B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....node u of T is associated with the subset N u N of points that are stored at the leaves of the subtree rooted at u. We preprocess N u for hemisphere reporting queries, where each query reports all points of N u lying inside a query hemisphere H ae S . By using a halfplane reporting structure [11], we can preprocess N u , in O(jN u j log jN u j) time, into a data structure of size O(jN u j) so that a hemisphere query can be answered in O(log jN u j t) time, where t is the output size. We attach this structure at u as its secondary structure. The total time spent in preprocessing N is ....

B. Chazelle, L. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

....segments allocated to v. Within Strip(v) the segments of E(v) can be viewed as lines since they cross Strip(v) completely. Let E (v) be the set of points dual to these lines. We store E (v) in an instance D(v) of the standard halfplane reporting (resp. counting) structure for R given in [10] (resp. 26] This structure uses O(m) space and has a query time of O(log m k v ) resp. O(m ) where m = jE(v)j and k v is the output size at v. To answer a query, we search in T using q s x coordinate. At each node v visited, we need to report or count the lines intersected by r. But, by ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76-90, 1985.

.... counting is close to optimal, because there is an t(n 2 ] log n) lower bound on the amount of storage of any data structure for half plane range counting with O(log n) query time [35] It is interesting that a much better solution can be obtained for half plane range reporting: Chazelle et al. [46] showed that one can achieve O(log n t k) query time using only linear storage. With this structure it is not possible, however, to do range counting without explicitly listing all the points, nor is it possible to extend the structure to triangular range searching. Trade offs and higher ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. In Proc. 2dth Annu. IEEE Sympos. Found. Cornput. Sci., pages 217-225, 1983.

....neighbors between lines and points: a) primal, nearest point neighbor to a line# (b) dual, vertical ray shooting in a line arrangement. minimum area triangle with s is the nearest vertical neighbor of l. This observation was used to develop O(n ) algorithms for the minimum triangle problem [7, 16, 15]. We can tighten this characterization as follows. Let 4pqr be the minimum area triangle, and assume that the vertical projection of r is between those of p and q. Then as before r is the nearest neighbor of line pq, but the vertical segment connecting r and line pq actually touches segment pq. ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

.... t) query cost, still with linear space. These techniques were adapted to external memory by the works of Agarwal et al. AAE 98] Kollios, Gunopoulos and Tsotras [KGT99] and Agarwal, Arge and Erickson [AAE00] Another series of techniques began with the work of Chazelle, Guibas and Lee [CGL85] who introduced to the problem the concept of arrangements. For the planar case, their technique achieves optimal time O(log n t) with linear space. However, generalizing the approach to higher dimensions, introduces non linear space, typically exponential to the problem dimension. For ....

B. Chazelle, L.J. Guibas, and D.T. Lee. The power of geometric duality. BIT, 25(1):76--90, 1985.

....node u of T is associated with the subset N u N of points that are stored at the leaves of the subtree rooted at u. We preprocess N u for hemisphere reporting queries, where each query reports all points of N u lying inside a query hemisphere H ae S 2 . By using a halfplane reporting structure [11], we can preprocess N u , in O(jN u j log jN u j) time, into a data structure of size O(jN u j) so that a hemisphere query can be answered in O(log jN u j t) time, where t is the output size. We attach this structure at u as its secondary structure. The total time spent in preprocessing N is O(jF ....

B. Chazelle, L. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

.... arrangement of lines in the plane; this last theorem states that the average value of the square of the number of vertices of a face of the arrangement is a O(1) this is a well known consequence of the linear bound on the complexity of the so called zone of a line in an arrangement of lines; see [10, 17, 13, 15] and [21, 4] for an higher dimensional analogue. To state our sum of squares theorem we need to introduce new operators. First we extend the de nition of the operator to the case where the set of obstacles is augmented with a set of bitangent obstacles : for H a set of pairwise disjoint ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76-90, 1985.

....of a query line has been the focus of extensive research in the past. Of course, this problem is of interest only when several queries are made. Typically some form of preprocessing takes place which allows each query to be answered in less than the brute force O(n) time. Chazelle, Guibas and Lee [CGL85] compute the convex layers of the given data set as a preprocessing step (in O(n log n) time using Chazelle s algorithm mentioned in chapter 2) Then they report every point on one side of a query line in O(k log n) time, where k is the number of points reported. Since we only need to count the ....

....L we have a sorted list of all intersection points on . ffl for every intersection point we have a radially sorted list of all lines intersecting the point. An arrangement of n lines may be constructed in Theta(n 2 ) time and space. This result was first obtained by Chazelle, Guibas and Lee [CGL85] and by Edelsbrunner, O Rourke and Seidel [EOS86] The proof of this result is described well in [O R95] The same algorithm may be used to construct an arrangement of line segments. A nice application of arrangements is for sorting all points about every point in a data set in O(n 2 ) time ....

B. Chazelle, L. Guibas, and D. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....for all points on one side of a query hyperplane. Halfspace range searching is the simplest form of non isothetic (non orthogonal) range searching. The problem was first considered in external memory by Franciosa and Talamo [83, 82] Based on an internal memory structure due to Chazelle et al. [55], Agarwal et al. 5] described an optimal O(log B N T=B) query and linear space structure for the 2 dimensional case. Using ideas from an internal memory result of Chan [50] they described a structure for the 3 dimensional case, answering queries in O(log B N T=B) expected I Os but requiring ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25(1):76--90, 1985.

....(i) as follows: Let a be any great arc of A or B and let G a be the great circle containing it. We add the G a s one by one to the initially empty arrangement. We compute the intersections of each G a with the current arrangement and retain only those that also belong to a. By the zone theorem [8], which is also applicable to great circles, due to central projection, this takes O(n) time per G a , hence O(n 2 ) time in all. The proof of part (ii) is as follows: A 0 has three types of vertices: a) vertices of A inside on R, b) vertices of R, and (c) intersections between arcs of A ....

....through the part of the current arrangement that is in the interior of R, from one intersection point to the other, and compute the intersection of G a with previously added great circles (type (a) vertices) We retain only those intersections that are also on the great arc a. By the zone theorem [8], applied to G a and the great circles, this takes O(n) time per G a . The time bound follows. 3.1 The constrained width problem The problem here is to minimize C wid (d) when d is restricted to lie within a convex polygonal region, R, on SS 2 , bounded by great arcs. Note that the antipodal V ....

Chazelle, B., Guibas, L., and Lee, D.T., (1985). The power of geometric duality, BIT, 25, 76-90. 19

....u of T is associated with the subset N u # N of points that are stored at the leaves of the subtree rooted at u. We preprocess N u for hemisphere reporting queries, where each query reports all points of N u lying inside a query hemisphere H # S 2 . By using a halfplane reporting structure [13], we can preprocess N u , in O( N u log N u ) time, into a data structure of size O( N u ) so that a hemisphere query can be answered in O(log N u t) time, where t is the output size. We attach this structure at u as its secondary structure. The total time spent in ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

....This can be seen by considering the problem of computing the combinatorial arrangement of a set of lines in the plane. There is an incremental algorithm that computes the arrangement by adding lines one by one, threading the new line through the arrangement of the previously inserted lines[12, 35, 30] (see also section 2.1) The algorithm uses a version of the orientation test to decide how the new line exits the current region of the arrangement. If the orientation test is implemented in floating point arithmetic, the answers may not always be reliable, and the algorithm must disambiguate in ....

B. Chazelle, L.J. Guibas, D.T. Lee, The power of geometric duality, Proc. 24th Annual Symp. Found. Comp. Science 217--225, 1983. 41

.... of the curves T i coincides with the arrangement of the curves fl Sigmai (up to some trivial details concerning the convex hull) To compute the arrangement of the curves T i we use the optimal incremental technique which have been developed for constructing arrangement of (pseudo)lines [2, 6, 9] (we omit trivial details concerning the 3n Gamma 3 touching points between the curves T i and T j for adjacent pseudotriangles T i and T j in the pseudo triangulation) however we have to be careful because the intersection of two pseudocircles T i and T j is not computable in O(1) time ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, pages 76--90, 1985.

....This observation not only simplifies the data structure but also gives better bounds in many cases, including halfspace range reporting. See [16, 52, 63, 75] for some applications of filtering search. An optimal halfspace reporting data structure in the plane was proposed by Chazelle et al. [74]. They compute convex layers L 1 ; Lm of S L i is the set of points lying on the boundary of the convex hull of S n S j i L j and store them in a linear size data structure, so that a query can be answered in O(log n k) time. Their technique does not extend to three dimensions. ....

....h S = one does not have to store simplex range searching structure at each node of the tree. Consequently, the query time and the size of the data structure can be improved slightly; see Table 3 for a summary of results. Problem d Size Query Time Source Reporting d = 2 n log n k [74] Emptiness d = 2 n log n [219] Reporting d = 3 n log n log n k [17] Emptiness d = 3 n log n [98] Reporting d 3 n log log n n 1 Gamma1=bd=2c polylog n k [179] Emptiness d 3 n n 1 Gamma1=bd=2c 2 O(log n) 179] Table 3. Asymptotic upper bounds for halfspace range searching in ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

....the preceding cell using crossing rules. If any DBG is not strongly connected, one of its strong components is a locally free subassembly in the corresponding direction. The cells in the plane and their adjacency relations can be computed in optimal Theta(k 2 ) time using a topological sweep [18, 27]. The cost of executing a crossing rule from cell f i to cell f j is proportional to the size of the crossing set C ij (or C ji ) Although a single C ij may include k contacts, each contact is only a member of crossing sets along its circle, and only those sets on the violating side of the ....

....c between P i and P j will produce a reciprocal constraint from P j to P i that must be added as well. The line corresponding to c can be incrementally added to the planar arrangement in O(k) time (O(k 4 ) time in the full rigid motion case) producing O(k) respectively O(k 4 ) new cells [18, 27]. In addition, the DBGs for all the motions violating constraint c need to be updated; these motions are given by the cells on the violating side of the constraint line. For each such cell f , the weight of the arc from P i to P j in G(f) must be increased by 1, a total of O(k 2 ) steps (O(k 5 ....

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B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....of only one cell in Delta(f) we obtain X Delta2 Delta(f) jS Delta j k f : 6.6.1) We now analyze the expected running time of the algorithm. We count the time spent during the ith stage in inserting the line i and then add this time over all stages of the algorithm. The zone theoreom [28, 40] implies that in Step 1 of the algorithm, we spend O(i) time in tracing i through A(L i 1 ) While processing an active face f of A(L i 1 ) that intersects i , for each cell Delta 2 Delta(f) we spend O(1) time in Step 1 and O(jS Delta j) time in Step 2. In Step 3, for each triangle s 2 ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

....of . In this case Omega; n) nodes of the tree are visited by the query algorithm. Similar performance degradation can be shown for the other mentioned structures. In the internal memory model, a two dimensional halfspace query can be answered in time O(log 2 N T ) time using O(N) space [12], but it may require O(log 2 N T ) I Os in terms of the external memory model. The only known external memory data structure with provably good query performance works in two dimensions, where it uses O(n p N)blocks of space and answers queries using optimal O(log B n t) I Os [20,21] 1.3 ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....geometry has been devoted to this fundamental problem called halfspace range reporting , a special case of range searching [4, 37, 49, 53, 55] Here are some of the major known results. First, halfspace range reporting in the planar case (d = 2) was solved optimally by Chazelle, Guibas, and Lee [28]. Their data structure takes linear space and answers a query in O(log n k) time, where k is the number of reported points. The preprocessing can be accomplished in O(n log n) time using Chazelle s algorithm for convex layers [22] Unfortunately, the approach does not generalize to higher ....

....furthermore, the space complexity can be improved. Using advanced tools and a larger preprocessing time, Appendix A.3 gives a modification of our data structure that is deterministic and uses only O(n log log n) space. 2. It is worthwhile to compare Chazelle, Guibas, and Lee s optimal method [28] with the specialization of our method in two dimensions. Ours seems easier to implement as convex layers are not involved. 3. Higher dimensional extensions are possible, although we do not see any significant improvements. By a standard lifting map, Corollaries 2.4 and 2.5 imply a new method for ....

B. Chazelle, L. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....perturbation of . In this case n) nodes of the tree are visited by the query algorithm. Similar performance degradation can be shown for the other mentioned structures. In the internal memory model, a two dimensional halfspace query can be answered in time O(log # N T ) time using O(N) space [14], but it may require O(log # N T ) I Os in terms of the external memory model. The only known external memory data structure with provably good query performance works in two dimensions, where it uses O(n # N) disk blocks of space and answers queries using optimal O(log # n t) I Os [24, 25] ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76-90.

....their faces. There are O(n 2 ) regions R ij , each with a constant number of edges. Therefore step 1 requires O(n 2 ) operations. A graph representing faces and adjacency relationships in an arrangement of m segments in the plane can be computed in O(m 2 ) steps and has O(m 2 ) faces [3, 5]. Here m = O(n 2 ) so the number of faces and the computing time for step 2 are O(n 4 ) Each region R ij has a constant number of boundary segments, so testing the initial face f 0 for inclusion in all regions requires O(n 2 ) operations. Finally, the depth first search steps over each ....

....in the average case and find practical bounds on its application. 6.1 Implementation A drawback of the algorithm above is the storage requirement: the arrangement may take O(n 4 ) space to store, which is impractical for complicated assemblies. The topological sweep line algorithm in [3, 5] sweeps over an arrangement of m lines in O(m) space and optimal O(m 2 ) time, but cannot be extended directly to the case of line segments instead of lines. Our implementation addresses these problems by performing a vertical line sweep [8, 13] over the arrangement of O(n 2 ) line segments. ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....for halfspace queries. The best halfspace counting data structure known requires roughly # # ) space to achieve logarithmic query time [17, 24 JEFF ERICKSON Table 2 Best known upper bounds for halfspace emptiness queries. Space Preprocessing Query Time Source # # 3 #(#) #(# log #) #(log #) [21, 3, 26] #(# ##### # log ##### #) #(# ##### # log ####### #) #(log #) 44] #(#) #(# ### ) #(# ######### 2 ##### # ## ) 39] # # 4 #(#) #(# log #) #(# ######### polylog #) 44] # # # # # ##### #(# polylog #) #( #polylog #)## ####### ) 44] 42] whereas, the same query time can be ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), pp. 76-90.

....perturbation of . In this case n) nodes of the tree are visited by the query algorithm. Similar performance degradation can be shown for the other mentioned structures. In the internal memory model, a two dimensional halfspace query can be answered in time O(log 2 N T ) time using O(N) space [14], but it may require O(log 2 N T ) I Os in terms of the external memory model. The only known external memory data structure with provably good query performance works in two dimensions, where it uses O(n p N) disk blocks of space and answers queries using optimal O(log B n t) I Os [24, ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76-90.

....for halfspace queries. The best halfspace counting data structure known requires roughly n d ) space to achieve logarithmic query time [17, 24 JEFF ERICKSON Table 2 Best known upper bounds for halfspace emptiness queries. Space Preprocessing Query Time Source d 3 O(n) O(n log n) O(log n) [21, 3, 26] O(n bd=2c = log bd=2c n) O(n bd=2c = log bd=2c n) O(log n) 44] O(n) O(n 1 ) O(n 1 1=bd=2c 2 O(log n) 39] d 4 O(n) O(n log n) O(n 1 1=bd=2c polylog n) 44] n s n bd=2c O(s polylog n) O( npolylog n) s 1=bd=2c ) 44] 42] whereas, the same query time can be ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), pp. 76-90.

....for its online version: Given a set of n points, preprocess it to answer halfspace emptiness (or reporting) queries. In two and three dimensions, wecan easily build a linear size data structure in O(n log n) time, that allows halfspace emptiness queries to be answered in logarithmic time [3, 10, 14]. In higher dimensions, a randomized algorithm due to Clarkson [12] answers halfspace emptiness queries in time O(log n) after O(n bd=2c ) 1 preprocessing time. Matousek [19] describes two halfspace emptiness data structures, one answering queries in time O(n 1;1=bd=2c polylog n) time ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....within it the set of back facets is the same. Let A 0 be the arrangement obtained by intersecting these cones with SS 2 . A 0 is essentially an arrangement of great circles on SS 2 , and hence has size O(n 2 ) and can be computed in time O(n 2 ) using the algorithm given in [7]. Next using the algorithm of Bose et al. 5] we compute a second arrangement, A 00 , which consists of portions of great circles (i.e. great arcs) on SS 2 such that all directions within a region of A 00 have the same extreme vertex. The desired arrangement, A, is the intersection of A ....

B. Chazelle, L.J. Guibas, and D.T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....3.1 may not be improvable. Consider the following well known problem in computational geometry: Problem 3. 2 (COLLINEARITY) Given V points in the plane, is there a straight line which contains three or more of these points The fastest algorithm known for COLLINEARITY takes O(V 2 ) time [11] and despite much effort this bound has not been improved. Recent work by Gajentaan and Overmars [7] suggests that the problem may indeed have an Omega Gamma V 2 ) lower bound in a reasonable model of computation, such as a real RAM, i.e. a Random Access Machine capable of operating on real ....

B.M. Chazelle, L.J. Guibas, and D.T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....number of researchers; see [2, 28] for surveys. In the planar case, O(n 1=2 polylog n) query time is attainable after O(n log n) preprocessing time [26] On the other hand, results on halfspace range reporting imply an output sensitive query time of O(log n k) with the same preprocessing time [9]. We point out that combining previous approaches properly yields a query time bound of O(k 1=2 n ) for any constant 0 in this planar case. The precise output sensitive bounds in higher dimensions, including preprocessing query time tradeoffs, are given by Theorem 4.1. Although the proof ....

B. Chazelle, L. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....geometry has been devoted to this fundamental problem called halfspace range reporting , a special case of range searching [4, 34, 44, 48, 49] Here are some of the major known results. First, halfspace range reporting in the planar case (d = 2) was solved optimally by Chazelle, Guibas, and Lee [25]. Their data structure takes linear space and answers a query in O(log n k) time, where k is the number of reported points. The preprocessing can be accomplished in O(n log n) time using Chazelle s algorithm for convex layers [19] Unfortunately, the approach does not generalize to higher ....

....furthermore, the space complexity can be improved. Using advanced tools and a larger preprocessing time, Appendix A.3 gives a modification of our data structure that is deterministic and uses only O(n log log n) space. 2. It is worthwhile to compare Chazelle, Guibas, and Lee s optimal method [25] with the specialization of our method in two dimensions. Ours seems easier to implement as convex layers are not involved. 3. Higher dimensional extensions are possible, although we do not see any significant improvements. By a standard lifting map, Corollaries 2.4 and 2.5 imply a new method for ....

B. Chazelle, L. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....scheme needed here. 18 P. J. de Rezende and D. T. Lee 5.1 Affine Geometry and Geometric Duality In this section, we will very briefly discourse on the subject of geometric duality in preparation for studying circular sorting in high dimensions. For further details, the reader is referred to [4, 5, 6, 9]. If A 1 and A 2 are affine spaces (possibly of different dimensions) we write A 1 k A 2 to denote that A 1 and A 2 are parallel. Definition 2 If U ae R d is any subset, its affine span hUi is defined by hUi = k P i=1 a i u i fi fi fi u i 2 U; k P i=1 a i = 1 ) Denote R d n ....

B. M. Chazelle, L. J. Guibas and D. T. Lee, "The Power of Geometric Duality," BIT, 25 (1985) 76--90.

....overlay of two arrangements A area and A ext . Faces of A area represent directions for which the back facets are the same (Section 2.2) Faces of A ext represent directions for which the extreme vertex is the same. A area is an arrangement of great circles and can be computed in O(n 2 ) time [6], while A ext is an arrangement of great arcs and can also be computed in O(n 2 ) time [4] Their overlay can be computed in O(n 2 ) additional time [15] Next, we pick a face, R, of A vol and generate a formula for the support volume for any direction d = xi yj zk in R, which is of the ....

....) time. Proof Let a be any great arc of A or B and let G a be the great circle containing it. We add the G a s one by one to the initially empty arrangement. We compute the intersections of each G a with the current arrangement and retain only those that also belong to a. By the zone theorem [6], which is also applicable to great circles, due to central projection, this takes O(n) time per G a , hence O(n 2 ) time in all. Lemma 3.2 Let R be a convex polygon on SS 2 composed of great and or small arcs and let A be an arrangement of O(n) great arcs. Then the overlay, A 0 , of R and ....

[Article contains additional citation context not shown here]

B. Chazelle, L. J. Guibas and D. T. Lee. The power of geometric duality. BIT, vol. 25, pages 76--90, 1985.

....This observation not only simplifies the data structure but also gives better bounds in many cases, including halfspace range reporting. See [15, 55, 66, 79] for some applications of filtering search. An optimal halfspace reporting data structure in the plane was proposed by Chazelle et al. [78]. They compute convex layers L 1 ; Lm of S, where L i is the set of points lying on the boundary of the convex hull of S n S j i L j , and store them in a linear size data structure, so that a query can be answered in O(log n k) time. Their technique does not extend to three ....

....whether h S = we do not have to store simplex range searching structure at each node of the tree. Consequently, the query time and the size of the data structure can be improved slightly; see Table 3 for a summary of results. Problem d Size Query Time Source Reporting d = 2 n log n k [78] Emptiness n log n [242] Reporting d = 3 n log n log n k [16] Emptiness n log n [103] Reporting d 3 n log log n n 1 Gamma1=bd=2c polylog n k [199] Emptiness n n 1 Gamma1=bd=2c 2 O(log n) 199] Table 3. Asymptotic upper bounds for halfspace range searching in near linear space. ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

....number of researchers; see [2, 27] for surveys. In the planar case, O(n 1=2 polylog n) query time is attainable after O(n log n) preprocessing time [25] On the other hand, results on halfspace range reporting imply an output sensitive query time of O(log n k) with the same preprocessing time [7]. We point out that combining previous approaches properly yields a query time bound of O(k 1=2 n ) for any constant 0 in this planar case. The precise output sensitive bounds in higher dimensions, including preprocessing query time tradeoffs, are given by Theorem 4.1. Although the proof ....

B. Chazelle, L. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....incrementally. Consider the problem of computing the line arrangements in the plane. Given is a set L of n straight lines in the plane, and we want to compute the partition of the plane induced by L. One obvious approach is to compute the partition iteratively by considering one line at a time[41]. As shown in Fig. 1, when line i is inserted, we need to traverse the regions that are intersected by the line and construct the new partition at the same time. One can show that the traversal and re partitioning of the intersected regions can be done in O(n) time per insertion, resulting in a ....

....and the half space range searching problems that have been well studied. A simplex range in k is a range whose boundary is specifed by k 1 hyperplanes. In 2 dimensions it is a triangle. The report mode half space range searching problem in the plane is optimally solved by Chazelle et al.[41] in Q(n) O(log n F) time and S(n) O(n) space, using geometric duality transform. But this method does not generalize to higher dimensions. For k = 3, Chazelle and Preparata[43] obtained an optimal O(log n F) time algorithm using O(n log n) space. In [2] Agarwal and Matousek obtained a more ....

B. Chazelle and L. J. Guibas and D. T. Lee, "The Power of Geometric Duality," BIT, 25 (1985), 76--90.

....n f on SS 2 . Let g f be the great circle bounding h f . Then A 0 is the arrangement of g f s for all f 2 P; it has size O(n 2 ) and it can be computed in O(n 2 ) time by mapping the great circles to straight lines in the plane via central projection [23] and then using the algorithm in [8]. Next we construct the arrangement, A 00 as follows: Recall that the facets of P are all triangles. Consider a facet f 2 P. Let its vertices be u; v and w. For what directions is u the highest point of f Consider the direction vu. Let h vu be the hemisphere having vu as its pole. Then, u will ....

....these cones on SS 2 as the arrangement, A 0 , of the great circles h f SS 2 , i.e. each cone is in 1 1 correspondence with a region of A 0 . Note that A 0 is composed of arcs of great circles, has size O(n 2 ) and can be computed in time O(n 2 ) using the algorithm given in [8]. It is obtained in a canonical form, where the edges incident at each vertex are in sorted order around the vertex. This allows the boundary of each face of A to be retrieved in time linear in its size. Next using the algorithm of Bose et al. 6] we compute a second arrangement, A 00 , of ....

B. Chazelle, L.J. Guibas, and D.T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....STRUCTURES WITH LOGARITHMIC QUERY TIME For the sake of simplicity, we first consider the halfspace range counting problem. Using a standard duality transform, this problem can be reduced to the Range Searching 11 TABLE 4 Halfspace range searching. d S(n) Q(n) Source Notes d = 2 n log n k [41] Reporting d = 3 n log n log n k [14] Reporting d = 3 n log n [52] Emptiness d 3 n log log n n 1 Gamma1=bd=2c log c n [79] Reporting d 3 n n 1 Gamma1=d 2 O(log n) 79] Emptiness following problem: Given a set H of n hyperplanes, determine the number of hyperplanes of H lying ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90. 26 Pankaj K. Agarwal

....computation, was only Omega (n log n) 138, 16] In Chapter 2, we show that, in the worst case, Omega (n d ) sidedness queries are required to decide, given n points in IR d , whether any d 1 lie on a common hyperplane. Since there is an algorithm that solves this problem in time O(n d ) [39, 68, 69], our lower bound is tight. Our lower bound follows from an extremely simple adversary argument, based on the construction of a set of points in general position with Omega (n d ) collapsible simplices, any one of which can be made degenerate without changing the result of any other sidedness ....

....set of points is in general position. A simple example of this type of problem is determining, given a set of points in the plane, whether any three of them are colinear. In 1983, van Leeuwen [151] asked for an algorithm to solve this problem in time o(n 2 log n) Chazelle, Guibas, and Lee [39] and Edelsbrunner, O Rourke, and Seidel [68] independently discovered an algorithm that runs in time and space O(n 2 ) by constructing the arrangement of lines dual to the input points. 1 Edelsbrunner et al. 68] also solved the higher dimensional version of this problem, which we call the ....

[Article contains additional citation context not shown here]

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....1 ffl Gammafl ) time, where fl = 1= 1 (d Gamma 1)bd=2c) The algorithm is a variant of Haussler and Welzl s [28] Their query time is O(n 1 ffl Gammafl 0 ) where fl 0 = 1= 1 d(d Gamma 1) This is independent of the answer size, however. These results do not improve the algorithm of [7] for halfplane queries; that algorithm requires O(n) storage, O(n log n) preprocessing, and O(A log n) query time. See also [43, 9] for recent related results. 1.2 Outline of the paper The remainder of this section gives an informal discussion of the ideas in this paper. The next section ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....hull conv(f 1 ; n g) This shows that computing intersections of halfspaces and computing the convex hulls are in fact equivalent problems. Notice that we have just applied a form of duality (or polarity) when we map a halfspace fx 2 E d : Delta x 1g to a point 2 E d . Duality [CGL85, Ede87] is extremely important in computational geometry as it allows one to transform a problem involving points to a problem involving halfspaces hyperplanes and vice versa. Sometimes we may gain more insight into the geometry of a problem by examining the problem in both its primal and dual setting. ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....is entirely different for halfspace queries. The best known halfspace counting data structure requires roughly Omega (n d ) space to achieve logarithmic query time [10, 25] whereas, the same query time can be achieved Space Preprocessing Query Time Source d = 2; 3 O(n) O(n log n) O(log n) [13, 3, 16] d 4 O(n bd=2c = log bd=2c n) O(n bd=2c = log bd=2c n) O(log n) 27] O(n) O(n 1 ) O(n 1 1=bd=2c 2 O(log n) 22] O(n) O(n log n) O(n 1 1=bd=2c polylog n) 27] n s n bd=2c O(spolylog n) O( npolylog n) s 1=bd=2c ) 27] Table 2. Best known upper bounds for halfspace ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT 25:76--90, 1985.

....partition graphs, and the space time tradeoffs to arbitrary trim partition graphs, in all dimensions. Corollary 7.6 is always an improvement (although a small one) over the lower bound in Corollary 4.3. 24 Jeff Erickson Space Preprocessing Query Time Source d = 2; 3 O(n) O(n log n) O(log n) [20, 3, 25] d 4 O(n bd=2c = log bd=2c n) O(n bd=2c = log bd=2c n) O(log n) 42] O(n) O(n 1 ) O(n 1 1=bd=2c 2 O(log n) 37] O(n) O(n log n) O(n 1 1=bd=2c polylog n) 42] n s n bd=2c O(spolylog n) O( npolylog n) s 1=bd=2c ) 42] Table 2. Best known upper bounds for ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT 25:76--90, 1985.

....neighbors between lines and points: a) primal, nearest point neighbor to a line; b) dual, vertical ray shooting in a line arrangement. minimum area triangle with s is the nearest vertical neighbor of l. This observation was used to develop O(n 2 ) algorithms for the minimum triangle problem [7, 16, 15]. We can tighten this characterization as follows. Let 4pqr be the minimum area triangle, and assume that the vertical projection of r is between those of p and q. Then as before r is the nearest neighbor of line pq, but the vertical segment connecting r and line pq actually touches segment pq. ....

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

....i, let ( k) denote the number of boundary faces in the arrangement A( n fkg) that are intersected by k . Observe that the sum ( k) equals the total number of edges bounding the boundary faces of A( Each such edge lies in the zone (in A( of one of the edges of s. Hence, by the Zone Theorem [9, 15], A( k) O(i) Since i is chosen randomly from the set L , i can be any of the lines 1 ; 2 ; i with equal probability. Therefore, the expected value E of s is E = A( k) O(1) Hence, the total number of pieces created in the ith stage is O(n) Summing over ....

....jS Delta j) Thus, 4.1) implies that the total time spent in processing f is O(jS Delta j log jS Delta j) O(k f log k f ) Let Z be the set of all active faces of A(L ) that are intersected by i . The total time spent in processing i is O(k f log k f ) Using the Zone Theorem [9, 15] and the theory of random sampling [13, 17] we can show that the expected value of k f is O(n log n) which implies the following theorem: Theorem 4.2 Let S be a set of n non intersecting triangles . We can compute a BSP for S of expected size O(n in expected time O(n Remark: Using ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

....intersects the interior of only one cell in (f ) we obtain X 2 (f) jS j k f : 4.1) We now analyze the expected running time of the algorithm. We count the time spent during the ith stage in inserting the line i and then add this time over all stages of the algorithm. The zone theorem [11, 17] implies that in Step 1 of the algorithm, we spend O(i) time in tracing i through A(L i 1 ) While processing an active face f of A(L i 1 ) that intersects i , for each cell 2 (f ) we spend O(1) time in Step 1 and O(jS j) time in Step 2. In Step 3, for each triangle s 2 S n F ....

B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76-90.

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B. Chazelle, L. J. Guibas, and D. T. Lee, The power of geometric duality, BIT, 25 (1985), 76--90.

No context found.

B. Chazelle, L. J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:76--90, 1985.

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Bernard Chazelle, Leonidas J. Guibas, and D. T. Lee. The power of geometric duality. BIT, 25:7690, 1985.

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B. Chazelle, L. Guibas, and D. Lee. The Power of Geometric Duality. Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, pp. 217-225, 1983.

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B. Chazelle, L. Guibas, and D. Lee. The power of geometric duality. Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, pp. 217-225, 1983.

No context found.

B. Chazelle, L.J. Guibas, and D.T. Lee, The Power of Geometric Duality, BIT 25 (1985), 76--90.

No context found.

B. Chazelle, L. Guibas, and D. T. Lee. The power of geometric duality. BIT 25 (1985) 76--90.

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