Parameterized Algorithms for dhitting Set: the Weighted Case Henning Fernau. Univ. Trier, FB 4 Abteilung Informatik Trier, Germany


 Susanna Quinn
 3 years ago
 Views:
Transcription
1 Parameterize Algorithms for Hitting Set: the Weighte Case Henning Fernau Trierer Forschungsberichte; Trier: Technical Reports Informatik / Mathematik No. 086, July 2008 Univ. Trier, FB 4 Abteilung Informatik Trier, Germany Theoretische Informatik Trier
2 Parameterize Algorithms for Hitting Set: the Weighte Case Henning Fernau 1 Univ. Trier, FB 4 Abteilung Informatik, Trier, Germany Abstract. We are going to analyze simple search tree algorithms for Weighte Hitting Set. Although the algorithms are simple, their analysis is technically rather involve. However, this approach allows us to even improve on elsewhere previously publishe running time estimates for the more restricte case of (unweighte) Hitting Set. 1 1 Introuction Our approach in general. We exhibit how to systematically esign an analyze search tree algorithms within the framework of parameterize algorithmics [3]. Here, we avocate a topown approach as oppose to a rather bottomup esign, because the resulting algorithms ten to be simpler than via the opposite approach, an they sometimes pretty much resemble heuristic pruning techniques as use in branchancut algorithms for solving har problems. Moreover, this approach is quite moular in the sense that it prouces algorithms whose search tree backbone, i.e., the branching pattern of the algorithm as such, is not affecte by the optimization techniques reflecte in what we will call heuristic priorities (accoring to which the branching is performe) an the employe reuction rules. This not only moularizes correctness proofs for such algorithms, but also favors rapi prototyping of implementations. We will exemplify this approach by eveloping an analyzing simple algorithms for Weighte Hitting Set (WHS) problems. No prior research on parameterize algorithms has been reporte for these problems. Problem statement. Weighte Hitting Set (WHS) can be viewe as a weighte vertex cover problem on hypergraphs. More formally, this problem can be state as follows: Given: A weighte hypergraph G = (V, E, w) with ege size boune by, i.e., e E( e ), an a weight function w : V [1, ) Parameter: a nonnegative integer k Question: Is there a (weighte) hitting set C of total weight of at most k, i.e., C V e E(C e ) an w(c) := x C w(x) k? 1 This is a full version of the paper presente in [9]. Most of the work on this material has been one while the author was with Univ. Tübingen, Germany, as well as with Univ. Hertforshire, Hatfiel, UK. The accoring support is gratefully acknowlege.
3 2 Why Hitting Set? Hitting Set problems show up in many places; e.g., Reiter s grounbreaking research on moelbase iagnosis [12,17] relates the automatic iagnosis of systems to Hitting Set. The thrive for minimum hitting sets is in that context motivate by the parsimony principle in two ways: (a) the simplest iagnosis tens to fin the actual cause, an (b) when the iagnosis implies exchanging (possibly) faulty components (as a consequence of a selfiagnosis of an autonomous system, e.g., in space), then a minimum hitting set might also be the cheapest repair solution; in that particular scenario, however, the weighte case seems to be even more interesting than the unweighte one. As a further application, in [10], connections between a twotree rawing problem that is important in bioinformatics an 4WHS are shown, where the weights reflect further natural restrictions from biological backgroun knowlege. The algorithmics of this paper can be immeiately transferre to both applications. Previous work. For the unweighte case (which is a special case of the weighte setting if all weights are equal to one), there is one publishe paper presenting a search tree algorithm for Unweighte Hitting Set (HS), > 2, from a parameterize perspective [13]. The exponential base of the running time estimate for these algorithms tens to 1 with growing, although in the simplest case = 3, it is still relatively far off from that boun: that basis is By an intricate case analysis of a comparatively complicate algorithm, they were able to arrive at an O (2.270 k ) algorithm for the (unweighte) 3HS problem (i.e., all weights equal one). This was improve in [6] to about O (2.179 k ) by using a similar methoology as explaine here for the weighte case. In fact, this result was recently improve by Wahlström [19] by a ifferent although relate methoology to O ( k ). Notice that we are ealing with search tree algorithms an apply a parameterize analysis of the search tree size. If we then say that the algorithm has O (f(k)) running time, where k is the parameter, this means that the search tree has size (number of leaves) O(f(k)), since the work in each search tree noe will be at worst polynomial in n. In actual fact, all analysis that follows will be a clever estimate on the size of the search tree. For the special case of 2HS, likewise known as Vertex Cover, in a kin of race (using more an more intricate case analysis) an O(1.285 k +kn)algorithm [1] has been obtaine. For 2WHS, likewise known as Weighte Vertex Cover, the best that was obtaine is on O (1.396 k ), see [14]. Our approach seems not to be suitable to tackle the case = 2. The results of this paper. As in the unweighte case [6], 2 our analysis is base on the introuction of a secon auxiliary parameter that allows us to account for gains obtaine by using appropriate reuction rules an heuristic priorities. This technique can be useful in other areas of parameterize algorithms, as we believe. We get the following table for the bases c of an O (c k ) algorithm for  WHS; the bases are better than those for the unweighte case publishe in [13]: 2 A revise version of that report is going to be publishe with Algorithmica.
4 c (1) General notions an efinitions. We introuce some terminology on hypergraphs as neee for Hitting Set. A hypergraph G = (V, E) is given by its finite set of vertices V an its set of (hyper)eges E, where a hyperege is a subset of V. The carinality e of a hyperege e is also calle its size. The carinality of the set of eges which contain the vertex v is calle the egree of v, written δ(v). 2 Heuristics an reuctions for Weighte Hitting Set 2.1 A simple branching algorithm Since each hyperege must be covere an the weights are all at least one, there exists a trivial O ( k )algorithm for WHS. simplewhs(g = (V, E, w), k, S): IF k > 0 AND G has some eges THEN choose some ege e; // to be refine S = ; // solution to be constructe FOREACH x e DO // recursively branch G = (V \ {x}, {e E x / e}); S = simplewhs(g, k w(x), S {x}) IF S failure THEN break return S ELSIF E = THEN return S ELSE return failure Obviously, the base of the exponential running time of this algorithm heavily epens on the necessary amount of branching. Observe that accoring to the problem specification, in a WHS instance, there might be eges of size up to alreay in the very beginning. Small eges may also be introuce later uring the run of the algorithm. A natural heuristic woul first branch on small eges. We woul therefore refine: simplewhs(g, k, S): IF k > 0 AND G has some eges THEN choose some ege e of smallest size;... // as before Can we make use of this heuristic priority in our analysis? We therefore now efine reuction rules which we will always exhaustively apply at the beginning of each recursive call. Moreover, we switch towars a binary branching at vertices (instea of branching on eges), as can be seen in Alg. WHSST below.
5 4 2.2 Reuction rules First reuction rule: vertex omination. The vertex omination rule that was use in [6,13] for the unweighte case is invali in full generality in the weighte case, but has to be replace by the following weighte vertex omination rule: If, for all eges e, x e implies y e an if w(y) w(x), then elete x. This reuction rule implies the following one (reuction rule for egreeonevertices): If x, y e with δ(x) = 1 an w(y) w(x), then remove x. The sounness of this rule is easily seen: the only reason for taking a vertex x into the hitting set, in a situation as escribe by the reuction rule, is that it might be cheap. Conserving expensive vertices makes no sense. This reuction rule immeiately implies: Lemma 1. In a reuce instance, there is no ege with more than one vertex of egree one. The next lemma is again an easy consequence from the weighte vertex omination rule an is of particular importance when > 3. Lemma 2. In a reuce instance, for any two eges e 1 an e 2, there is at most one x e 1 e 2 with δ(x) = 2. Other rules state in [6] literally transfer to the weighte case: Secon reuction rule: ege omination. An ege e is ominate by another ege f if f e. Then, we elete e, since covering f will automatically also cover e. Thir reuction rule: small eges. Delete all eges of size one an place the corresponing vertices into the hitting set. The small ege rule, together with the vertex omination rule, proves the nonexistence of isolate eges in the following precise sense: Lemma 3. In a reuce instance, there is no ege e such that all vertices x e have egree one. Fourth reuction rule: ege cover rule. If G contains a component C that is of maximum vertex egree two, then resolve C in polynomial time. This (last) rule is justifie by the following lemma: Lemma 4. If G is a weighte hypergraph of maximum vertex egree of two, then a minimum weighte hitting set can be foun in polynomial time. Proof. To G, there correspons an egeweighte graph G whose vertices are the eges of G an whose vertexajacency relation is the egeajacency relation of G. Then, a minimum weighte hitting set of G correspons to a minimum weighte ege cover of G that can be compute in polynomial time.
6 5 2.3 Branching rules an their analysis The iea of making favorable branches first has also another bearing, this time on the way we are going to analyze the search tree algorithm, base on an auxiliary parameter l. Let T l (k), l 0 enote the size (more precisely, the number of leaves) of the search tree when assuming that at least l eges in the given instance (with parameter k) have a size of (at most) 1. The intuition is that T 3 (k) woul escribe a situation which is more like ( 1)WHS than T 2 (k). The unerlying iea is that search trees with many small eges are smaller than search trees with only a few; hence: k : T l (k) T l+1 (k). (2) Regaring an upper boun on the size T (k) of the search tree of the whole problem, we can equate T (k) = T 0 (k) by following the same intuition. Eq. (2) also shows that, upon analyzing a T l situation, we can always assume that there are exactly l eges that have a size of at most 1, an these small eges o have a size of exactly 1. Our algorithm will make choices with the bias of what we will call heuristic priorities. They can be refine if necessary along the analysis of the algorithm. The simplest list to start with might contain a single rule that shoul be intuitively clear: Choose a vertex of highest egree within an ege of smallest size. We will upate the list of priorities whenever necessary. WHSST(G = (V, E, w), k, S): exhaustively apply reuction rules; IF k > 0 THEN IF E = THEN return S; choose some vertex x accoring to the heuristic priorities S = ; // solution to be constructe E = { e E x / e }; S = WHSST((V \ {x}, E ), k w(x), S {x}); IF S ==failure THEN E = { e \ {x} e E }; S = WHSST((V \ {x}, E ), k, S); return S ELSIF G contains some eges or k < 0 return failure ELSE // G contains no eges an k is zero return S In the very beginning, given the instance (G, k), we call WHSST(G, k, ). We assume that reuction rules may also change the parameter value k an the solution S. The algorithm is quite generic: the list of reuction rules may grow an we might also change the heuristic priorities. The simple binary branching structure of WHSST enables a straightforwar inuctive proof of its correctness:
7 6 Theorem 1. If the reuction rules are correct, then WHSST(G, k, ) either returns a correct hitting set to the WHS instance (G, k) or it returns failure, if there is no solution of size at most k. Proof. The proof is by straightforwar inuction on the number of vertices of the hypergraph, similar to the unweighte case consiere in [6]. 3 A simple branching analysis We will now unertake a simple analysis, only consiering T 0, T 1 an (partially) T 2 an T 3. Lemma 5. T 0 (k) T 0 (k 1) + T 3 (k). Proof. Whenever we select an ege of size to branch on (accoring to the heuristic priorities), we can fin an ege that contains a vertex x of egree three or larger ue to the ege cover rule. One branch is that x is put into the hitting set. This reuces the amissible weight by at least one. If x is not put into the hitting set, then at least three new eges of size two are create. T 1 branching. In the next lemma, we show a first step into a strategy which will finally give us better branching behaviors. Namely, we try to exploit the effect of reuction rules triggere in ifferent subcases. This alreay necessitates a refinement in the choice of heuristic priorities: within a smallest ege e of size j <, we prefer branching at x e that maximizes the number of incient eges of size j + 1. Lemma 6. T 1 (k) max{t 0 (k 1) + T 1 (k 1) + T 2 (k 1) + ( 4)T 3 (k 1), T 0 (k 1) + T 1 (k 1) + (j 2)T 2 (k 1) + ( 2 (2j + 1) + (j 2 + j))t 0 (k 2) : j = 2, 3,..., 2} if 4; if T 0 (k) ( 1) k or if = 3, this may be simplifie: T 1 (k) T 0 (k 1) + ( 2)T 1 (k 1). Proof. The instance G has an ege e = {x 1, x 2,..., x 1 } of size ( 1). Case 1. If there is an ege f of size such that 2 j := e f 2 (this can only happen if 4), then we woul first branch at the vertices in e f; ue to weighte vertex omination, at least one of the j branches that take one of the vertices of e f into the hitting set is an T 1 (k 1)branch an j 2 are even T 2 (k 1)branches (or better). If none of the vertices from e f goes into the hitting set, then, in orer to cover e, there are 1 j many possibilities left, an in orer to cover f, there are j remaining possibilities. This explains the other (( j) 1)( j) many T 0 (k 2)branches. We can neglect these cases in our time analysis when assuming T 0 (k) ( 1) k. For reaability, we refer for this analysis to the appenix. Case 2. If the previous case oes not occur, then for all eges f e, e f 1. Assume that x 1 is the vertex of maximum egree in e, so that we branch at x 1. If δ(x 1 ) = 1, we can eterministically resolve the case with the reuction rules (apply 1 times the weighte vertex omination rule an then the small ege
8 7 rule) an get one T 0 (k 1)branch. This is obviously better than the inequality claime in the lemma. Therefore, we can now assume that δ(x 1 ) 2. If we take x 1 into the hitting set, then we get a T 0 (k 1)branch. If we o not take x 1 into the hitting set, we create one new ege e 1 of size ( 1) an we get the ege e = e \ {x 1 } of size ( 2). In the next recursive call, e is the ege of smallest size. There is no other ege of that size, since Case 1 i not apply. We therefore continue branching at the vertex (say x 2 ) of maximum egree in e. Again, δ(x 2 ) = 1 is better than the case we are going to pursue next. If δ(x 2 ) 2, then we again have two cases: either we take x 2 into the hitting set or not. If x 2 goes into the hitting set, then this is a T 1 (k 1)branch; namely, since Case 1 i not apply, x 2 / e 1, so that the small ege e 1 will be preserve. If x 2 oes not go into the hitting set, then there will be a new ege e 2 of size ( 1) ( new ue to ege omination). In the next recursive call, e = e \ {x 1, x 2 } is the ege of smallest size. The argument continues an shows that branches of type T j (k 1) will show up, for j = 2, 3,..., 2. This shows the claim, taking into account that T j (k 1) T 3 (k 1) for j 3 ue to Eq. (2). Estimating branching numbers. By using the inequality T 3 (k) T 2 (k) T 1 (k), Lemmas 5 an 6 yiel: T 0 (k) T 0 (k 1) + T 1 (k) (3) T 1 (k) T 0 (k 1) + ( 2)T 1 (k 1) With c being the largest positive real root of the characteristic polynomial x 2 x + 2, i.e., c = = + ( 2) (4) we can see that by setting T 0 (k) = c k an T 1 (k) = (c 1)c k 1, the inequalities system (3) can be solve. The larger, the closer c gets to 1. Hence: T (k) 2.62 k 3.42 k 4.31 k 5.24 k 9.13 k k Obviously, this is worse than what Nieermeier an Rossmanith got in [13] for the (general) unweighte case (ue to the lack of the vertex omination rule in full generality), but shows the same limit behavior (when is large). Can we o better? Let us give a simple trial to incorporate T 2 an T 3 into the analysis in the special case of Weighte 3Hitting Set. 4 Weighte 3Hitting Set We will use subscripts in the functions that escribe the search tree sizes to inicate this special case. We branch accoring to the following heuristic priorities. Let s be the size of the smallest ege in the instance G = (V, E, w).
9 8 Let E s be the collection of smallest size eges. (P 3 1) Let the set of (first) branching caniates B be e E s e. (P 3 2) If e is a smallest ege that is isjoint with all other e E s, refine B = e. (P 3 3) If no such isolate smallest ege exists, then upate B to collect the vertices of maximum egree in the hypergraph ( e E s e, E s ). (P 3 4) Select x B to be a vertex of maximum egree in G. It is easy to check that the analyses of Lemmas 5 an 6 are still vali uner these heuristic priorities. Lemma 7. T 2 3 (k) max{t 1 3 (k 1) + T 2 3 (k 1), T 0 3 (k 1) + T 0 3 (k 2)} Proof. We consier first the situation that the two eges e 1 an e 2 of size two are isjoint (see (P 3 2)). Then, basically the analysis of Lemma 6 applies, showing the claim. More precisely, we have T3 2 (k) T3 1 (k 1) + T3 2 (k 1). Otherwise, e 1 e 2, i.e., e 1 = {x, y} an e 2 = {x, z}. Accoring to the heuristic priority (P 3 3), we branch at x. If we take x into the hitting set, we get a T3 0 (k 1)branch. Not taking x into the hitting set enforces y an z into the hitting set, which is a T3 0 (k 2)branch. T Lemma 8. T (k 1) + T3 0 (k 2), (k) max T3 0 (k 1) + T3 0 (k 3), T3 2 (k 1) + T3 3 (k 1) Proof. If there is a ege e of size two that has nonempty intersection with any other ege of size two, ue to (P 3 2) we branch on e without estroying the at least two other eges of size two. The reasoning given in Lemma 6 therefore yiels the upper boun T 2 3 (k 1) + T 3 3 (k 1) in this case. If the first case oes not apply, the all eges of size two are connecte. Let e 1, e 2, e 3 be three connecte eges of size two. If x e 1 e 2 e 3 exists, then we branch at x ue to (P 3 3). This gives the (trivial) upper boun of T 0 3 (k 1) + T 0 3 (k 3). Otherwise, we branch at some x containe in two small eges ue to (P 3 3); w.l.o.g.: x e 1 e 2. Since x / e 3, the case that we take x into the hitting set is inee a T 1 3 (k 1)branch. This explains the upper boun T 1 3 (k 1) + T 0 3 (k 2). Theorem 2. Weighte 3Hitting Set can be solve in time O ( k ). The algebra justifying this claim can be foun in the following subsection. We only mention for the reaer that likes to skip this section in a first rea that the exact solution of the inequalities system can be escribe by the largest positive root c 3 of the polynomial x 3 2x 2 x + 1, which then gives T3 0 (k) = c k 3, T 1 3 = c k 3/(c 3 1), T 2 3 (k) = c k 3/(c 3 1) 2, an T 3 3 (k) = c k 1 3 (c 3 1). This worst case is realize when all T 3 branches are accoring to the T 1 3 (k 1) + T 0 3 (k 2) estimate. Improving on that particular case woul not help too much, however, since the other extreme cases show also branching behaviors worse than 2.2 k. Observe that this also means that a search tree in the T 3 (k)case is only about half the size of a search tree in the T 0 (k)case.
10 9 4.1 The algebra for Weighte 3Hitting Set In the following, we suppress the subscript 3, since we are only ealing with this case. Let us first show some algebra in case that we only analyze up to T 2 (k), i.e., if we put T 3 (k) = T 2 (k). What branching behavior o we observe in either case in Lemma 7? 1. If T 2 (k) T 1 (k 1) + T 2 (k 1), we get (by Lemma 5), (T 0 (k) T 0 (k 1)) = T 1 (k 1) + (T 0 (k 1) T 0 (k 2)), which gives as a recurrence By Lemma 6, T 1 (k 1) = T 0 (k) 2T 0 (k 1) + T 0 (k 2). T 0 (k + 1) 2T 0 (k) + T 0 (k 1) = T 0 (k 1) + T 0 (k) 2T 0 (k 1) + T 0 (k 2). Therefore, 0 = T 0 (k + 1) 3T 0 (k) + 2T 0 (k 1) T 0 (k 2). This is resolve by T 0 (k) k. 2. T 2 (k) T 0 (k 1)+T 0 (k 2) gives immeiately the characteristic polynomial x 2 2x 1 with largest positive real root x = ; this is hence the worst case here. It might be surprising at first glance that we treate the weak inequalities as if they were equalities. This is base on experience: taking this approach we were always able to come up with a solution to the envisage system of inequalities. However, these computations shoul be justifie by further analysis. Since this is only an (encouraging) intermeiate result, we refrain from giving such valiation here; we will however give it in the general case. For Weighte 3Hitting Set, we erive the following recurrences: T 0 (k) T 0 (k 1) + T 3 (k) T 1 (k) T 0 (k 1) + T 1 (k 1) T 2 (k) max{t 1 (k 1) + T 2 (k 1), T 0 (k 1) + T 0 (k 2)} T 1 (k 1) + T 0 (k 2), T 3 (k) max T 0 (k 1) + T 0 (k 3), T 2 (k 1) + T 3 (k 1) How o we arrive at possible solutions? Firstly, we try to solve extreme cases that are obtaine by treating the weak inequalities as if they were equalities an by iscussing all possible combinations as escribe by the maximum operator. In our case, this woul in principle result in six systems of equations.
11 10 However, since T 2 shows up only in one place in a righthan sie of a T 3 (k)  inequality, we only get four cases. It is noteworthy to see that, when assuming T 0 (k) = c k, (for all situations) the first equation gives T 3 (k) = c k 1 (c 1) an the secon equation gives T 1 (k) = c k /(c 1). Notice that finally we are looking for an overall solution for all T j, i.e., T j (k) = α j c k 3 with c 3 an the α j still to be etermine. Our consierations so far entail: α 0 = 1 α 1 = 1/(c 3 1) α 2 α 3 = (c 3 1)/c 3 T 0 (k) = T 0 (k 1) + T 3 (k) T 1 (k) = T 0 (k 1) + T 1 (k 1) (T 2 (k) = max{t 1 (k 1) + T 2 (k 1), T 0 (k 1) + T 0 (k 2)}) T 3 (k) = T 1 (k 1) + T 0 (k 2) Obviously, the function T 2 (k) oes not come into play if we are primarily intereste in looking for solutions for T 0 (k) in this case. Plugging in the expressions for T 0 (k), T 1 (k) an T 3 (k) in the last equation yiels: c k 1 (c 1) = c k 1 /(c 1) + c k 2. After multiplication with (c 1)c 2 k an some reorering, this becomes the characteristic polynomial: 0 = c(c 1) 2 c (c 1) = c 3 2c 2 c + 1. Its largest positive real root can be boune by from above. As can be seen, this is the claime worst case. T 0 (k) = T 0 (k 1) + T 3 (k) T 1 (k) = T 0 (k 1) + T 1 (k 1) (T 2 (k) = max{t 1 (k 1) + T 2 (k 1), T 0 (k 1) + T 0 (k 2)}) T 3 (k) = T 0 (k 1) + T 0 (k 3) The same solution strategy provies: c k 1 (c 1) = c k 1 + c k 3. Multiplication with c 3 k an some reorering yiels: 0 = c 3 2c 2 1;
12 11 the largest positive real root can be boune by from above. In the following two cases, we have to istinguish the two ifferent upper bouns for T 2. T 0 (k) = T 0 (k 1) + T 3 (k) T 1 (k) = T 0 (k 1) + T 1 (k 1) T 2 (k) = T 1 (k 1) + T 2 (k 1) T 3 (k) = T 2 (k 1) + T 3 (k 1) From T 1 (k) = c k /(c 1), we can euce from the thir equation: T 2 (k) = c k /((c 1) 2 ). Therefore, the last equation gives: c k (c 1) = c k 1 /((c 1) 2 ) + c k 1 (c 1). Multiplication with c 1 k (c 1) 2 results in: 0 = (c 1) 4 c = c 4 4c 3 + 6c 2 5c + 1, whose largest positive real root can be estimate by A irect plugin yiels: T 0 (k) = T 0 (k 1) + T 3 (k) T 1 (k) = T 0 (k 1) + T 1 (k 1) T 2 (k) = T 0 (k 1) + T 0 (k 2) T 3 (k) = T 2 (k 1) + T 3 (k 1) T 0 (k) = T 0 (k 1) + T 3 (k) = T 0 (k 1) + T 2 (k 1) + T 3 (k 1) This gives again = T 0 (k 1) + T 0 (k 2) + T 0 (k 3) + (T 0 (k 1) T 0 (k 2)) = 2T 0 (k 1) + T 0 (k 3) 0 = c 3 2c 2 1; the largest positive real root can be boune by from above. So, the worst scenario gives the characteristic polynomial c 3 2c 2 c + 1, whose largest positive real root c 3 can be boune from above by We haven t yet etermine α 2. From T 2 (k) T 1 (k 1)+T 2 (k 1), we woul get α 2 = 1/(c 3 1) 2, an from T 2 (k) T 0 (k 1) + T 0 (k 2), we arrive at α 2 = (c 3 +1)/c 2 3. However, c 2 3 = (c 3 +1)(c 3 1) 2 = (c 2 3 1)(c 3 1) = c 3 3 c 2 3 c 3 +1 is true, since c 3 is a root of the characteristic polynomial, so that both (seemingly ifferent) values of α 2 are in fact equal. In other wors, we foun:
13 12 α 0 = 1 α 1 = 1/(c 3 1) 0.80 α 2 = (c 3 + 1)/c α 3 = (c 3 1)/c After having obtaine these numbers, we shoul verify that inee T 0 (k) = c k 3, T 1 = c k 3/(c 3 1), T 2 (k) = c k 2 3 (c 3 +1), an T 3 (k) = c k 1 3 (c 3 1) satisfies the whole system of inequalities. This amounts in showing that the extreme case we foun is inee maximizing all righthan sie maxima functions. In fact, our reasoning with etermining α 2 in the preceing paragraph alreay shows that T 1 (k 1) + T 2 (k 1) = T 0 (k 1) + T 0 (k 2) for our functions. We still have to eal with the T 3 (k) inequalities. 1. To show: T 1 (k 1) + T 0 (k 2) T 0 (k 1) + T 0 (k 3). Substituting the functions we erive an multiplying with c 3 k (c 1) leaves us to show: c 2 + (c 2 c) (c 3 c 2 ) + (c 1) 0 c 3 3c 2 + 2c 1 which is true for c = c To show: T 1 (k 1) + T 0 (k 2) T 2 (k 1) + T 3 (k 1). Substituting the functions we erive an iviing by c k 2 leaves us to show: c c c2 + c c 2 + c2 c = 1 + c2 c c This in turn means we have to verify (again) for c = c 3 : 0 c 3 3c 2 + 2c 1. 5 Weighte Hitting Set with 4 How well o our consierations transfer to the more general case? We analyze possible T 2 branches in what follows. Since the obtaine bases are quite satisfactory, we refrain from analyzing the T 3 branches. In our analysis, we apply the following heuristic priorities to a given (reuce) instance G = (V, E, w): Let s be the size of the smallest ege in the instance G = (V, E, w). Let E s be the collection of smallest size eges. (P 1) Let the set of (first) branching caniates B be e E s e. (P 2) Define G B = (B, E s ) an upate B to be the set of vertices in G B of maximum egree. (P 3) Choose a vertex x B of maximum egree in G. One can check that Lemmas 5 an 6 are still vali when assuming these heuristic priorities. Since in our opinion solving Hitting Set for larger is of less practical importance, we will efer some etails of the following analysis to the appenix. Analyzing T 2. We will istinguish several cases in what follows: Lemma 9. Let e 1 an e 2 be two eges of size 1. If e 1 e 2 =, then we can estimate T 2(k) T 1(k 1) + ( 2)T 2 (k 1).
14 13 This can be basically inherite from Lemma 6 ue to ege omination. As we will see, this is the secon worst case branching. Being the simplest case, we give some etails. As justifie in the appenix, we solve the next set of equations: T 0 (k) = T 0 (k 1) + T 2 (k) (5) T 1 (k) = T 0 (k 1) + T 1 (k 1) + ( 3)T 2 (k 1) T 2 (k) = T 1 (k 1) + ( 2)T 2 (k 1) This yiels, after some algebra: 0 = T 0 (k + 1) T 0 (k) + ( 1)T 0 (k 1) T 0 (k 2). (6) Theorem 3. Let c enote the largest positive real root of the polynomial x 3 x 2 + ( 1)x 1. Then T 0 (k) = ck, T 1 (k) = α,1c k with α,1 = (c + 2)(c 1)/c an T 2 (k) = α,2c k with α,2 = (c 1)/c solve the system (5). The following table lists some of the exponential bases c for (5): c (7) Lemma 10. Let e 1 an e 2 be two eges of size 1. If e 1 e 2 = j {1, 2,..., 2}, then we can estimate T 2 (k) T 0 (k 1) + T 1 (k 1) + (j 2)T 2 (k 1) + ( 1 j) 2 T 0 (k 2), thereby assuming that T 0 (k) ( 1)k, i.e., c 1. Proof. The priorities (P 1) an (P 2) let us branch at a vertex x e 1 e 2. If j > 1, the weighte vertex omination rule moreover guarantees that there is a vertex of egree at least three in e 1 e 2, an (P 3) will select one such vertex x for branching. Hence, when x is not taken into the hitting set, then we gain at least one ege of size 1 if j > 1 ue to vertex omination, see Lemma 2, since we will continue selecting vertices within e 1 e 2 accoring to (P 2). The case that eges that intersect with e 1 e 2 might contain more than one vertex in this intersection turns out not to be the worst case (assuming ( 1) k as a lower boun of our approach) along the lines of Lemma 6. If e 1 e 2 is exhauste, then in the case that we take none of the vertices from e 1 e 2 into the hitting set, we are left with two very small eges e 1 = e 1 \ e 2 an e 2 = e 2 \ e 1. P 1 lets us continue branching at say e 1. Having selecte x e 1 to go into the hitting set, e 2 will be the smallest ege (of size ( 1 j)), an hence P 1 continues to branch on e 2 in the next recursion step. This explains that we get (very grossly estimate) ( 1 j) 2 many T 0 (k 2)branches. In orer to prove Theorem 4, the following technical lemma is important:
15 14 Lemma 11. If j > 1 an > 3, then T 1 (k 1) + ( 2)T 2 (k 1) T 0 (k 1) + T 1 (k 1) + (j 2)T 2 (k 1) + ( 1 j) 2 T 0 (k 2) for T 0 (k) = ck an T 2 (k) = ck c k 1 with 1 c, inepenent of T 1. We nee a somewhat stronger result (compare to Lemma 10) in the case j = 1 that escribes our worst case (for > 4): Lemma 12. Let e 1 an e 2 be two eges of size 1. If e 1 e 2 = 1, then we can estimate T 2 (k) T 0 (k 1) + ( 3)T 1 (k 2) + [( 2)( 3) + 1]T 0 (k 2). Moreover, T 1 (k 1)+( 2)T 2 (k 1) T 0 (k 1)+( 3)T 1 (k 2)+[( 2)( 3)+1]T 0 (k 2) for T l as efine in Theorem 4 below. Proof. We only explain the branching in what follows (for the algebra, see the appenix). Assume that {x} = e 1 e 2. x is selecte for branching accoring to (P 1). If x oes not go into the hitting set, then we may continue branching on e 1. The claim is that, for any y e 1 \ {x} (with one possible exception, if δ(y) = 1 for some y e 1 ; but ue to Lemma 1, there is at most one vertex of egree one in e 1 an (P 3) avois branching at that vertex), there is an ege e y e 1 with y e y such that there is a vertex z y e 2 \ e 1 with z y / e y. For, if (e 2 \{x}) e y, then the ege omination rule woul have triggere. The branch that takes y an z y into the hitting set is a T 1 (k 2)branch (possibly better). Theorem 4. WHS can be solve in time O (c k ), where c is the largest positive root of the characteristic polynomial x 4 3x 3 ( 2 5+5)x 2 +x+( 2 6+9). Some values of c are liste below: c (8) Is it worthwhile trying to further improve on the exponential bases as erive in this section? In principle, yes of course; however, one woul nee a ifferent approach for substantial improvements: (a) the seconworst case is only slightly better than the worst case that we analyze, an (b) with growing, the lower boun ( 1) assume in (some) estimates is alreay quite well approximate. The most interesting case that remains seems to be = 4, which we tackle in the following separate section.
16 Hitting Set We are claiming that Lemma 9 actually provies the worst case for 4WHS, base on a eeper analysis an (again) slightly change heuristic priorities (which we will not make explicit in this case but which will become clear from the analysis). So, we are going to show in the remainer of this section the following result: Theorem 5. Let c 4 enote the largest positive real root of the polynomial x 3 4x 2 +3x 1. Then T 0 4 (k) = c k 4, T 1 4 (k) = α 4,1 c k 4 with α 4,1 = (c 4 2)(c 4 1)/c 4 an T 2 4 (k) = α 4,2 c k 4 with α 4,2 = (c 4 1)/c 4 solve the system (12). Moreover, O (c k 4) is an upper boun on the running time of our algorithm for solving Weighte 4Hitting Set. We can boun c 4 from above by As we have alreay seen before, the worst case (from above) we have to eal with is the case of two eges e 1, e 2 with e 1 = e 2 = 3 an {x} = e 1 e 2. We will analyze two subcases: (a) e 1, e 2: e i (e i \ {x}) but e i e 3 i = for i = 1, 2. (b) e 1, e 2 with e i (e i \ {x}) : e i e 3 i as well, for i = 1, 2. In case (a), we can branch as follows: Taking x into the hitting set gives a T 0 4 (k 1)branch. Otherwise, ue to the conition, let us continue branching at {x 1 } = e 1 e 1. (Observe that ue to weighte vertex omination, not all e 1 satisfying the conition (a) may contain {x 1, x 2 } = e 1 \ {x}.) Similarly, there is some {y 1 } = e 2 e 2. So, if we take both x 1 an y 1 into the hitting set, we get a T 0 4 (k 2)branch. If we take y 1 into the hitting set but not x 1, we must select x 2. Since y 1 / e 1 by (a), this is a T 1 4 (k 2)branch. Similarly, taking x 1 into the hitting set but not y 1 is a T 1 4 (k 2)branch. If neither x 1 nor y 1 go into the hitting set, then we gain two new small eges ue to conition (a), so this is even a T 2 4 (k 2)branch. Altogether, we have erive in case (a): T 2 4 (k) T 0 4 (k 1) + T 0 4 (k 2) + 2T 1 4 (k 2) + T 2 4 (k 2). (9) In case (b), there must be an ege e with (e 1 \{x}) e an (e 2 \{x}) e, since otherwise (i.e., if the forall conition is vacuously satisfie) the weighte vertex omination rule woul trigger an result in the following branching: T 2 4 (k) T 0 4 (k 1) + 2T 0 4 (k 2). We will see that the case we are going to consier will result in a branching that is strictly worse, so that we can neglect this case. Moreover, we can also assume that (e 1 \ {x}) e = 1, for if not, the weighte vertex omination rule woul trigger in the case that x is not taken into the hitting set. This means that one of the two vertices from (e 1 \ {x}) e must go into the hitting set. Moreover, since e is now hit, two subcases arise: either both vertices from e 1 \ {x} are containe in e an there are no eges that contain vertices from e 2 \ {x} apart from e 2 an possibly e; then, the weighte vertex omination rule woul trigger once more an altogether yiel the branch we alreay observe before, namely: T 2 4 (k) T 0 4 (k 1) + 2T 0 4 (k 2);
17 16 or, say y 1 e 2 \ {x} is containe in one other ege e y besies e 2 an e, an no vertex from e 1 \ {x} is containe in e y. Branching at y 1 woul hence gain us one small ege at least in the situation that y 1 is not going into the hitting set. Altogether, we get as estimate: T 2 4 (k) T 0 4 (k 1) + T 0 4 (k 2) + T 1 4 (k 2). This is again always better than the general estimate that we erive now. So, we can assume now that {x 1 } = (e 1 \ {x}) e an that {y 1 } = (e 2 \ {x}) e. Assume we start branching at x 1. Let us call {x 2 } = e 1 \ {x, x 1 }. We istinguish two subcases regaring {y 2 } = (e 2 \ {x, y 1 }): (i) δ(y 2 ) = 1 an (ii) δ(y 2 ) 2. Case (i) is again split into two cases: (ia) {e E y 1 e, x 1 / e} = 1 an (ib) {e E y 1 e, x 1 / e} > 1. In case (ia), if x 1 is taken into the hitting set, we will elete either y 1 or y 2 ue to the weighte vertex omination rule, an then this ege is resolve by the small ege rule. This gives a T 0 4 (k 2)branch. If x 1 is not taken into the hitting set, x 2 must be in. Moreover, if we continue branching at y 2, we get a T 0 4 (k 2)branch when y 1 goes into the hitting set an two T 0 4 (k 3)branches when not y 1 but y 2 is in the hitting set, since then one of the two remaining vertices from e must be in the hitting set, too. Overall, we get in this case: T 2 4 (k) T 0 4 (k 1) + 2T 0 4 (k 2) + 2T 0 4 (k 3). (Notice that this is strictly speaking a tight analysis for δ(y 1 ) = 2.) Again, this is not the worst case to consier. In case (ib), if x 1 is taken into the hitting set, we might take y 1 into the hitting set. This gives a T 0 4 (k 2)branch. If y 1 oes not go into the hitting set, then y 2 will, giving a T 1 4 (k 2)branch (gaining a small ege by the case assumption). If x 1 is not going into the hitting set but x 2, we get one T 0 4 (k 2)branch an two T 1 4 (k 3)branches. This yiels: T 2 4 (k) T 0 4 (k 1) + 2T 0 4 (k 2) + T 1 4 (k 2) + 2T 1 4 (k 3); again, this is not the worst case to consier. In case (ii), we can assume that there is an ege e y that contains y 2 but none of the vertices from {x, y 1, x 1 }. Namely, since δ(y 2 ) 2, there is at least one ege e y besies e 2 that contains y 2. As can be seen, δ(y 2 ) = 2 is the worst case we assume henceforth. If y 1 e y, the weighte ege omination rule woul have triggere, yieling a better branching as analyze before. If x 1 e y, we have a situation analyze uner case (i) above. We branch as follows: If x 1 goes into the hitting set, we can gain a small ege in the case that y 2 oes not go into the hitting set (assume we continue branching at y 2 ); hence, this is one T 0 4 (k 2)branch an one T 1 4 (k 2)branch. If x 1 is not in the hitting set, let us continue branching at y 1. By a simple analysis, we get one T 0 4 (k 2) an two T 0 4 (k 3)branches (similar as in the previous cases). This means: T 2 4 (k) T 0 4 (k 1) + 2T 0 4 (k 2) + T 1 4 (k 2) + 2T 0 4 (k 3). (10) We will show now that the case that e 1 e 2 = is in fact the worst case that yiels the estimate T 2 4 (k) = T 1 4 (k 1) + 2T 2 4 (k 1).
18 17 We first consier Eq. (9): we have to fin an upper boun for T 2 4 (k) + T 0 4 (k 1) + T 0 4 (k 2) + 2T 1 4 (k 2) + T 2 4 (k 2). With the settings of the functions as formulate in Theorem 5, this expression means: c k 4 + 2c k c k (c 4 2)(c 4 1)c k c k 2 4 c k 3 4. After multiplication with c 3 k 4, we get the following chain: c c c (c 4 2)(c 4 1) 1 = c c c c 2 4 6c = [ c c 2 4 3c 4 + 1] c The expression in square brackets vanishes, since c 4 is a root of the characteristic polynomial mentione in Theorem 5, an the inequality follow from 3 c 4. Let us consier Eq. (10): we will fin an upper boun for T 2 4 (k) + T 0 4 (k 1) + 2T 0 4 (k 2) + T 1 4 (k 2) + 2T 0 4 (k 3) uner the settings of the functions as formulate in Theorem 5. This means we have to upperboun: c k 4 + 2c k c k (c 4 2)(c 4 1)c k c k 3 4. After multiplication with c 3 k 4, we get the following chain: c c c (c 4 2)(c 4 1) + 2 = c c 2 4 c = [ c c 2 4 3c 4 + 1] c c c c 4 = c 4 (3 c 4 ) 0 The expression in square brackets vanishes, since c 4 is a root of the characteristic polynomial mentione in Theorem 5, an the two inequalities follow from 3 c 4. 7 Conclusions Let us allow first one final remark regaring the correctness of our analysis: Remark 1. The astute reaer might have notice that there are possible situations say with two or more eges of size 1 where the reuction rules actually estroy all of them an replace them by one ege of size 2 or smaller. However, it can be easily verifie that, for all 3 an l 2 as far as analyze, T l (k) ( 2)k. This is base on the assumption c 1. Therefore, this omission in the analysis will never affect the worst case running time claime in the paper.
19 18 We have evelope an analyze a novel, topown methoology for parameterize search tree algorithms. Up to now, we have applie this methoology to Hitting Set [6], biplanarization problems [8] (thereby improving on the constants erive in [4]), linear arrangement problems (in the long version of [7]) an to Weighte Hitting Set (this paper). A further natural caniate for applying this technique woul be (Weighte) Directe Feeback Vertex / Arc Set in Tournaments, as consiere in [15], as well as variants thereof [2]. In orer to apply this metho, we nee a kin of secon auxiliary parameter in the problem which we try to improve on in case the main parameter cannot be improve upon binary branching. In the case of (Weighte) Hitting Set, the number of eges of small size is such an auxiliary parameter. Our results show that this methoology is a quite powerful tool of algorithm analysis. For example, while the gap between the running times of the (very sophisticate) best search tree algorithms for Weighte Vertex Cover an for Vertex Cover [1,14] o iffer significantly (both algorithms being approximately of the same complexity), this paper shows that with our analysis metho of a comparatively simple algorithm for 3WHS, we can even (slightly) improve on the previous analysis of a much more sophisticate algorithm for Unweighte 3HS [13]. It may be interesting to compare the way the analysis of the recurrences guie by the auxiliary parameter is unertaken in this paper with the analysis metho of Wahlström [18] or with Eppstein s quasiconvex metho [5]. It woul be also interesting to see this approach applie to other, ifferent problems with accoringly ifferent auxiliary parameters. More generally speaking, there seems to be a recent thrive in Exact Algorithmics towars simple algorithms. The Minimum Dominating Set algorithm of Fomin, Granoni an Kratsch is only one more example (see [11]) that incientally also uses a (special) Hitting Set algorithm. This irection of research certainly brings practical an theoretical research on attacking har problems closer together, since one coul also envisage a kin of interplay between algorithm analysis an algorithm testing in the near future. Can an appropriate analysis then explain certain observe phenomena of the implementation? The moular ecomposition of such an algorithm into the actual recursive search tree backbone an the reuction rules an (in particular) the heuristic priorities also opens up a whole area of experimental algorithmics: uner which circumstances (or, in a more theoretical formulation: for which classes of hypergraphs) is a certain set of rules the most successful? Can this be prove? Due to the simple overall structure of the algorithms, also an analysis of expecte running times (possibly aing coin tossing into the heuristic priorities) might be possible. Incientally, improvements in parameterize algorithms for Hitting Set also entail improvements in exact algorithms for Minimum Hitting Set, measure in terms of number of vertices: in the case of 3Hitting Set, the use of the algorithm exhibite in [6] improve Wahlström s algorithm [18] from
20 19 O ( n ) own to O ( n ) (personal communication by Wahlström ating from 2005). The results of this paper will immeiately entail new running time bouns for exact algorithms for Minimum Weighte Hitting Set. For example, along the lines sketche by Raman, Saurabh an Sikar, in [16], we get an exact algorithm for Minimum Weighte 4Hitting Set that runs in time O (1.97 n ), using our parameterize Weighte 4Hitting Set algorithm. References 1. J. Chen, I. A. Kanj, an W. Jia. Vertex cover: further observations an further improvements. Journal of Algorithms, 41: , M. Dom, J. Guo, F. Hüffner, R. Nieermeier, an A. Truß. Fixeparameter tractability results for feeback set problems in tournaments. In T. Calamoneri, I. Finocchi, an G. F. Italiano, eitors, Conference on Algorithms an Complexity CIAC, volume 3998 of LNCS, pages Springer, R. G. Downey an M. R. Fellows. Parameterize Complexity. Springer, V. Dujmović, M. R. Fellows, M. Hallett, M. Kitching, G. Liotta, C. McCartin, N. Nishimura, P. Rage, F. A. Rosamon, M. Suerman, S. Whitesies, an D. R. Woo. A fixeparameter approach to 2layer planarization. Algorithmica, 45: , D. Eppstein. Quasiconvex analysis of backtracking algorithms. In Proc. 15th Symp. Discrete Algorithms SODA, pages ACM an SIAM, January H. Fernau. A topown approach to searchtrees: Improve algorithmics for 3 Hitting Set. Technical Report TR04073, Electronic Colloquium on Computational Complexity ECCC, An upate version has been accepte to Algorithmica. 7. H. Fernau. Parameterize algorithmics for linear arrangement problems. In U. Faigle, eitor, CTW 2005: Workshop on Graphs an Combinatorial Optimization, pages University of Cologne, Germany, Long version to appear in Discrete Applie Mathematics. 8. H. Fernau. Twolayer planarization: improving on parameterize algorithmics. In P. Vojtáš, M. Bieliková, B. CharronBost, an O. Sýkora, eitors, SOFSEM, volume 3381 of LNCS, pages Springer, H. Fernau. Parameterize algorithms for hitting set: the weighte case. In T. Calamoneri, I. Finocchi, an G. F. Italiano, eitors, Conference on Algorithms an Complexity CIAC, volume 3998 of LNCS, pages Springer, H. Fernau, M. Kaufmann, an M. Poths. Comparing trees via crossing minimization. In R. Ramanujam an Saneep Sen, eitors, Founations of Software Technology an Theoretical Computer Science FSTTCS 2005, volume 3821 of LNCS, pages Springer, F. V. Fomin, F. Granoni, an D. Kratsch. Measure an conquer: omination a case stuy. In L. Caires, G. F. Italiano, L. Monteiro, C. Palamiessi, an M. Yung, eitors, Automata, Languages an Programming, 32n International Colloquium, ICALP, volume 3580 of LNCS, pages Springer, J. e Kleer, A. K. Mackworth, an R. Reiter. Characterizing iagnoses an systems. Artificial Intelligence, 56: , R. Nieermeier an P. Rossmanith. An efficient fixeparameter algorithm for 3Hitting Set. Journal of Discrete Algorithms, 1:89 102, R. Nieermeier an P. Rossmanith. On efficient fixe parameter algorithms for weighte vertex cover. Journal of Algorithms, 47:63 77, 2003.
nparameter families of curves
1 nparameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,
More informationFirewall Design: Consistency, Completeness, and Compactness
C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an XiangYang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 787121188,
More informationFactoring Dickson polynomials over finite fields
Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University
More informationA Generalization of Sauer s Lemma to Classes of LargeMargin Functions
A Generalization of Sauer s Lemma to Classes of LargeMargin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationOn Adaboost and Optimal Betting Strategies
On Aaboost an Optimal Betting Strategies Pasquale Malacaria 1 an Fabrizio Smerali 1 1 School of Electronic Engineering an Computer Science, Queen Mary University of Lonon, Lonon, UK Abstract We explore
More information19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes
First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is socalle because we rearrange the equation
More informationUNIFIED BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH
UNIFIED BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH OLIVIER BERNARDI AND ÉRIC FUSY Abstract. This article presents unifie bijective constructions for planar maps, with control on the face egrees
More informationModelling and Resolving Software Dependencies
June 15, 2005 Abstract Many Linux istributions an other moern operating systems feature the explicit eclaration of (often complex) epenency relationships between the pieces of software
More informationAsymptotically Faster Algorithms for the Parameterized face cover Problem
Asymptotically Faster Algorithms for the Parameterized face cover Problem Faisal N. AbuKhzam 1, Henning Fernau 2 and Michael A. Langston 3 1 Division of Computer Science and Mathematics, Lebanese American
More informationMSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUMLIKELIHOOD ESTIMATION
MAXIMUMLIKELIHOOD ESTIMATION The General Theory of ML Estimation In orer to erive an ML estimator, we are boun to make an assumption about the functional form of the istribution which generates the
More informationOptimal Control Policy of a Production and Inventory System for multiproduct in Segmented Market
RATIO MATHEMATICA 25 (2013), 29 46 ISSN:15927415 Optimal Control Policy of a Prouction an Inventory System for multiprouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational
More informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More information2 HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;
More informationThe oneyear nonlife insurance risk
The oneyear nonlife insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on nonlife insurance reserve risk has been evote to the ultimo risk, the risk in the
More informationChapter 2 Review of Classical Action Principles
Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationIntegral Regular Truncated Pyramids with Rectangular Bases
Integral Regular Truncate Pyramis with Rectangular Bases Konstantine Zelator Department of Mathematics 301 Thackeray Hall University of Pittsburgh Pittsburgh, PA 1560, U.S.A. Also: Konstantine Zelator
More informationA New Evaluation Measure for Information Retrieval Systems
A New Evaluation Measure for Information Retrieval Systems Martin Mehlitz martin.mehlitz@ailabor.e Christian Bauckhage Deutsche Telekom Laboratories christian.bauckhage@telekom.e Jérôme Kunegis jerome.kunegis@ailabor.e
More informationPurpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of
Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose
More informationData Center Power System Reliability Beyond the 9 s: A Practical Approach
Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of missioncritical
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms
More informationState of Louisiana Office of Information Technology. Change Management Plan
State of Louisiana Office of Information Technology Change Management Plan Table of Contents Change Management Overview Change Management Plan Key Consierations Organizational Transition Stages Change
More informationJON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT
OPTIMAL INSURANCE COVERAGE UNDER BONUSMALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,
More informationLecture 17: Implicit differentiation
Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we
More informationA Comparison of Performance Measures for Online Algorithms
A Comparison of Performance Measures for Online Algorithms Joan Boyar 1, Sany Irani 2, an Kim S. Larsen 1 1 Department of Mathematics an Computer Science, University of Southern Denmark, Campusvej 55,
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationWhich Networks Are Least Susceptible to Cascading Failures?
Which Networks Are Least Susceptible to Cascaing Failures? Larry Blume Davi Easley Jon Kleinberg Robert Kleinberg Éva Taros July 011 Abstract. The resilience of networks to various types of failures is
More informationarcsine (inverse sine) function
Inverse Trigonometric Functions c 00 Donal Kreier an Dwight Lahr We will introuce inverse functions for the sine, cosine, an tangent. In efining them, we will point out the issues that must be consiere
More information2r 1. Definition (Degree Measure). Let G be a rgraph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by,
Theorem Simple Containers Theorem) Let G be a simple, rgraph of average egree an of orer n Let 0 < δ < If is large enough, then there exists a collection of sets C PV G)) satisfying: i) for every inepenent
More informationLABORATORY COMPACTION TEST Application
1 LABORATORY COMPACTION TEST Application Compaction of soil is probably one of the largest contributors to the site ork economy; in other orsit is important. Large sums of money are spent on it every
More informationAn intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations
This page may be remove to conceal the ientities of the authors An intertemporal moel of the real exchange rate, stock market, an international ebt ynamics: policy simulations Saziye Gazioglu an W. Davi
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationCh 10. Arithmetic Average Options and Asian Opitons
Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options
More informationDepartment of Mathematical Sciences, University of Copenhagen. Kandidat projekt i matematik. Jens Jakob Kjær. Golod Complexes
F A C U L T Y O F S C I E N C E U N I V E R S I T Y O F C O P E N H A G E N Department of Mathematical Sciences, University of Copenhagen Kaniat projekt i matematik Jens Jakob Kjær Golo Complexes Avisor:
More informationSensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm
Sensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,
More information20122013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 08
01013 Enhance Instructional Transition Guie Unit 08: Exponents an Polynomial Operations (18 ays) Possible Lesson 01 (4 ays) Possible Lesson 0 (7 ays) Possible Lesson 03 (7 ays) POSSIBLE LESSON 0 (7 ays)
More informationDifferentiability of Exponential Functions
Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an
More informationMinimumEnergy Broadcast in AllWireless Networks: NPCompleteness and Distribution Issues
MinimumEnergy Broacast in AllWireless Networks: NPCompleteness an Distribution Issues Mario Čagal LCAEPFL CH05 Lausanne Switzerlan mario.cagal@epfl.ch JeanPierre Hubaux LCAEPFL CH05 Lausanne Switzerlan
More informationLecture L253D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L253D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general threeimensional
More informationImproved division by invariant integers
1 Improve ivision by invariant integers Niels Möller an Torbjörn Granlun Abstract This paper consiers the problem of iviing a twowor integer by a singlewor integer, together with a few extensions an
More informationRisk Management for Derivatives
Risk Management or Derivatives he Greeks are coming the Greeks are coming! Managing risk is important to a large number o iniviuals an institutions he most unamental aspect o business is a process where
More informationWitt#5e: Generalizing integrality theorems for ghostwitt vectors [not completed, not proofread]
Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5e: Generalizing integrality theorems for ghostwitt vectors [not complete, not proofrea In this note, we will generalize most of
More informationKater Pendulum. Introduction. It is wellknown result that the period T of a simple pendulum is given by. T = 2π
Kater Penulum ntrouction t is wellknown result that the perio of a simple penulum is given by π L g where L is the length. n principle, then, a penulum coul be use to measure g, the acceleration of gravity.
More informationMathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition
Mathematical Moels of Therapeutical Actions Relate to Tumour an Immune System Competition Elena De Angelis (1 an PierreEmmanuel Jabin (2 (1 Dipartimento i Matematica, Politecnico i Torino Corso Duca egli
More informationWeb Appendices of Selling to Overcon dent Consumers
Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the
More informationAn Introduction to Eventtriggered and Selftriggered Control
An Introuction to Eventtriggere an Selftriggere Control W.P.M.H. Heemels K.H. Johansson P. Tabuaa Abstract Recent evelopments in computer an communication technologies have le to a new type of largescale
More informationNotes on tangents to parabolas
Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,
More information! # % & ( ) +,,),. / 0 1 2 % ( 345 6, & 7 8 4 8 & & &&3 6
! # % & ( ) +,,),. / 0 1 2 % ( 345 6, & 7 8 4 8 & & &&3 6 9 Quality signposting : the role of online information prescription in proviing patient information Liz Brewster & Barbara Sen Information School,
More informationRules for Finding Derivatives
3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to
More informationLecture 13: Differentiation Derivatives of Trigonometric Functions
Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the
More information(We assume that x 2 IR n with n > m f g are twice continuously ierentiable functions with Lipschitz secon erivatives. The Lagrangian function `(x y) i
An Analysis of Newton's Metho for Equivalent Karush{Kuhn{Tucker Systems Lus N. Vicente January 25, 999 Abstract In this paper we analyze the application of Newton's metho to the solution of systems of
More informationUnsteady Flow Visualization by Animating EvenlySpaced Streamlines
EUROGRAPHICS 2000 / M. Gross an F.R.A. Hopgoo Volume 19, (2000), Number 3 (Guest Eitors) Unsteay Flow Visualization by Animating EvenlySpace Bruno Jobar an Wilfri Lefer Université u Littoral Côte Opale,
More informationOptimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs
Applie Mathematics ENotes, 8(2008), 194202 c ISSN 16072510 Available free at mirror sites of http://www.math.nthu.eu.tw/ amen/ Optimal Control Of Prouction Inventory Systems With Deteriorating Items
More informationM147 Practice Problems for Exam 2
M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work
More informationDiffraction Grating Equation with Example Problems 1
Emmett J. Ientilucci, Ph.D. Digital Imaging an Remote Sensing Laboratory Rochester Institute of Technology December 18, 2006 Diffraction Grating Equation with Example Problems 1 1 Grating Equation In Figure
More informationUsing Stein s Method to Show Poisson and Normal Limit Laws for Fringe Subtrees
AofA 2014, Paris, France DMTCS proc. (subm., by the authors, 1 12 Using Stein s Metho to Show Poisson an Normal Limit Laws for Fringe Subtrees Cecilia Holmgren 1 an Svante Janson 2 1 Department of Mathematics,
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationA BlameBased Approach to Generating Proposals for Handling Inconsistency in Software Requirements
International Journal of nowlege an Systems Science, 3(), 7, JanuaryMarch 0 A lamease Approach to Generating Proposals for Hanling Inconsistency in Software Requirements eian Mu, Peking University,
More informationPROBLEMS. A.1 Implement the COINCIDENCE function in sumofproducts form, where COINCIDENCE = XOR.
724 APPENDIX A LOGIC CIRCUITS (Corrispone al cap. 2  Elementi i logica) PROBLEMS A. Implement the COINCIDENCE function in sumofproucts form, where COINCIDENCE = XOR. A.2 Prove the following ientities
More informationHull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5
Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. OnePerio Binomial Moel Creating synthetic options (replicating options)
More informationA Data Placement Strategy in Scientific Cloud Workflows
A Data Placement Strategy in Scientific Clou Workflows Dong Yuan, Yun Yang, Xiao Liu, Jinjun Chen Faculty of Information an Communication Technologies, Swinburne University of Technology Hawthorn, Melbourne,
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationA Universal Sensor Control Architecture Considering Robot Dynamics
International Conference on Multisensor Fusion an Integration for Intelligent Systems (MFI2001) BaenBaen, Germany, August 2001 A Universal Sensor Control Architecture Consiering Robot Dynamics Frierich
More informationThe most common model to support workforce management of telephone call centers is
Designing a Call Center with Impatient Customers O. Garnett A. Manelbaum M. Reiman Davison Faculty of Inustrial Engineering an Management, Technion, Haifa 32000, Israel Davison Faculty of Inustrial Engineering
More informationView Synthesis by Image Mapping and Interpolation
View Synthesis by Image Mapping an Interpolation Farris J. Halim Jesse S. Jin, School of Computer Science & Engineering, University of New South Wales Syney, NSW 05, Australia Basser epartment of Computer
More informationIntroduction to Integration Part 1: AntiDifferentiation
Mathematics Learning Centre Introuction to Integration Part : AntiDifferentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More information9.3. Diffraction and Interference of Water Waves
Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately
More informationReading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14.
7 Shocks Reaing: Ryen chs 3 & 4, Shu chs 5 & 6 For the enthusiasts, Shu chs 3 & 4 A goo article for further reaing: Shull & Draine, The physics of interstellar shock waves, in Interstellar processes; Proceeings
More informationMeasures of distance between samples: Euclidean
4 Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate
More informationPythagorean Triples Over Gaussian Integers
International Journal of Algebra, Vol. 6, 01, no., 5564 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok
More informationMathematics Review for Economists
Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school
More informationImage compression predicated on recurrent iterated function systems **
1 Image compression preicate on recurrent iterate function systems ** W. Metzler a, *, C.H. Yun b, M. Barski a a Faculty of Mathematics University of Kassel, Kassel, F. R. Germany b Faculty of Mathematics
More informationUCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 9 Paired Data. Paired data. Paired data
UCLA STAT 3 Introuction to Statistical Methos for the Life an Health Sciences Instructor: Ivo Dinov, Asst. Prof. of Statistics an Neurology Chapter 9 Paire Data Teaching Assistants: Jacquelina Dacosta
More informationA Case Study of Applying SOM in Market Segmentation of Automobile Insurance Customers
International Journal of Database Theory an Application, pp.2536 http://x.oi.org/10.14257/ijta.2014.7.1.03 A Case Stuy of Applying SOM in Market Segmentation of Automobile Insurance Customers Vahi Golmah
More informationSearch Advertising Based Promotion Strategies for Online Retailers
Search Avertising Base Promotion Strategies for Online Retailers Amit Mehra The Inian School of Business yeraba, Inia Amit Mehra@isb.eu ABSTRACT Web site aresses of small on line retailers are often unknown
More informationBOSCH. CAN Specification. Version 2.0. 1991, Robert Bosch GmbH, Postfach 50, D7000 Stuttgart 1. Thi d t t d ith F M k 4 0 4
Thi t t ith F M k 4 0 4 BOSCH CAN Specification Version 2.0 1991, Robert Bosch GmbH, Postfach 50, D7000 Stuttgart 1 The ocument as a whole may be copie an istribute without restrictions. However, the
More informationProduct Differentiation for SoftwareasaService Providers
University of Augsburg Prof. Dr. Hans Ulrich Buhl Research Center Finance & Information Management Department of Information Systems Engineering & Financial Management Discussion Paper WI99 Prouct Differentiation
More informationShort Cycles make Whard problems hard: FPT algorithms for Whard Problems in Graphs with no short Cycles
Short Cycles make Whard problems hard: FPT algorithms for Whard Problems in Graphs with no short Cycles Venkatesh Raman and Saket Saurabh The Institute of Mathematical Sciences, Chennai 600 113. {vraman
More informationSupporting Adaptive Workflows in Advanced Application Environments
Supporting aptive Workflows in vance pplication Environments Manfre Reichert, lemens Hensinger, Peter Daam Department Databases an Information Systems University of Ulm, D89069 Ulm, Germany Email: {reichert,
More informationThe Inefficiency of Marginal cost pricing on roads
The Inefficiency of Marginal cost pricing on roas Sofia GrahnVoornevel Sweish National Roa an Transport Research Institute VTI CTS Working Paper 4:6 stract The economic principle of roa pricing is that
More information1 Derivatives of Piecewise Defined Functions
MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.
More informationMinimizing Makespan in Flow Shop Scheduling Using a Network Approach
Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Amin Sahraeian Department of Inustrial Engineering, Payame Noor University, Asaluyeh, Iran 1 Introuction Prouction systems can be ivie
More informationIf you have ever spoken with your grandparents about what their lives were like
CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth The question of growth is nothing new but a new isguise for an ageol issue, one which has always intrigue an preoccupie economics:
More informationRUNESTONE, an International Student Collaboration Project
RUNESTONE, an International Stuent Collaboration Project Mats Daniels 1, Marian Petre 2, Vicki Almstrum 3, Lars Asplun 1, Christina Björkman 1, Carl Erickson 4, Bruce Klein 4, an Mary Last 4 1 Department
More informationBOSCH. CAN Specification. Version 2.0. 1991, Robert Bosch GmbH, Postfach 30 02 40, D70442 Stuttgart
CAN Specification Version 2.0 1991, Robert Bosch GmbH, Postfach 30 02 40, D70442 Stuttgart CAN Specification 2.0 page 1 Recital The acceptance an introuction of serial communication to more an more applications
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationEven Faster Algorithm for Set Splitting!
Even Faster Algorithm for Set Splitting! Daniel Lokshtanov Saket Saurabh Abstract In the pset Splitting problem we are given a universe U, a family F of subsets of U and a positive integer k and the objective
More informationTrigonometry CheatSheet
Trigonometry CheatSheet 1 How to use this ocument This ocument is not meant to be a list of formulas to be learne by heart. The first few formulas are very basic (they escen from the efinition an/or Pythagoras
More informationy or f (x) to determine their nature.
Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic
More information11 CHAPTER 11: FOOTINGS
CHAPTER ELEVEN FOOTINGS 1 11 CHAPTER 11: FOOTINGS 11.1 Introuction Footings are structural elements that transmit column or wall loas to the unerlying soil below the structure. Footings are esigne to transmit
More informationDetecting Possibly Fraudulent or ErrorProne Survey Data Using Benford s Law
Detecting Possibly Frauulent or ErrorProne Survey Data Using Benfor s Law Davi Swanson, Moon Jung Cho, John Eltinge U.S. Bureau of Labor Statistics 2 Massachusetts Ave., NE, Room 3650, Washington, DC
More informationCrossOver Analysis Using TTests
Chapter 35 CrossOver Analysis Using ests Introuction his proceure analyzes ata from a twotreatment, twoperio (x) crossover esign. he response is assume to be a continuous ranom variable that follows
More informationSensitivity Analysis of Nonlinear Performance with Probability Distortion
Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 2429, 214 Sensitivity Analysis of Nonlinear Performance with Probability Distortion
More informationDiscrete Mathematics, Chapter 5: Induction and Recursion
Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Wellfounded
More informationDigital barrier option contract with exponential random time
IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun
More informationAnalysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs
Analysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
More informationCALCULATION INSTRUCTIONS
Energy Saving Guarantee Contract ppenix 8 CLCULTION INSTRUCTIONS Calculation Instructions for the Determination of the Energy Costs aseline, the nnual mounts of Savings an the Remuneration 1 asics ll prices
More informationHeatAndMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar
Aalborg Universitet HeatAnMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar Publishe in: International Journal of Heat an Mass Transfer
More information