Computing the depth of an arrangement of axisaligned rectangles in parallel


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1 Computing the depth of an arrangement of axisaigned rectanges in parae Hemut At Ludmia Scharf Abstract We consider the probem of computing the depth of the arrangement of n axisaigned rectanges in the pane, which is the maximum number of rectanges containing a common point. For this probem we give a sequentia O(n og n) time agorithm, and a parae agorithm with running time O(og 2 n) in the cassica PRAM mode. We aso describe how to adopt the paraeization to a shared memory machine mode with a fixed number of processing units. 1 Introduction In this paper we consider a basic geometric probem: how to compute the depth of the arrangement of n axisaigned rectanges in R 2. The depth of a point p is defined as the number of rectanges containing p, and the depth of the arrangement is the maximum depth over a points in R 2, or, equivaenty, it is the maximum number of rectanges that contain a common point, see Figure 1 for an exampe. Figure 1: Arrangement of axisparae rectanges with ces of maxima depth 4 (shaded). This research was partiay supported by the German Research Foundation (DFG) in the project Parae agorithms in computationa geometry with focus on shape matching, contract number AL 253/71. Institute of Computer Science, Freie Universität Berin, 1
2 We describe a parae agorithm for this probem for a shared memory parae machine mode. The agorithm has a running time O(og 2 n) and tota work O(n og 2 n) in the cassica EREWPRAM mode. We aso describe how to adopt the paraeization to a more reaistic assumption of having a fixed number k of processing units in a shared memory machine, which fits better the modern muticore processors. The current trend in the microprocessor industry is to increase the performance in computing not by increasing the CPU cock rates but by mutipe CPU cores working on shared memory and common cache. This trend in the hardware deveopment makes the design of parae agorithms once again an active topic in the agorithmic community. In this paper we first describe a sequentia O(n og n) time agorithm for computing the depth of the arrangement of axisaigned rectanges in the pane. Namey, we construct a baanced search tree on the xcoordinates of the corners of rectanges and then perform a pane sweep aong the yaxis, whie updating the box coverage information in the tree. To our knowedge no O(n og n) agorithm for the depth probem has been given before expicity, athough the resut is somehow fokore knowedge in the computationa geometry community. In fact, the scheme of agorithms given for soving Kee s measure probem (KMP), i.e., computing the voume of the union of n axisparae boxes, can be used for computing the depth of arrangements. For the twodimensiona KMP Bentey described but did not pubish such an agorithm [2] the idea of which is given in [6], however. Our agorithm is simiar to some extent. We deveop and describe it in detai, mosty in order to derive from it in Section 3 the efficient parae agorithm for the probem. For the appications of the depth computation probem we mention two exampes: One is a geometric pattern matching probem. For two mpoint sets A and B in R 2 finding a transformation minimizing the directed L Hausdorff distance from A to B can be reduced to finding the depth in an arrangement of O(m 2 ) boxes. Another exampe is a custering probem: For a given set of n points in R 2 and a given radius r find a L disk of radius r containing the argest number of points, that is, the densest custer of radius r. This custering probem is dua to determining the deepest point in the arrangement of n boxes with side ength 2r. For genera shape (agebraic) regions, not just axisaigned rectanges, there is no better agorithm known for computing the depth of their arrangement than to construct the compete arrangement and then to traverse it. For arrangements of disks the probem is known to be 3SUM hard [1], so subquadratic agorithms are not ikey to exist. For an arrangement of axisaigned boxes in higher dimensions Chan [3] describes a sequentia agorithm with running time O((n d/2 / og d/2 1 n) og d/2 og n) for d 3. 2 Sequentia Agorithm In this section we describe the sequentia agorithm for computing the depth of the arrangement of axisaigned rectanges. The genera idea is the foowing: For a given set of n axisaigned rectanges we buid a baanced binary search tree T on the xcoordinates of the vertica sides of the rectanges, so that a xcoordinates are in the eaves of the tree. Let x 1, x 2,..., x 2n be the xcoordinates of the vertica sides of the rectanges in sorted order. With the eaf abeed with x i, i = 1,..., 2n 1, we associate the interva [x i, x i+1 ). The ast eaf, abeed with x 2n, is associated with the interva [x 2n, x 2n ]. With an interna node v we associate the union of the intervas 2
3 of its chidren. Space requirement for the tree is inear in the number of rectanges. Next we perform a topdown sweep aong the yaxis. Each sweepine event, i.e., a y coordinate of the top or bottom of a rectange, has two xcoordinates a and b, with a, b {x 1,..., x 2n } and a < b, of the vertica sides of the corresponding rectange and an event vaue d associated with it. The event vaue is d = 1 if it is the top of the rectange (the rectange is opened ) and d = 1 if it is the bottom (the rectange is cosed ). To process a sweepine event we traverse the tree T from the root node to the eaves abeed a and b. In the nodes of the tree we want to count the number of rectanges covering the associated xinterva, and to update this information with each yevent. Of course, we cannot store this information directy and update it for a covered nodes for each rectange, since that coud make up to inear time per update. Instead, we maintain in every interna node v for the current state of the sweepine in counters and r the number of rectanges covering the interva of the eft and right chid of v that were opened minus the number of ones cosed since the ast traversa of that chid. In other words, the counters and r store the additive update of the information about how many open rectanges cover the interva of the eft and right chid, respectivey, at the current position of the sweepine. Additionay, counters m and r m store the maximum of this additive update for the eft and right chid, respectivey, since the ast traversa of the corresponding chid node. Every eaf node contains counters c and c m, which keep track of the current and maximum coverage of the associated interva during the sweep. The information in counters, r, m and r m is exacty as described above if the node v is traversed, and thus updated, by the current sweepine event. For subsequent events that do not traverse v the information may get outdated. Thus, the counters of the interna node v store the updates that happened between the ast traversa of the corresponding chid node and the ast traversa of v. The counters c and c m in the eaf nodes are ony updated for the open and coseevents of the corresponding rectange. During each traversa of v by one of the searches the counter vaues are propagated from v to its chid on the search path in temporary counters t and t m, which are initiay set to 0. I.e., once we updated v as described beow and move to its eft (right) chid, t and t m are set to the vaues of v. and v. m (v.r and v.r m ), and then v., v. m (v.r, v.r m ) are reset to 0. Thus, when we enter the chid node w the counter t is the additive change since the ast update of w of the number of open rectanges that competey cover the interva of w; t m is either the maximum vaue of that change between the ast update of w and the current event, or 0 if the additive updates were a negative. An update of an interna node v is performed sighty differenty depending on whether both xcoordinates a and b associated with the event are contained in the subtree rooted at v (see Procedure 1: SearchBoth), or the search paths for a and b spit earier in the tree and the subtree of v contains ony a (see Procedure 2: SearchLeft) or ony b (procedure SearchRight). If both a and b are contained in the subtree rooted at v we need to update the counters of v ony with the vaues propagated from the parent node, without considering the event vaue. For and r we simpy add the vaue of t (ines 3 and 4 of procedure SearchBoth). The maxcounters ( m and r m ) are set to the maximum of their od vaue, and the sum of the od counter ( or r, resp.) and t m (ines 1 and 2 of procedure SearchBoth). If both a and b are contained in the same, say eft, subtree of v then t and t m are set to and m, the search for both xcoordinates is continued in the eft subtree, and the counters and m are reset to 0. If the paths to a and b spit in the node v, we perform two separate 3
4 searches in the eft and right subtrees (ines 12 and 13) v. m = max(v. m, v. + t m ) v.r m = max(v.r m, v.r + t m ) v. = v. + t v.r = v.r + t if a < b v.x then SearchBoth (v.eft, a, b, v., v. m ) v. = v. m = 0 if v.x < a < b then SearchBoth (v.right, a, b, v.r, v.r m ) v.r = v.r m = 0 if a v.x < b then SearchLeft(v.eft, a, v., v. m ) SearchRight(v.right, b, v.r, v.r m ) v. = v. m = v.r = v.r m = 0 Procedure 1 : SearchBoth(v, a, b, t, t m ) Notice that the update instructions for the counters make sure that the stored vaues are not the absoute numbers of simutaneousy opened rectanges but an additive update to the information stored in the corresponding subtree. To iustrate this remark we consider an exampe of a (sub)sequence of update events for a node v starting from some event, that resets the counters and m to 0, that is, the event updates the eft chid of v with the information gathered so far, see Figure 2. Assume that the node v now gets the foowing events: two cosing, three opening, and one cosing, where the eft chid is not traversed by the events, but the corresponding rectanges cover the interva of the eft chid. Then the vaues of and m counters are updated as shown in Figure 2. At the end of this subsequence the interva of the eft chid is covered by exacty as many rectanges as at the beginning: two od rectanges were cosed, three new opened and one new cosed, thus = 0. However, during this subsequence, between the ast two events, the interva of the eft chid was covered by one more rectange than in the beginning. This additiona coverage is maxima over the whoe subsequence, therefore, m = 1. Once the search path for a and b spits in some vertex, we know that the current rectange spans a intervas of the right subtrees of the eft search path, i.e., path to a, and a intervas of the eft subtrees of the right search path, i.e., path to b. Therefore, for a node v on the eft (right) search path we aso add the event vaue d to the counter r (). When we reach the eaves containing a and b we can update the current and maximum depth of the associated intervas. The updates are described in the procedure SearchLeft for the search path of a. The search for b is performed in a procedure caed SearchRight, which is anaogous to procedure SearchLeft and, therefore, is not expicity given here. After a sweepine events have been processed, the depth of the arrangement is determined as the maximum of the c m counters of the eaf nodes. The correctness of the agorithm is based on the foowing observation: Once we reach the bottom of a rectange, the counter c m in the eaf node abeed with the xcoordinate of its eft vertica boundary contains the maximum coverage of the associated xinterva between the highest ycoordinate and the ycoordinate of the bottom of the rectange. Since, ceary, 4
5 ... events 1st event of the subsequence m 0 0 sweep direction cose cose open open open cose interva of v interva of the eft chid of v Figure 2: An exampe of a subsequence of sweepine events and the corresponding updates of the counters and m of an interna node v if a v.x and v is an interna node then v.r m = max(v.r m, v.r + t m, v.r + t + d) v.r = v.r + t + d SearchLeft(v.eft, a, v. + t, max(v. m, v. + t m )) v. = v. m = 0 if a > v.x then v. m = max(v. m, v. + t m ); v. = v. + t SearchLeft(v.right, a, v.r + t, max(v.r m, v.r + t m )) v.r = v.r m = 0 if a = v.x and v is a eaf then c m = max (c m, c + t m, c + t + d) c = c + t + d Procedure 2 : SearchLeft(v, a, t, t m ) 5
6 every maximay covered ce has a eft vertica boundary that is a part of a eft boundary of a rectange, and thus covers at east one eaf interva, we capture at east one ce with maximum depth this way, i.e., store its depth in a counter c m of one of the eaves. If we want not ony to compute the depth, but aso get a point with maximum depth, we can additionay store a ycoordinate for each maxcounter. This ycoordinate has to be updated with the yvaue of that event which resuts in the counter update. The time needed to construct the tree and to sort the yevents is O(n og n). Each of the 2n events is processed in O(og n) time. Theorem 1 summarizes the resut of this section: Theorem 1. The depth of an arrangement of n axisaigned rectanges in R 2 can be computed in time O(n og n) with O(n) additiona memory. 3 Parae Agorithm To enabe a parae execution of the agorithm we maintain socaed history ists in the nodes of the tree T. A history ist of a node v contains an entry for each event of the sweepine that traverses the node v, i.e., for each ycoordinate of a top or bottom side of a rectange spanning the interva [a, b] in xdirection, such that a or b is contained in the subtree rooted at v. A history entry α of an interna node contains its timestamp the ycoordinate (or the rank of the ycoordinate) of the sweepine event, the corresponding xvaues, the event vaue d and the counters, r, m, r m, as described in Section 2, and resetfags ρ, ρ r. The reset fags indicate whether the vaues of the eft or right counters, respectivey, survive unti the next event: The vaue of ρ /ρ r in the entry α is 0 if the event of α causes the traversa of the eft/right subtree, since in this case the counter vaues are propagated to the subtree and wi be reset in the node v. Otherwise, the vaue is 1. Additionay, every history event has a pointer to the corresponding event, i.e., the event with the same ycoordinate, in the parent node. A history entry of a eaf node contains ony its ycoordinate, the event vaue d, and two counters c and c m. Every yevent appears in at most two nodes of each eve of the tree and requires constant space. Thus, the space for the tree is O(n og n). A information of a history event, except for the counter vaues, m, r, r m, is known at the construction time of the tree and can be set during the tree construction. Now we can fi out the counter vaues in the history ists starting with the root node down to the eaves. The eft/right counters in a root node events are set to 0. For an interna node v et α (i) be the ith history event of v and et α (j) be the corresponding history event in the parent node of v, i.e., y (i) = y (j). Then the counters of α (i) are computed according to the foowing rues: Let t (i) and t (i) m denote the vaues of r (j) and r (j) m parent node, and vaues of (j) and (j) m of the event α (j) if v is a right chid of its otherwise. If α (i) contains both xcoordinates a and b 6
7 associated with y (i) then set (i) = (i 1) ρ (i 1) + t (i) (1) { } m (i) = max m (i 1) ρ (i 1), (i 1) ρ (i 1) + t (i) m (2) r (i) = r (i 1) ρ (i 1) r + t (i) (3) { } r m (i) = max r m (i 1) ρ (i 1) r, r (i 1) ρ (i 1) r + t (i) m, (4) where the vaues with the highindex (i 1) denote the vaues of the history event preceding α (i) in the node v. Aso if a history event α (i) of a node v contains ony a, and a is in the right subtree of v, or if α (i) contains ony b, and b is in the eft subtree of v, the counters are set as above. That is, the vaues from the parent node v are propagated to the corresponding chid node but the interva associated with the chid is not competey covered by the rectange causing the event, and thus, we do not need to consider the event vaue d. In case α (i) contains ony a, and a is in the eft subtree of v, then the compete right subtree is covered by the current rectange. Therefore, the right counters are incremented by the event vaue d (i) : r (i) = r (i 1) ρ (i 1) r + t (i) + d (i) (5) { r m (i) = max r m (i 1) ρ (i 1) r, r (i 1) ρ (i 1) r + t (i) m, r (i)}. (6) The eft counters are updated as in equations (1), (2). In case α (i) contains ony b, and b is in the right subtree of v the eft counters must be adjusted anaogousy: (i) = (i 1) ρ (i 1) + t (i) + d (i) (7) { m (i) = max m (i 1) ρ (i 1), (i 1) ρ (i 1) + t (i) m, (i)}. (8) and the right counters are updated as in equations (3), (4). The counters of the ith entry in a eaf node are updated as foows: c (i) = c (i 1) + t (i) + d (i) (9) { c (i) m = max c (i 1) m, c (i 1) + t m (i), c (i 1) + t (i) + d (i)}. (10) Then, after a events have been processed, each eaf node stores the maxima coverage of its associated interva up to the position where the (ast) rectange with the corresponding vertica side was cosed. The depth of the arrangement is then the maximum over the c m counters of the eaves. So we coud buid the tree T and then traverse it evebyeve starting from the root node to the eaves, and nodebynode within one eve, setting the counters in a history events. Thus, we woud have a sequentia agorithm with running time in O(n og n) as before but with O(n og n) memory usage. Parae impementation on a PRAM. For the parae agorithm we assume that there are O(n) processors on a EREWPRAM machine avaiabe. Then sorting of the corner points 7
8 of the rectanges once by ycoordinates and once by xcoordinates can be performed in O(og n) time, i.e., O(n og n) tota work, using, for exampe, the sorting agorithm by Coe [4]. In the foowing we assume that a xcoordinates of the vertica sides of the rectanges are distinct, which simpifies the anaysis and the description. With carefu consideration of technica detais the anaysis can be extended to the genera setting within the same time and tota work bounds as beow. The tree T without the history ists can be buid straightforwardy in time O(og n) on presorted xcoordinates of the vertica sides of the rectanges. The unsorted history ists for each eve of the tree can be constructed in constant time per eve with O(n) processing units: We assign one processor to every history event. Every processor writes an entry for its event to the history ists of the corresponding two eaf nodes. Then, each processor creates evebyeve parent entries for its event in the history ists of the nodes on the path from the two eaves to the root node. At the end of this process the history ists contain entries with correcty set timestamps (the ycoordinates), pointers to the parent events, the event vaue d, and, for the interna nodes, the reset switches ρ, ρ r. The counter vaues remain open. Severa processors can write their entries to the history ist of the same node in parae since, if we organize the history ists as arrays, every processor independenty can easiy compute the index of its entry in the ist. The tota time for the construction of the unsorted history ists is O(og n). Now we can sort a history ists by timestamps in parae. The tota size of the history ists in one eve is at this stage 2n and the tota size of a history ists in the tree is O(n og n). We can sort the history ists evebyeve, processing a ists of one eve in parae in time O(og n) per eve with O(n) processors. Then, for the compete tree, the construction time of the sorted history ists is O(og 2 n). The computation of the eft/right counters in the event entries is performed evebyeve starting with the root node down to the eaves. We first observe that the computation of the counters (i) and r (i) according to equations (1), (3), (5), (7) corresponds to a prefix sum computation: Consider an interna node v and its eft counter of the ith history entry (i). Let j be the highest index i with ρ (j) = 0. Then the vaue of (i) is the sum i k=j+1 (t(k) +d (k) ), where d (k) is set to 0 if the computation of (k+1) foows equation (1). Thus, if we can subdivide the counters of a history ist into subsequences corresponding to bocks of ones terminated by a zero of the ρ switches, then we can in a first step set each (i) to t (i) or t (i) +d (i), respectivey, and then perform parae prefix sum computations on the subsequences. For the subdivision into subsequences we can again use the parae prefix sum computation. First, we invert the vaues in the ρ sequence, shift it by one to the right, and add a preceding 0, i.e., ˆρ (i) = 1 ρ (i 1) for i > 1 and ˆρ (1) = 0. Now we can compute the prefix sum of the ˆρ vaues. The bocks of equa vaues in this new sequence correspond exacty to the subsequences in the sequence. Then in the prefix sum computation of the vaues we ony need to consider those entries that have the same ˆρ vaue. The rcounter vaues are computed anaogousy. Prefix sum computation can be performed in O(og n) time [5].The tota size of a prefix sum ists of one eve is O(n), and there are O(og n) eves. So we need O(og 2 n) time in tota. For the computation of the maxcounters according to equations (2), (4) or (6), (8), e.g., for m (i), we need the vaues m (i 1), (i 1), t m, and possiby (i). A of these vaues, except for m (i 1), are computed by now. Observe, that these equations are of the form u (i) = 8
9 max ( u (i 1) ρ (i 1), v (i)) (, where, e.g., in (6) u (i) = r m (i) and v (i) = max r i 1 ρ (i 1) + t (i) m, r ). (i) So the u (i) are prefix maxima of the previousy computed v (i) and can be computed in O(og n) time anaogousy to the prefix sums. We summarize the preceding sketch of the parae agorithm and its anaysis: Theorem 2. The depth of an arrangement of n axisaigned rectanges in R 2 can be computed on a EREWPRAM with O(n) processing units in time O(og 2 n). Parae impementation for a fixed number k of processors with shared memory: Sorting of the x and ycoordinates of the vertica and horizonta sides of the rectanges can obviousy be performed on a kprocessor machine in time O( n k og n). Then we have to spit the work performed by the agorithm between k processors. For this purpose we spit the tree construction into k subtrees, each containing at most 2n/k xcoordinates. Each of the subtrees is constructed by one processor sequentiay. Afterwards, the k subtrees are combined into a singe tree by adding a tree of height og k on top of the subtrees. The tree construction incudes the history ists except for the vaues of the counters r,, r m, m in the interna nodes, and the counters c, c m in the eaves. For the computation of the counters in the history ists we appy the same idea: for the top tree we appy parae prefix sum computation by processing the history ists in bocks of at most k eements. There are O(n/k) such bocks in each eve, and each bock is processed in O(og k) time. Thus, the og k eves of the top tree can be processed in O( n k og2 k) time. For the k subtrees we appy the sequentia agorithm to find the maximum depth in each subtree. The size of the subtrees is O(n/k), thus, the time for the remaining eves is O( n k og n). The tota time is then O( n k (og2 k + og n)). Summarizing, we have: Coroary 1. The depth of an arrangement of n axisparae rectanges can be computed in parae by k processors with shared memory in time O(n/k(og 2 k + og n)). 4 Future Work Athough the depth computation of a set of rectanges in R 2 is an interesting probem on its own, we pan to deveop parae agorithms for higher dimensiona depth computation. Further, we are interested in an impementation of the agorithm presented here, and possiby agorithms for higher dimensiona probems, and in their experimenta evauation. The impementations shoud be performed for currenty avaiabe parae hardware patforms, such as muticore CPUs and genera purpose GPUs. References [1] B. Aronov and S. HarPeed. On approximating the depth and reated probems. SIAM J. Comput., 38(3): , [2] J. L. Bentey. Agorithms for Kee s rectange probems. Unpubished notes, [3] T. M. Chan. A (sighty) faster agorithm for Kee s measure probem. In SCG 08: Proceedings of the twentyfourth annua symposium on Computationa geometry, pages , New York, NY, USA, ACM. 9
10 [4] R. Coe. Parae merge sort. SIAM J. Comput., 17(4): , [5] D. W. Hiis and G. L. Steee. Data parae agorithms. Communications of the ACM, December [6] J. van Leeuwen and D. Wood. The measure probem for rectanguar ranges in dspace. J. Agorithms, 2(3): ,
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