J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389--418, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....during the motion) Clearly there can be a near quadratic number of combinatorial changes to the configuration function. But a local and compact structure may not be possible. The standard many faces construction [25] as well as Erickson s lower bounds on the static version of this problem [27], suggest that a natural certificate set size for this problem is Theta(n 4=3 ) By using ffl net techniques we can give a kinetic data structure that processes a near quadratic number of events and has a proof size of O(n 5=3 ) 4] It would be quite interesting to do better. Several ....

J. Erickson. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 127--137, 1995.

....hits. Using using the best known unidirectional ray shooting data structure, due to Agarwal and Sharir [3] we obtain an algorithm with running time O##n#n ### p ### #p# log # n#, where n is the number of triangles and p is the number of pixels. Erickson s lower bound for Hopcroft s problem [18] suggests that this algorithm is close to optimal in the worst case, even for the simpler problem of deciding whether any ray hits a triangle. In practice, rayshooting queries are answered by walking through a decomposition of space determined by the triangles, such as an octtree [21] ....

J. Erickson. New lower bounds for Hopcroft's problem. ######## ####### ##### 16:389-418, 1996.

....situation, e.g. the set of almost unit distances among n points. The problem is also very similar to Hopcroft s problem, which asks to determine the existence of at least one point hyperplane incidence, whereas we wish to nd all. Especially interesting in this connection is the model of Erickson [7] for lower bounds in a class of algorithms for Hopcroft s problem. There he also constructs systems of points and lines with a complicated incidence structure which cannot be coded eciently: his function d (n; m) is the maximum over all sets of n points, m hyperplanes of the size of the ....

J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389-418, 1996.

....i.e. points, in this case) queries are points, and the answer to a query is whether or not it is contained in the database. The problem can be solved in O(1) probes per query via hashing. Our lower bounds show that a similar result for the multi dimensional generalization is impossible. Erickson [21] proves lower bounds on the related Hopcroft s problem of deciding for a set of points and a set of hyperplanes whether or not there is a point that lies on one of the hyperplanes. This is not considered as a data structure problem, and the bounds are on the computation time as a function of the ....

J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389--418, 1996.

....graph that describes a recursive decomposition of space. Our model is powerful enough to describe most, if not all 1 , known data structures for these problems. Partition graphs have been previously used to study offline range searching problems such as Hopcroft s point line incidence problem [19, 18]. We summarize our results below. In each of these results and throughout the paper, s denotes space, p denotes preprocessing time, and t denotes worst case query time. For comparison, the best known upper bounds are listed in Table 1. For a review of range searching techniques and results, see ....

....Thus, results of Bronnimann, Chazelle, and Pach [6] immediately imply that st d = Omega ( n= log n) d (d 1) d 1) This lower bound applies with high probability to a randomly generated set of points. ffl Lower bounds on the complexity of Hopcroft s point hyperplane incidence problem [19, 18] imply the worst case bounds pt (d 2) d 1) 2 = Omega (n d ) and pt 2= d 1) Omega (n (d 2) d ) These lower bounds match known upper bounds up to polylogarithmic factors when d = 2, p = O(npolylogn) or t = O(polylog n) 21, 25] These results require restrictions on the partition ....

[Article contains additional citation context not shown here]

J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom. 16:389--418, 1996.

....i.e. points, in this case) queries are points, and the answer to a query is whether or not it is contained in the database. The problem can be solved in O(1) probes per query via hashing. Our lower bounds show that a similar result for the multidimensional generalization is impossible. Erickson [20] proves lower bounds on the related Hopcroft s problem of deciding for a set of points and a set of hyperplanes whether or not there is a point that lies on one of the hyperplanes. This is not considered as a data structure problem, and the bounds are on the computation time as a function of the ....

J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389--418, 1996.

....to check the distance to each corresponding site. One would not expect that segment queries would be as efficient as point queries. In fact, one could use segment queries to solve Hopcroft s problem: determine whether any point from a given set of n points lies on any of n given lines. Erickson [10] has shown that any algorithm that can be implemented in a computational model based on partition trees, which includes our algorithms, must take Omega Gamma n 4=3 ) time to solve Hopcroft s problem. We found it surprising, therefore, that the nearest foreign neighbor problem for disjoint ....

....may intersect. We do not into precise detail because the technology is more standard and the running times are asymptotically slower. As mentioned in the introduction, the general problem can be used to solve Hopcroft s problem: given n points and n lines, does any point lie on any line. Erickson [10] has shown that any algorithm that can be implemented in a computational model based on partition trees, which includes our algorithms, must take Omega Gamma n 4=3 ) time to solve Hopcroft s problem. When the points are given in advance and the query segments are given on line, then we can ....

Jeff Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389--418, 1996.

....a set of n points and a set of n lines; the fastest known algorithm for Hopcroft s problem is due to Matousek [15, 24] and runs in O(n 4=3 2 O(log n) time. For a certain general class of algorithms, a lower bound of Omega (n 4=3 ) for this problem was recently given by Erickson [19]. Given this situation, we were motivated to seek additional conditions which make easier the task of reporting the purple intersections. A natural condition is that of connectedness for each of the monochromatic inputs, as introduced above. This condition often pertains in situations where the ....

J. Erickson. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 127--137, 1995.

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J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom. 16:389--418, 1996.

....is at least as hard as Hopcroft s problem: given a set of points and lines in the plane, does any point lie on a line The main idea of the reduction is to replace each point and line with an arbitrarily thin spike. In light of this reduction and Erickson s# ) lower bound for Hopcroft s problem [12], an algorithm that detects intersections in o(n ) worst case time appears unlikely. In practice, one of the most popular techniques for intersection detection uses a hierarchy of bounding volumes. For a given placement of two disjoint polyhedra, the algorithms refine their hierarchies only to ....

J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom. 16:389--418, 1996.

....hits. Using using the best known unidirectional ray shooting data structure, due to Agarwal and Sharir [3] we obtain an algorithm with running time O( n n 2=3 p 2=3 p) log 3 n) where n is the number of triangles and p is the number of pixels. Erickson s lower bound for Hopcroft s problem [18] suggests that this algorithm is close to optimal in the worst case, even for the simpler problem of deciding whether any ray hits a triangle. In practice, rayshooting queries are answered by walking through a decomposition of space determined by the triangles, such as an octtree [21] ....

J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom. 16:389-418, 1996.

....functions are considered in [138] Neither the semigroup arithmetic model nor the pointer machine model can be used to study the complexity of emptiness queries. A few results on hyperplane and halfspace emptiness queries are presented in the partition graph model, recently developed by Erickson [113, 114, 115]. Informally, a partition graph describes a recursive decomposition of both primal and dual space into connected subsets. See Section 4.2 for a discussion on duality. A partition graph is a directed acyclic graph with constant outdegree, with a single source, called the root, and several ....

....for online and offline emptiness query problems, in the partition graph model of computation. His techniques were first applied to Hopcroft s problem Given a set of n points and m lines, does any point lie on a line for which he obtained a lower bound of Omega Gamma n 2=3 m 2=3 ) [114], almost matching the best known upper bound O(n 2=3 m 2=3 2 O(log (n m) due to Matou sek [183] Slightly better lower bounds are known for higher dimensional versions of Hopcroft s problem [114, 113] but for the special case n = m, the best known lower bound is still ....

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J. Erickson, New lower bounds for Hopcroft's problem, Discrete Comput. Geom., 16 (1996), 389--418.

....The halfspace emptiness problem asks, given a set of points and a set of halfspaces, whether any halfspace contains a point. In this paper, we derivenewlower bounds for the time required to solve this problem, generalizing earlier lower bounds for Hopcroft s pointline incidence problem [16]. In this paper, we will consider the following formulation of the problem: Given a set of points and hyperplanes, is every pointaboveevery hyperplane Using linear programming [13, 21, 24, 25] we can decide in linear time whether the union of a set of halfspaces is IR d . If it is, then every ....

....of the halfspace emptiness problem in IR 5 , matching known upper bounds up to polylogarithmic factors. We obtain marginally larger bounds in dimensions 9 and higher. Our lower bounds apply to polyhedral partitioning algorithms, a restriction of the class of partitioning algorithms introduced in [16]. Informally, a polyhedral partitioning algorithm covers space with a constant number of constant complexity polyhedra, determines whichpoints and halfspaces intersect each polyhedron, and recursively solves the resulting subproblems. The basic approach is the same as the one used to provelower ....

[Article contains additional citation context not shown here]

J. Erickson. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 127--137, 1995.

....several regions with nice properties and recursively constructing a data structure for each region. Range queries are answered with such a data structure by performing a depth first search through the resulting recursive space partition. The partition graph model, recently introduced by Erickson [117, 118, 119], formalizes this divide and conquer approach, at least for hyperplane and halfspace range searching data structures. The partition graph model can be used to study the complexity of emptiness queries, unlike the semigroup arithmetic and pointer machine models, in which such queries are trivial. ....

....offline emptiness query problems, in the partition graph model of computation. His techniques were first applied to Hopcroft s problem Given a set of n points and m lines, does any point lie on a line for which he obtained a lower bound of Omega Gamma n log m n 2=3 m 2=3 m log n) [118], almost matching the best known upper bound O(n log m n 2=3 m 2=3 2 O(log (n m) m log n) due to Matousek [203] Slightly better lower bounds are known for higher dimensional versions of Hopcroft s problem [118, 117] but for the special case n = m, the best known lower bound is ....

[Article contains additional citation context not shown here]

J. Erickson, New lower bounds for Hopcroft's problem, Discrete Comput. Geom., 16 (1996), 389--418.

.... to within polylogarithmic factors [3] and improves the previous best lower bound of Omega Gamma n log m m log n) 1] Our lower bounds apply to polyhedral partitioning algorithms, a restriction of the class of partitioning algorithms introduced to study the complexity of Hopcroft s problem [2]. Informally, a polyhedral partitioning algorithm covers space with a constant number of constant complexity polyhedra (typically simplices or combinatorial cubes) determines which points and which halfspaces intersect each polyhedron, and recursively solves the resulting subproblems. All known ....

....restriction to constantcomplexity polyhedra is necessary for the lower bound to hold, since the halfspace emptiness problem can be solved in linear time in the unrestricted partitioning algorithm model. To prove the new lower bound, we use almost exactly the same approach as for Hopcroft s problem [2]. Let P be a set of points and H a set of This research was done while the author was a student in the Computer Science Division at U. C. Berkeley and was partially supported by a GAANN fellowship. An extended abstract of this paper will appear in the proceedings of the 37th FOCS. The paper is ....

J. Erickson. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 127--137, 1995.

....most, if not all, known hyperplane range searching data structures. 2 Partition graphs were originally introduced to study the complexity of Hopcroft s problem Given a set of points and hyperplanes, does any hyperplane contain a point and similar offline geometric searching problems [29]. We summarize our results below. In each of these results and throughout the paper, s denotes space, p denotes preprocessing time, and t denotes worst case query time. For comparison, the best known upper bounds are listed in Table 1. For a thorough overview of range searching techniques, ....

....log n) d (d 1) d 1) This lower bound applies with high probability to a randomly generated set of points. This is the first nontrivial lower bound for hyperplane emptiness queries in any model of computation. ffl We generalize earlier lower bounds on the complexity of Hopcroft s problem [29] for the special case of polyhedral partition graphs. Specifically, we prove that in the worst case, the time to preprocess n points in IR d and perform k hyperplane emptiness queries is Omega (n log k n 1 2=d(d 1) k 2= d 1) n 2= d 1) k 1 2=d(d 1) k log n) even if the query ....

[Article contains additional citation context not shown here]

J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom. 16:389--418, 1996.

....The halfspace emptiness problem asks, given a set of points and a set of halfspaces, whether any halfspace contains a point. In this paper, we derive new lower bounds for the time required to solve this problem, generalizing earlier lower bounds for Hopcroft s pointline incidence problem [16]. In this paper, we will consider the following formulation of the problem: Given a set of points and hyperplanes, is every point above every hyperplane Using linear programming [13, 21, 24, 25] we can decide in linear time whether the union of a set of halfspaces is IR d . If it is, then ....

....of the halfspace emptiness problem in IR 5 , matching known upper bounds up to polylogarithmic factors. We obtain marginally larger bounds in dimensions 9 and higher. Our lower bounds apply to polyhedral partitioning algorithms, a restriction of the class of partitioning algorithms introduced in [16]. Informally, a polyhedral partitioning algorithm covers space with a constant number of constant complexity polyhedra, determines which points and halfspaces intersect each polyhedron, and recursively solves the resulting subproblems. The basic approach is the same as the one used to prove lower ....

[Article contains additional citation context not shown here]

J. Erickson. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 127--137, 1995.

....and Cook reduction in the theory of NP completeness [30] On the Relative Complexities of Some Geometric Problems 3 compute the answer. While this model works quite well for studying range counting problems, it is not at all applicable to decision or optimization problems. Similarly, Erickson [24] has proven Omega Gamma n 4=3 ) lower bounds for a number of problems, including Hopcroft s problem (Problem A) and unit distance detection (Problem G) in what he calls the partitioning algorithm model. Informally, a partitioning algorithm splits the plane up into a constant number of regions, ....

....points are on each line by building a general purpose range searching data structure over the points and querying it once for each line, must take Omega Gamma n 4=3 ) time. More recent results of Chazelle [14] imply the same lower bound for the offline counting version. Finally, Erickson [24] has shown a lower bound of Omega Gamma n 4=3 ) for any partitioning algorithm that solves the original decision problem. Hopcroft s problem is a special case of a large number of other more general problems, including the following. We leave the reductions as easy exercises for the reader. ....

[Article contains additional citation context not shown here]

J. Erickson. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 127--137, 1995.

....proven a number of lower bounds for online and offline range counting problems in the Fredman Yao semigroup arithmetic model [16] While this model works quite well for studying this sort of counting problem, it is not at all applicable to decision or optimization problems. Similarly, Erickson [15] has proven Omega Gamma n 4=3 ) lower bounds for a number of problems, including Hopcroft s problem (Problem A) and unit distance detection (Problem F) in what he calls the partitioning algorithm model. This model is specifically tailored towards a specific class of incidence detection ....

....Problems Problem F. Unit Distance Detection: Given a set of points in the plane, is any pair of points at unit distance Best known upper bound: O(n 4=3 log 2 n) 18] Erickson s lower bound for the unit distance problem follows immediately from his lower bound proof for Hopcroft s problem [15]. Nevertheless, we are unable to show that detecting unit distances is harder, or easier, than detecting point line incidences, nor are we able to show that both are harder than some third simpler problem. Both problems are special cases of several other problems, such as point circle incidence ....

J. Erickson. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom,, 1995. To appear.

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J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389--418, 1996.

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J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389-418, 1996.

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J. Erickson. New lower bounds for Hopcroft's problem. Discrete Comput. Geom., 16:389-418, 1996.

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Erickson, J. New lower bounds for Hopcroft's problem. In Proc. 11th Annu. ACM Sympos. Comput. Geom. (1995), pp. 127-137. 24

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