this stri is comletely seen by the guards in revious stris and the rocess laces no s K 1 -restricted guard in s K 1. In this way, any guard ḡ in Ḡ can be associated to at most two s j-restricted guards. Hence, the number of Tye 1 s j -restricted guards, and thus also the number of Tye 2 right guards, is at most 2 Ḡ = 2 G. Next, reflect the set H K back along the y-axis to become a guard set U of P. We claim that U is uer serial with at most 2 G s i -restricted guards, for stris s i, 0 i < K, and at most 3 G left guards. This follows since 16

17 a stri s j in P when reflected back becomes a stri s i in P, with i = K j 1. Each s j -restricted guard ḡ j in H K lies on the right edge of s j and sees the uer boundary of s j so the corresonding s i -restricted guard g i in U lies on the left edge of s i and sees the uer boundary of s i. Finally, the right guards of H K on the left edges of stris in P corresond to left guards of U in right edges of stris in P. The number of guards has not changed so U is uer serial as claimed. By choosing G to be an otimal guard cover for P, we have that G = OPT, thus roving the lemma. We can, using the same roof technique, show a corresonding lemma for lower serial guards Aroximation Factor Next, we establish the aroximation factor of the algorithm. LEMMA 4.6 The algorithm GUARD-MONOTONE-POLYGON laces at most 30OPT guards in P, where OPT is the size of the smallest guard cover for P. PROOF: To bound the total number of guards, we establish the number of guards laced by Stes 5 7 throughout the iterations of algorithm GUARD-MONOTONE-POLYGON. To do so, we comare the number of guards laced in each ste with the size of an uer and a lower serial guard cover. Let G U be the set of guards assigned in Stes 5 7 in the iterations of the algorithm when the selection in Ste 4 makes X = U and X = D; see Figure 10. Similarly, let G D be the guards assigned when X = D and X = U. Consider first the set G U and order the guard triles in this set from left to right, G U = {g 1, g 1, ĝ 1, g 2, g 2, ĝ 2,...}. In iteration i of the loo, our algorithm erforms Stes 5 7 with X = U and laces guards g j, g j and ĝ j, all having the same x-coordinate, with j i being the roer index in G U. The next time the algorithm erforms Stes 5 7 with X = U it laces guards g j+1, g j+1 and ĝ j+1. Let G i be the set of guards laced in iterations 1 to i by the algorithm. The guards g j, g j and ĝ j are the rightmost guards in G i. We comare the number of guards in G U with the size of an uer serial guard cover U and show that G U contains at most 3 U guards. To do this, we construct a secondary guard set H incrementally starting with the emty guard set H 0. For every index j > 0 we go through the guard triles in G U as follows: If U has an s-restricted guard g in the interval between g j and g j+1 (in the case of j = 0 we consider the leftmost end oint s of P to be the imaginary guard g 0 ), for some stri s, then we let H j := {g j, g j, ĝ j} H j 1, if j > 0. If U has no s-restricted guard in the interval between g j and g j+1, then this whole interval is contained in a stri s associated to U. This means that the uer base v U (G i ) is either seen by the s-restricted guard g s to the left of g j or by a left guard in U in the uer shadow shd U (G i ). If g s sees v U (G i ) then the shortest ath from g s to t, the rightmost oint of P, crosses the vertical line through g j at a oint. Let H j := {g j, g j, } H j 1, i.e., we exchange the guard ĝ j for a guard at. We claim that g s does not see v U (H j ) but this follows immediately by Lemma 2.1. Furthermore, u U (H j ) is not to the right of u U (G i ), since by our algorithm ĝ j is laced so that u U (G i ) is as far to the right as ossible. If g s does not see v U (G i ) then we let H j := {g j, g j, ĝ j} H j 1. We have that v U (G i ) = v U (H j ). The set H constructed according to the rules given above obeys three imortant criteria: 17

18 1. H = G U, 2. for each j, either there is an s-restricted guard between g j and g j+1 or no s-restricted guard g s in U sees the uer base v U (H j ), 3. u U (H j ) is not to the right of u U (G i ), where i is the iteration index when guards g j, g j and ĝ j are laced. Let us count the number of guards in H. If two subsequent triles {g j, g j, j} and {g j+1, g j+1, j+1} in H have an s-restricted guard g s in the interval between them, then we associate the trile {g j, g j, j} to g s. We call such an association an α-association. By construction, an s-restricted guard g s in U can only be α-associated to a guard trile in H once. If two subsequent triles {g j, g j, j} and {g j+1, g j+1, j+1} in H do not have any s-restricted guard g s in the interval between them, then we know that the uer base v U (H j ) is seen by a left guard g in U in the uer shadow shd U (H j ) and we associate the trile {g j, g j, j} to g. We call such an association a β-association. Since the uer sear s U (H j ) and s U (H j ) obey the rerequisites of Lemma 4.4, for any j j, the two uer shadows shd U (H j ) and shd U (H j ) do not intersect. This means that a left guard g can only be β- associated to a guard trile in H once. From this we can deduce that the number of guard triles in H is at most the number of s-restricted guards and left guards in U together. From Lemma 4.5, we know that this is at most 5OPT. By a comletely symmetrical argument we can construct a set of guard triles H of the same size as the set G D and deduce that the number of guard triles in this set is also bounded by 5OPT. The total number of guards constructed by our algorithm is therefore bounded by G U + G D = H + H 3 U + 3 D 30OPT, as claimed. 4.3 Comutation Comuting Kernel Exansions Let, q and r be three oints in P. We let VP B (, q, r) denote the art of the visibility olygonvp() between the oints q and r. Let R be a ossibly disconnected olygonal region in P having m vertices and assume that the vertices are ordered v 1,...,v m from left to right. We claim the following lemma. LEMMA 4.7 ke(r) = m 1 i=1 i VP B (v j, v i, v i+1 ) m VP R (v j, v m ). j=1 j=1 PROOF: From Definition 4.1 we have that ke(r) = { v 1 R L () VP()} where R L () is the art of R to the left of. Let be a oint between v i and v i+1. We show that is in ke(r) if and only if i j=1 VP(v j). Assume first that i j=1 VP(v j). In this case, there is a vertex v j of R such that does not see v j. Hence, there is a oint in R to the left of, the vertex v j, not seen by, so is not in ke(r) roving the first imlication of the equivalence. 18

19 q l q v j l q R q q r Figure 16: Illustrating the roof of Lemma 4.7. Assume next that ke(r). In this case, there is a oint q R L () not seen from. Consider the shortest ath SP(, q). Let [, q] be the last segment and let [r, ] be the enultimate segment of SP(, q). Since does not see q, SP(, q) consists of at least two segments and because of the monotonicity of P, the oint is a vertex of P to the left of. Let l q be the maximal line segment interior to P from through q to. The segment l q contains a maximal subsegment l q comletely contained in R. Let q and q be the two end oints of l q, with q to the left of q ; see Figure 16. Assume that [r,, ] forms a right turn. Follow the boundary of R in counterclockwise order from q to q, above the segment [, ], until the first vertex v j of R is encountered. Since v j is above [, ] and to the left of, the oint does not see v j and j i. If [r,, ] forms a left turn we can make a symmetric argument to show that there is a vertex of R to the left of not seen by. We have thus roved both directions of the equivalence. Lemma 4.7 gives us a method to comute the kernel exansion of a region. We begin by ordering the vertices of the region from left to right. For each vertex, in order, we comute the visibility olygon [15, 22, 25] and establish the aroriate intersections in successive order; see Figure 17. The comlexity of the algorithm is O(m log m + mn), where m is the number of vertices of R and n is the number of vertices of P. If R is monotone, the comlexity reduces to O(mn) since the sorting of the vertices of R can be done in linear time. The algorithm reeatedly comutes intersections between two monotone olygons and combines the result with the left art established in revious iterations. The intersection between two monotone olygons having n and n vertices resectively can be comuted in O(n + n ) time with a lane swee algorithm. Using the linear time intersection algorithm we can successively comute the intersections between visibility olygons, obtaining the kernel exansions of the aroriate ocket regions, i.e., the sears of each tye. Since the number of ocket vertices is at most linear in total, we have the following lemma. LEMMA 4.8 A sear in a monotone olygon can be comuted in quadratic time Comuting the Next Guard Consider Ste 7 of algorithm GUARD-MONOTONE-POLYGON. In an iteration, just before we reach Ste 7, we have a artial guard set G with the rightmost guards at g and g and we are suosed to lace a third guard ĝ on the same vertical line in such a way that the sear ti of tye X with resect to G {ĝ} is as far to the 19

20 Algorithm Inut: COMPUTE-KERNEL-EXPANSION A region R in a monotone olygon P Outut: The kernel exansion ke(r) 1 Order the vertices of R from left to right; v 1,..., v m 2 Let K := VP R(v 1) 3 for each vertex v i in order from v 2,..., v m do 3.1 Comute VP R(v i) 3.2 Let K L(v i) and K R(v i) be the two arts of K to the left and right of v i resectively 3.3 Let K := K L(v i) (K R(v i) VP R(v i)) endfor 4 return K End COMPUTE-KERNEL-EXPANSION Figure 17: The algorithm for comuting kernel exansions. l g r r ê y = ax + b q q e u X(G { }) u X(G {}) Figure 18: Illustrating the movement of u X(G {}). right as ossible. Let l g be the vertical line through g and g. The line l g intersects U at U and D at D. The algorithm emulates a sliding rocess whereby a oint slides along l g from U to D and we maintain the sear ti u X (G {}) as a function of, continuously udating the sear ti as moves along l g. To accurately detect for which oint that the oint u X (G {}) is rightmost, we let the x-coordinate of u X (G {}) be a function of the y-coordinate of. We denote this function by x X (y), where y corresonds to the arameter of the vertical line l g = (1 y) U + y D. The sear ti u X (G {}) is adjacent to two edges e and e of the corresonding sear and, if u X (G {}) moves when moves along l g, at least one of these edges must move as moves. One of the two edges, say e, extends towards the left, reaching a oint r on a boundary edge ê of X. The edge e, on the other hand, can either coincide with the oosite boundary X (when u X (G {}) lies on X) or it extends towards the left, touching a vertex of the boundary X before it reaches a vertex v of a tye X ocket. In the most general case, both r and v move as moves. Consider first the movement of r on ê, where ê is a segment on the line y = ax + b. The suorting segment [, r] touches X at a oint q and the other suorting segment [u X (G {}), r] touches X at a oint q ; see Figure 18. We want to establish the equation of the line coinciding with [u X (G {}), r] in terms of the y-coordinate 20

21 r r q q q q q q q q (a) (b) (c) Figure 19: The cases for arameter change. of. If we let r be a function of, we have that r is the intersection oint between the lines y = y() y(q) x() x(q) x + y(q) y() y(q) x() x(q) x(q) and y = ax + b. So, by setting the two linear functions equal, we obtain the coordinates of the oint r. The two coordinates are each the ratio between two affine functions in y(), i.e., ( c y() + d r = y() + h, c y() + d ), y() + h where c, d, h, c, and d, are constants deendent on a, b and q. Hence, the line through r and q can be established to be y = g(y())x + k(y()), where g(y) = (αy + β)/(y + γ) and k(y) = (α y + β )/(y + γ), for constants α, β, γ, α and β. With similar calculations we can establish the other suorting line that intersects the vertex v of a tye X ocket to have the equation y = g (y())x + k (y()) as a function of y(). The function x X (y()) is the x-coordinate of the intersection oint between the two suorting lines, i.e., giving us g(y)x X (y) + k(y) = g (y)x X (y) + k (y), x X (y) = k (y) k(y) g(y) g (y) = Ay2 + By + C A y 2 + B y + C, where the constants A, B, C, A, B and C only deend on the oints of contact that the four suorting lines corresonding to visibility olygon edges make with the boundary. The constant arameters A, B, C, A, B and C can change value as the suorting lines make contact on different vertices and edges of the olygon and ocket boundaries. We are interested in comuting these oints of arameter change to be able to udate the function x X (y) aroriately; see Figure 19. These occur when: the convex vertex of an edge of VP(G {}) adjacent to a ocket becomes incident to two vertices on the olygon boundary of P; see Figures 19(a) and (b). the convex vertex of an edge of VP(u X (G {})) adjacent to a ocket becomes incident to two vertices on the olygon boundary of P or to one vertex of the boundary of P and one vertex of a tye X ocket of G ; see Figures 19(a) and (c). We can establish a suerset of these oints on l g that we call the rimary event oints by comuting the visibility olygon of each boundary and ocket vertex to the right of l g and obtaining the, at most two, intersection 21

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11)

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11) Point Location Prerocess a lanar, olygonal subdivision for oint location ueries. = (18, 11) Inut is a subdivision S of comlexity n, say, number of edges. uild a data structure on S so that for a uery oint

More information

The Online Freeze-tag Problem

The Online Freeze-tag Problem The Online Freeze-tag Problem Mikael Hammar, Bengt J. Nilsson, and Mia Persson Atus Technologies AB, IDEON, SE-3 70 Lund, Sweden mikael.hammar@atus.com School of Technology and Society, Malmö University,

More information

Problem Set 6 Solutions

Problem Set 6 Solutions ( Introduction to Algorithms Aril 16, 2004 Massachusetts Institute of Technology 6046J/18410J Professors Erik Demaine and Shafi Goldwasser Handout 21 Problem Set 6 Solutions This roblem set is due in recitation

More information

Solutions to Problem Set 3

Solutions to Problem Set 3 Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Comuter Science October 3 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised October 8, 2005, 979 minutes Solutions

More information

A Modified Measure of Covert Network Performance

A Modified Measure of Covert Network Performance A Modified Measure of Covert Network Performance LYNNE L DOTY Marist College Deartment of Mathematics Poughkeesie, NY UNITED STATES lynnedoty@maristedu Abstract: In a covert network the need for secrecy

More information

2.1 Simple & Compound Propositions

2.1 Simple & Compound Propositions 2.1 Simle & Comound Proositions 1 2.1 Simle & Comound Proositions Proositional Logic can be used to analyse, simlify and establish the equivalence of statements. A knowledge of logic is essential to the

More information

Assume a height value is associated with each point. " A triangulation of the points defines a piecewiselinear surface of triangular patches.

Assume a height value is associated with each point.  A triangulation of the points defines a piecewiselinear surface of triangular patches. A triangulation of set of oints in the lane is a artition of the convex hull to triangles whose vertices are the oints, and are emty of other oints." There are an exonential number of triangulations of

More information

Survey of Terrain Guarding and Art Gallery Problems

Survey of Terrain Guarding and Art Gallery Problems Survey of Terrain Guarding and Art Gallery Problems Erik Krohn Abstract The terrain guarding problem and art gallery problem are two areas in computational geometry. Different versions of terrain guarding

More information

where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians =

where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians = Introduction to Modeling 3.6-1 3.6 Sine and Cosine Functions The general form of a sine or cosine function is given by: f (x) = asin (bx + c) + d and f(x) = acos(bx + c) + d where a, b, c, and d are constants

More information

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

More information

1 Gambler s Ruin Problem

1 Gambler s Ruin Problem Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins

More information

Computational Geometry

Computational Geometry Motivation Motivation Polygons and visibility Visibility in polygons Triangulation Proof of the Art gallery theorem Two points in a simple polygon can see each other if their connecting line segment is

More information

SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID

SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID International Journal of Comuter Science & Information Technology (IJCSIT) Vol 6, No 4, August 014 SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID

More information

Computational Geometry

Computational Geometry Motivation 1 Motivation Polygons and visibility Visibility in polygons Triangulation Proof of the Art gallery theorem Two points in a simple polygon can see each other if their connecting line segment

More information

Introduction to NP-Completeness Written and copyright c by Jie Wang 1

Introduction to NP-Completeness Written and copyright c by Jie Wang 1 91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use time-bounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of

More information

United Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane

United Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane United Arab Emirates University College of Sciences Deartment of Mathematical Sciences HOMEWORK 1 SOLUTION Section 10.1 Vectors in the Plane Calculus II for Engineering MATH 110 SECTION 0 CRN 510 :00 :00

More information

Assignment 9; Due Friday, March 17

Assignment 9; Due Friday, March 17 Assignment 9; Due Friday, March 17 24.4b: A icture of this set is shown below. Note that the set only contains oints on the lines; internal oints are missing. Below are choices for U and V. Notice that

More information

A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION

A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION 9 th ASCE Secialty Conference on Probabilistic Mechanics and Structural Reliability PMC2004 Abstract A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION

More information

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks 6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

More information

The Optimal Sequenced Route Query

The Optimal Sequenced Route Query The Otimal Sequenced Route Query Mehdi Sharifzadeh, Mohammad Kolahdouzan, Cyrus Shahabi Comuter Science Deartment University of Southern California Los Angeles, CA 90089-0781 [sharifza, kolahdoz, shahabi]@usc.edu

More information

2 Random variables. The definition. 2b Distribution function

2 Random variables. The definition. 2b Distribution function Tel Aviv University, 2006 Probability theory 2 2 Random variables 2a The definition Discrete robability defines a random variable X as a function X : Ω R. The robability of a ossible value R is P ( X =

More information

X How to Schedule a Cascade in an Arbitrary Graph

X How to Schedule a Cascade in an Arbitrary Graph X How to Schedule a Cascade in an Arbitrary Grah Flavio Chierichetti, Cornell University Jon Kleinberg, Cornell University Alessandro Panconesi, Saienza University When individuals in a social network

More information

Static and Dynamic Properties of Small-world Connection Topologies Based on Transit-stub Networks

Static and Dynamic Properties of Small-world Connection Topologies Based on Transit-stub Networks Static and Dynamic Proerties of Small-world Connection Toologies Based on Transit-stub Networks Carlos Aguirre Fernando Corbacho Ramón Huerta Comuter Engineering Deartment, Universidad Autónoma de Madrid,

More information

An Analysis of Reliable Classifiers through ROC Isometrics

An Analysis of Reliable Classifiers through ROC Isometrics An Analysis of Reliable Classifiers through ROC Isometrics Stijn Vanderlooy s.vanderlooy@cs.unimaas.nl Ida G. Srinkhuizen-Kuyer kuyer@cs.unimaas.nl Evgueni N. Smirnov smirnov@cs.unimaas.nl MICC-IKAT, Universiteit

More information

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method Universiteit-Utrecht * Deartment of Mathematics Otimal a riori error bounds for the Rayleigh-Ritz method by Gerard L.G. Sleijen, Jaser van den Eshof, and Paul Smit Prerint nr. 1160 Setember, 2000 OPTIMAL

More information

ENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS

ENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS ENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS Liviu Grigore Comuter Science Deartment University of Illinois at Chicago Chicago, IL, 60607 lgrigore@cs.uic.edu Ugo Buy Comuter Science

More information

Confidence Intervals for Capture-Recapture Data With Matching

Confidence Intervals for Capture-Recapture Data With Matching Confidence Intervals for Cature-Recature Data With Matching Executive summary Cature-recature data is often used to estimate oulations The classical alication for animal oulations is to take two samles

More information

24: Polygon Partition - Faster Triangulations

24: Polygon Partition - Faster Triangulations 24: Polygon Partition - Faster Triangulations CS 473u - Algorithms - Spring 2005 April 15, 2005 1 Previous lecture We described triangulations of simple polygons. See Figure 1 for an example of a polygon

More information

1 Point Inclusion in a Polygon

1 Point Inclusion in a Polygon Comp 163: Computational Geometry Tufts University, Spring 2005 Professor Diane Souvaine Scribe: Ryan Coleman Point Inclusion and Processing Simple Polygons into Monotone Regions 1 Point Inclusion in a

More information

lecture 25: Gaussian quadrature: nodes, weights; examples; extensions

lecture 25: Gaussian quadrature: nodes, weights; examples; extensions 38 lecture 25: Gaussian quadrature: nodes, weights; examles; extensions 3.5 Comuting Gaussian quadrature nodes and weights When first aroaching Gaussian quadrature, the comlicated characterization of the

More information

2016 Physics 250: Worksheet 02 Name

2016 Physics 250: Worksheet 02 Name 06 Physics 50: Worksheet 0 Name Concets: Electric Field, lines of force, charge density, diole moment, electric diole () An equilateral triangle with each side of length 0.0 m has identical charges of

More information

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens.

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens. Pinhole Otics Science, at bottom, is really anti-intellectual. It always distrusts ure reason and demands the roduction of the objective fact. H. L. Mencken (1880-1956) OBJECTIVES To study the formation

More information

Homework Exam 1, Geometric Algorithms, 2016

Homework Exam 1, Geometric Algorithms, 2016 Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3-dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the

More information

Theoretical comparisons of average normalized gain calculations

Theoretical comparisons of average normalized gain calculations PHYSICS EDUCATIO RESEARCH All submissions to PERS should be sent referably electronically to the Editorial Office of AJP, and then they will be forwarded to the PERS editor for consideration. Theoretical

More information

A New Method for Eye Detection in Color Images

A New Method for Eye Detection in Color Images Journal of Comuter Engineering 1 (009) 3-11 A New Method for Eye Detection in Color Images Mohammadreza Ramezanour Deartment of Comuter Science & Research Branch Azad University, Arak.Iran E-mail: Mr.ramezanoor@gmail.com

More information

The Magnus-Derek Game

The Magnus-Derek Game The Magnus-Derek Game Z. Nedev S. Muthukrishnan Abstract We introduce a new combinatorial game between two layers: Magnus and Derek. Initially, a token is laced at osition 0 on a round table with n ositions.

More information

On Multicast Capacity and Delay in Cognitive Radio Mobile Ad-hoc Networks

On Multicast Capacity and Delay in Cognitive Radio Mobile Ad-hoc Networks On Multicast Caacity and Delay in Cognitive Radio Mobile Ad-hoc Networks Jinbei Zhang, Yixuan Li, Zhuotao Liu, Fan Wu, Feng Yang, Xinbing Wang Det of Electronic Engineering Det of Comuter Science and Engineering

More information

Approximation Algorithms for Art Gallery Problems

Approximation Algorithms for Art Gallery Problems Approximation Algorithms for Art Gallery Problems Subhas C. Nandy (nandysc@isical.ac.in) Advanced Computing and Microelectronics Unit Indian Statistical Institute Kolkata 700108, India. Outline 1 Introduction

More information

Chapter 3. Special Techniques for Calculating Potentials. r ( r ' )dt ' ( ) 2

Chapter 3. Special Techniques for Calculating Potentials. r ( r ' )dt ' ( ) 2 Chater 3. Secial Techniues for Calculating Potentials Given a stationary charge distribution r( r ) we can, in rincile, calculate the electric field: E ( r ) Ú Dˆ r Dr r ( r ' )dt ' 2 where Dr r '-r. This

More information

A Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations

A Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations A Simle Model of Pricing, Markus and Market Power Under Demand Fluctuations Stanley S. Reynolds Deartment of Economics; University of Arizona; Tucson, AZ 85721 Bart J. Wilson Economic Science Laboratory;

More information

The Graphical Method. Lecture 1

The Graphical Method. Lecture 1 References: Anderson, Sweeney, Williams: An Introduction to Management Science - quantitative aroaches to decision maing 7 th ed Hamdy A Taha: Oerations Research, An Introduction 5 th ed Daellenbach, George,

More information

Week 2 Polygon Triangulation

Week 2 Polygon Triangulation Week 2 Polygon Triangulation What is a polygon? Last week A polygonal chain is a connected series of line segments A closed polygonal chain is a polygonal chain, such that there is also a line segment

More information

Fourier Analysis of Stochastic Processes

Fourier Analysis of Stochastic Processes Fourier Analysis of Stochastic Processes. Time series Given a discrete time rocess ( n ) nz, with n :! R or n :! C 8n Z, we de ne time series a realization of the rocess, that is to say a series (x n )

More information

Effect Sizes Based on Means

Effect Sizes Based on Means CHAPTER 4 Effect Sizes Based on Means Introduction Raw (unstardized) mean difference D Stardized mean difference, d g Resonse ratios INTRODUCTION When the studies reort means stard deviations, the referred

More information

Synopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE

Synopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Develoment FRANCE Synosys There is no doubt left about the benefit of electrication and subsequently

More information

Loglikelihood and Confidence Intervals

Loglikelihood and Confidence Intervals Stat 504, Lecture 3 Stat 504, Lecture 3 2 Review (contd.): Loglikelihood and Confidence Intervals The likelihood of the samle is the joint PDF (or PMF) L(θ) = f(x,.., x n; θ) = ny f(x i; θ) i= Review:

More information

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.

More information

ERDŐS SZEKERES THEOREM WITH FORBIDDEN ORDER TYPES. II

ERDŐS SZEKERES THEOREM WITH FORBIDDEN ORDER TYPES. II ERDŐS SEKERES THEOREM WITH FORBIDDEN ORDER TYPES. II GYULA KÁROLYI Institute of Mathematics, Eötvös University, Pázmány P. sétány /C, Budaest, H 7 Hungary GÉA TÓTH Alfréd Rényi Institute of Mathematics,

More information

It is important to be very clear about our definitions of probabilities.

It is important to be very clear about our definitions of probabilities. Use Bookmarks for electronic content links 7.6 Bayesian odds 7.6.1 Introduction The basic ideas of robability have been introduced in Unit 7.3 in the book, leading to the concet of conditional robability.

More information

Lecture Notes The Fibonacci Sequence page 1

Lecture Notes The Fibonacci Sequence page 1 Lecture Notes The Fibonacci Sequence age De nition: The Fibonacci sequence starts with and and for all other terms in the sequence, we must add the last two terms. F F and for all n, + + + So the rst few

More information

POISSON PROCESSES. Chapter 2. 2.1 Introduction. 2.1.1 Arrival processes

POISSON PROCESSES. Chapter 2. 2.1 Introduction. 2.1.1 Arrival processes Chater 2 POISSON PROCESSES 2.1 Introduction A Poisson rocess is a simle and widely used stochastic rocess for modeling the times at which arrivals enter a system. It is in many ways the continuous-time

More information

4 Perceptron Learning Rule

4 Perceptron Learning Rule Percetron Learning Rule Objectives Objectives - Theory and Examles - Learning Rules - Percetron Architecture -3 Single-Neuron Percetron -5 Multile-Neuron Percetron -8 Percetron Learning Rule -8 Test Problem

More information

Computational Finance The Martingale Measure and Pricing of Derivatives

Computational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure 1 Comutational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure The Martingale measure or the Risk Neutral robabilities are a fundamental concet

More information

An important observation in supply chain management, known as the bullwhip effect,

An important observation in supply chain management, known as the bullwhip effect, Quantifying the Bullwhi Effect in a Simle Suly Chain: The Imact of Forecasting, Lead Times, and Information Frank Chen Zvi Drezner Jennifer K. Ryan David Simchi-Levi Decision Sciences Deartment, National

More information

A 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction Stark-Heegner Theorem. Let d > 0 be a square-free integer then Q( d) has

A 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction Stark-Heegner Theorem. Let d > 0 be a square-free integer then Q( d) has A 60,000 DIGIT PRIME NUMBER OF THE FORM x + x + 4. Introduction.. Euler s olynomial. Euler observed that f(x) = x + x + 4 takes on rime values for 0 x 39. Even after this oint f(x) takes on a high frequency

More information

Math 5330 Spring Notes Prime Numbers

Math 5330 Spring Notes Prime Numbers Math 5330 Sring 206 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating

More information

Predicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products

Predicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products Predicate Encrytion Suorting Disjunctions, Polynomial Equations, and Inner Products Jonathan Katz Amit Sahai Brent Waters Abstract Predicate encrytion is a new aradigm for ublic-key encrytion that generalizes

More information

THE ROTATION INDEX OF A PLANE CURVE

THE ROTATION INDEX OF A PLANE CURVE THE ROTATION INDEX OF A PLANE CURVE AARON W. BROWN AND LORING W. TU The rotation index of a smooth closed lane curve C is the number of comlete rotations that a tangent vector to the curve makes as it

More information

Variations on the Gambler s Ruin Problem

Variations on the Gambler s Ruin Problem Variations on the Gambler s Ruin Problem Mat Willmott December 6, 2002 Abstract. This aer covers the history and solution to the Gambler s Ruin Problem, and then exlores the odds for each layer to win

More information

Algorithms for Constructing Zero-Divisor Graphs of Commutative Rings Joan Krone

Algorithms for Constructing Zero-Divisor Graphs of Commutative Rings Joan Krone Algorithms for Constructing Zero-Divisor Grahs of Commutative Rings Joan Krone Abstract The idea of associating a grah with the zero-divisors of a commutative ring was introduced in [3], where the author

More information

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z: NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two oerations defined on them, addition and multilication,

More information

Memory management. Chapter 4: Memory Management. Memory hierarchy. In an ideal world. Basic memory management. Fixed partitions: multiple programs

Memory management. Chapter 4: Memory Management. Memory hierarchy. In an ideal world. Basic memory management. Fixed partitions: multiple programs Memory management Chater : Memory Management Part : Mechanisms for Managing Memory asic management Swaing Virtual Page relacement algorithms Modeling age relacement algorithms Design issues for aging systems

More information

TRANSCENDENTAL NUMBERS

TRANSCENDENTAL NUMBERS TRANSCENDENTAL NUMBERS JEREMY BOOHER. Introduction The Greeks tried unsuccessfully to square the circle with a comass and straightedge. In the 9th century, Lindemann showed that this is imossible by demonstrating

More information

Monotone Partitioning. Polygon Partitioning. Monotone polygons. Monotone polygons. Monotone Partitioning. ! Define monotonicity

Monotone Partitioning. Polygon Partitioning. Monotone polygons. Monotone polygons. Monotone Partitioning. ! Define monotonicity Monotone Partitioning! Define monotonicity Polygon Partitioning Monotone Partitioning! Triangulate monotone polygons in linear time! Partition a polygon into monotone pieces Monotone polygons! Definition

More information

Large-Scale IP Traceback in High-Speed Internet: Practical Techniques and Theoretical Foundation

Large-Scale IP Traceback in High-Speed Internet: Practical Techniques and Theoretical Foundation Large-Scale IP Traceback in High-Seed Internet: Practical Techniques and Theoretical Foundation Jun Li Minho Sung Jun (Jim) Xu College of Comuting Georgia Institute of Technology {junli,mhsung,jx}@cc.gatech.edu

More information

Discrete Stochastic Approximation with Application to Resource Allocation

Discrete Stochastic Approximation with Application to Resource Allocation Discrete Stochastic Aroximation with Alication to Resource Allocation Stacy D. Hill An otimization roblem involves fi nding the best value of an obective function or fi gure of merit the value that otimizes

More information

Lecture Notes: Discrete Mathematics

Lecture Notes: Discrete Mathematics Lecture Notes: Discrete Mathematics GMU Math 125-001 Sring 2007 Alexei V Samsonovich Any theoretical consideration, no matter how fundamental it is, inevitably relies on key rimary notions that are acceted

More information

Alpha Channel Estimation in High Resolution Images and Image Sequences

Alpha Channel Estimation in High Resolution Images and Image Sequences In IEEE Comuter Society Conference on Comuter Vision and Pattern Recognition (CVPR 2001), Volume I, ages 1063 68, auai Hawaii, 11th 13th Dec 2001 Alha Channel Estimation in High Resolution Images and Image

More information

HYPOTHESIS TESTING FOR THE PROCESS CAPABILITY RATIO. A thesis presented to. the faculty of

HYPOTHESIS TESTING FOR THE PROCESS CAPABILITY RATIO. A thesis presented to. the faculty of HYPOTHESIS TESTING FOR THE PROESS APABILITY RATIO A thesis resented to the faculty of the Russ ollege of Engineering and Technology of Ohio University In artial fulfillment of the requirement for the degree

More information

Polygon Triangulation

Polygon Triangulation Polygon Triangulation A polygonal curve is a finite chain of line segments. Line segments called edges, their endpoints called vertices. A simple polygon is a closed polygonal curve without self-intersection.

More information

Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W

Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the

More information

CABRS CELLULAR AUTOMATON BASED MRI BRAIN SEGMENTATION

CABRS CELLULAR AUTOMATON BASED MRI BRAIN SEGMENTATION XI Conference "Medical Informatics & Technologies" - 2006 Rafał Henryk KARTASZYŃSKI *, Paweł MIKOŁAJCZAK ** MRI brain segmentation, CT tissue segmentation, Cellular Automaton, image rocessing, medical

More information

Automatic Search for Correlated Alarms

Automatic Search for Correlated Alarms Automatic Search for Correlated Alarms Klaus-Dieter Tuchs, Peter Tondl, Markus Radimirsch, Klaus Jobmann Institut für Allgemeine Nachrichtentechnik, Universität Hannover Aelstraße 9a, 0167 Hanover, Germany

More information

Stat 134 Fall 2011: Gambler s ruin

Stat 134 Fall 2011: Gambler s ruin Stat 134 Fall 2011: Gambler s ruin Michael Lugo Setember 12, 2011 In class today I talked about the roblem of gambler s ruin but there wasn t enough time to do it roerly. I fear I may have confused some

More information

Failure Behavior Analysis for Reliable Distributed Embedded Systems

Failure Behavior Analysis for Reliable Distributed Embedded Systems Failure Behavior Analysis for Reliable Distributed Embedded Systems Mario Tra, Bernd Schürmann, Torsten Tetteroo {tra schuerma tetteroo}@informatik.uni-kl.de Deartment of Comuter Science, University of

More information

FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES

FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES AVNER ASH, LAURA BELTIS, ROBERT GROSS, AND WARREN SINNOTT Abstract. We consider statistical roerties of the sequence of ordered airs obtained by taking

More information

APPENDIX A Mohr s circle for two-dimensional stress

APPENDIX A Mohr s circle for two-dimensional stress APPENDIX A ohr s circle for two-dimensional stress Comressive stresses have been taken as ositive because we shall almost exclusively be dealing with them as oosed to tensile stresses and because this

More information

Risk in Revenue Management and Dynamic Pricing

Risk in Revenue Management and Dynamic Pricing OPERATIONS RESEARCH Vol. 56, No. 2, March Aril 2008,. 326 343 issn 0030-364X eissn 1526-5463 08 5602 0326 informs doi 10.1287/ore.1070.0438 2008 INFORMS Risk in Revenue Management and Dynamic Pricing Yuri

More information

Minimizing the Communication Cost for Continuous Skyline Maintenance

Minimizing the Communication Cost for Continuous Skyline Maintenance Minimizing the Communication Cost for Continuous Skyline Maintenance Zhenjie Zhang, Reynold Cheng, Dimitris Paadias, Anthony K.H. Tung School of Comuting National University of Singaore {zhenjie,atung}@com.nus.edu.sg

More information

Tangent and normal lines to conics

Tangent and normal lines to conics 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

The impact of metadata implementation on webpage visibility in search engine results (Part II) q

The impact of metadata implementation on webpage visibility in search engine results (Part II) q Information Processing and Management 41 (2005) 691 715 www.elsevier.com/locate/inforoman The imact of metadata imlementation on webage visibility in search engine results (Part II) q Jin Zhang *, Alexandra

More information

On Software Piracy when Piracy is Costly

On Software Piracy when Piracy is Costly Deartment of Economics Working aer No. 0309 htt://nt.fas.nus.edu.sg/ecs/ub/w/w0309.df n Software iracy when iracy is Costly Sougata oddar August 003 Abstract: The ervasiveness of the illegal coying of

More information

Joint Production and Financing Decisions: Modeling and Analysis

Joint Production and Financing Decisions: Modeling and Analysis Joint Production and Financing Decisions: Modeling and Analysis Xiaodong Xu John R. Birge Deartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208,

More information

Softmax Model as Generalization upon Logistic Discrimination Suffers from Overfitting

Softmax Model as Generalization upon Logistic Discrimination Suffers from Overfitting Journal of Data Science 12(2014),563-574 Softmax Model as Generalization uon Logistic Discrimination Suffers from Overfitting F. Mohammadi Basatini 1 and Rahim Chiniardaz 2 1 Deartment of Statistics, Shoushtar

More information

Local Connectivity Tests to Identify Wormholes in Wireless Networks

Local Connectivity Tests to Identify Wormholes in Wireless Networks Local Connectivity Tests to Identify Wormholes in Wireless Networks Xiaomeng Ban Comuter Science Stony Brook University xban@cs.sunysb.edu Rik Sarkar Comuter Science Freie Universität Berlin sarkar@inf.fu-berlin.de

More information

FINDING ALL WEAKLY-VISIBLE CHORDS OF A POLYGON IN LINEAR TIME

FINDING ALL WEAKLY-VISIBLE CHORDS OF A POLYGON IN LINEAR TIME Nordic Journal of Computing 1(1994), 433 457. FINDING ALL WEAKLY-VISIBLE CHORDS OF A POLYGON IN LINEAR TIME GAUTAM DAS PAUL J. HEFFERNAN GIRI NARASIMHAN Mathematical Sciences Department The University

More information

12. Visibility Graphs and 3-Sum Lecture on Monday 9 th

12. Visibility Graphs and 3-Sum Lecture on Monday 9 th 12. Visibility Graphs and 3-Sum Lecture on Monday 9 th November, 2009 by Michael Homann 12.1 Sorting all Angular Sequences. Theorem 12.1 Consider a set P of n points in the plane.

More information

Computational Geometry [csci 3250]

Computational Geometry [csci 3250] Computational Geometry [csci 3250] Laura Toma Bowdoin College Polygon Triangulation Polygon Triangulation The problem: Triangulate a given polygon. (output a set of diagonals that partition the polygon

More information

The Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+

The Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+ The Cubic Formula The quadratic formula tells us the roots of a quadratic olynomial, a olynomial of the form ax + bx + c. The roots (if b b+ 4ac 0) are b 4ac a and b b 4ac a. The cubic formula tells us

More information

More Properties of Limits: Order of Operations

More Properties of Limits: Order of Operations math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f

More information

Hidden Mobile Guards in Simple Polygons

Hidden Mobile Guards in Simple Polygons CCCG 2012, Charlottetown, P.E.I., August 8 10, 2012 Hidden Mobile Guards in Simple Polygons Sarah Cannon Diane L. Souvaine Andrew Winslow Abstract We consider guarding classes of simple polygons using

More information

Complex Conjugation and Polynomial Factorization

Complex Conjugation and Polynomial Factorization Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.

More information

Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

More information

Triangulation. Is this always possible for any simple polygon? If not, which polygons are triangulable.

Triangulation. Is this always possible for any simple polygon? If not, which polygons are triangulable. Triangulation Partition polygon P into non-overlapping triangles using diagonals only. Is this always possible for any simple polygon? If not, which polygons are triangulable. Does the number of triangles

More information

A Brief Introduction to Design of Experiments

A Brief Introduction to Design of Experiments J. K. TELFORD D A Brief Introduction to Design of Exeriments Jacqueline K. Telford esign of exeriments is a series of tests in which uroseful changes are made to the inut variables of a system or rocess

More information

MATHEMATICAL BACKGROUND

MATHEMATICAL BACKGROUND Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a

More information

Simulation and Verification of Coupled Heat and Moisture Modeling

Simulation and Verification of Coupled Heat and Moisture Modeling Simulation and Verification of Couled Heat and Moisture Modeling N. Williams Portal 1, M.A.P. van Aarle 2 and A.W.M. van Schijndel *,3 1 Deartment of Civil and Environmental Engineering, Chalmers University

More information

Computational Geometry. Lecture 1: Introduction and Convex Hulls

Computational Geometry. Lecture 1: Introduction and Convex Hulls Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,

More information

Stability Improvements of Robot Control by Periodic Variation of the Gain Parameters

Stability Improvements of Robot Control by Periodic Variation of the Gain Parameters Proceedings of the th World Congress in Mechanism and Machine Science ril ~4, 4, ianin, China China Machinery Press, edited by ian Huang. 86-8 Stability Imrovements of Robot Control by Periodic Variation

More information

2D Modeling of the consolidation of soft soils. Introduction

2D Modeling of the consolidation of soft soils. Introduction D Modeling of the consolidation of soft soils Matthias Haase, WISMUT GmbH, Chemnitz, Germany Mario Exner, WISMUT GmbH, Chemnitz, Germany Uwe Reichel, Technical University Chemnitz, Chemnitz, Germany Abstract:

More information