A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment

We consider the problem of planning the motion of an arbitraryk-sided polygonal robotB, free to translate and rotate in a polygonal environmentV bounded byn edges. We present an algorithm that constructs a single component of the free configuration space ofB in timeO((kn) ^2+ɛ), for any ɛ>0. This algorithm, combined with some standard techniques in motion planning, yields a solution to the underlying motion-planning problem, within the same running time. Work on this paper by Dan Halperin has been supported by a Rothschild Postdoctoral Fellowship, by a grant from the Stanford Integrated Manufacturing Association (SIMA), by NSF/ARPA Grant IRI-9306544, and by NSF Grant CCR-9215219. Work on this paper by Micha Sharir has been supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary and extended version of the paper has appeared as: D. Halperin and M. Sharir, Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment,Proc. 34th IEEE Symp. on Foundations of Computer Science, 1993, pp. 382–391. Part of the work on the paper was carried out while Dan Halperin was at Tel Aviv University.     

Almost tight upper bounds for the single cell and zone problems in three dimensions

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement ofn low-degree algebraic surface patches in 3-space. We show that this complexity isO(n ^2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries. This extends several previous results, almost settles a 9-year-old open problem, and has applications to motion planning of general robot systems with three degrees of freedom. As a corollary of the above result, we show that the overall complexity of all the three-dimensional cells of an arrangement ofn low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-calledzone of σ in the arrangement) isO(n ^2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries. Work on this paper by the first author has been supported by a Rothschild Postdoctoral Fellowship, by a grant from the Stanford Integrated Manufacturing Association (SIMA), by NSF/ARPA Grant IRI-9306544, and by NSF Grant CCR-9215219. Work on this paper by the second author has been supported by NSF Grants CCR-91-22103 and CCR-93-111327, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the Israel Science Fund administered by the Israeli Academy of Sciences.     

Euclidean minimum spanning trees and bichromatic closest pairs

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inE ^d in timeO(F _d (N,N) log ^d N), whereF _d (n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inE ^d . IfF _d (N,N)=Ω(N ^1+ε), for some fixed ɛ>0, then the running time improves toO(F _d (N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)^2/3+m log² n+n log² m) inE ₃, which yields anO(N ^4/3 log^4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE ³. Ind≥4 dimensions we obtain expected timeO((nm)^{1−1/([d/2]+1)+ε}+m logn+n logm) for the bichromatic closest pair problem andO(N ^{2−2/([d/2]+1)ε}) for the Euclidean minimum spanning tree problem, for any positive ɛ. The first, second, and fourth authors acknowledge support from the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center under NSF Grant STC 88-09648. The second author's work was supported by the National Science Foundation under Grant CCR-8714565. The third author's work was supported by the Deutsche Forschungsgemeinschaft under Grant A1 253/1-3, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”. The last two authors' work was also partially supported by the ESPRIT II Basic Research Action of the EC under Contract No. 3075 (project ALCOM).     

Vertical decomposition of arrangements of hyperplanes in four dimensions

We show that, for any collection ℋ ofn hyperplanes in ℜ⁴, the combinatorial complexity of thevertical decomposition of the arrangementA(ℋ) of ℋ isO(n ⁴ logn). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them. Work on this paper by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Leonidas Guibas was also supported by National Science Foundation Grant CCR-9215219. Work by Micha Sharir was also supported by National Science Foundation Grant CCR-91-22103, and by grants from the G.I.F.—the German Isreali Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences. Work by Jiří Matouŝek was done while he was visiting Tel Aviv University, and its was partially supported by a Humboldt Research Fellowship. Work on this paper by Dan Halperin was carried out while he was at Tel Aviv University.     

Points and triangles in the plane and halving planes in space

We prove that for any setS ofn points in the plane andn ^3−α triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn ^3−3α/(c log⁵ n) of the triangles. This implies that any set ofn points in three-dimensional space defines at most halving planes. Work on this paper by Boris Aronov and Rephael Wenger has been supported by DIMACS under NSF Grant STC-88-09648. Work on this paper by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-87-14565. Micha Sharir has been supported by ONR Grant N00014-87-K-0129, by NSF Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Israeli National Council for Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

On the minimality of polygon triangulation

The problem of triangulating a polygon using the minimum number of triangles is treated. We show that the minimum number of triangles required to partition a simplen-gon is equal ton+2w –d – 2, wherew is the number of holes andd is the maximum number of independent degenerate triangles of then-gon. We also propose an algorithm for constructing the minimum triangulation of a simple hole-freen-gon. The algorithm takesO(nlog² n+DK ²) time, whereD is the maximum number of vertices lying on the same line in then-gon andK is the number of minimally degenerate triangles of then-gon.     

A new graph triconnectivity algorithm and its parallelization

We present a new algorithm for finding the triconnected components of an undirected graph. The algorithm is based on a method of searching graphs called open ear decomposition. A parallel implementation of the algorithm on a CRCW PRAM runs inO(log² n) parallel time usingO(n+m) processors, wheren is the number of vertices andm is the number of edges in the graph.A preliminary version of this paper was presented at the19th Annual ACM Symposium on Theory of Computing, New York, NY, May 1987.Supported by NSF Grant DCR 8514961.Supported by NSF Grant ECS 8404866 and the Semiconductor Research Corporation Grant 86-12-109.     

An optimal convex hull algorithm in any fixed dimension

We present a deterministic algorithm for computing the convex hull ofn points inE ^d in optimalO(n logn+n ^⌞d/2⌟ ) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n ^⌜d/2⌝ ) time. This research was supported in part by the National Science Foundation under Grant CCR-9002352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. A preliminary version of this paper has appeared in “An optimal convex hull algorithm and new results on cuttings”,Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science, October 1991, pp. 29–38. The convex hull algorithm given in the present paper, although similar in spirit, is considerably simpler than the one given in the proceedings.     

Diameter,width, closest line pair,and parametric searching

We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improved solutions for them. We obtain, for any fixed ε>0, anO(n ^1+ε) algorithm for computing the diameter of a point set in 3-space, anO(^8/5+ε) algorithm for computing the width of such a set, and onO(n ^8/5+ε) algorithm for computing the closest pair in a set ofn lines in space. All these algorithms are deterministic. Work by Bernard Chazelle was supported by NSF Grant CCR-90-02352. Work by Herbert Edelsbrunner was supported by NSF Grant CCR-89-21421. Work by Leonidas Guibas and Micha Sharir was supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Micha Sharir was also supported by ONR Grant N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

A generalized hypergreedy algorithm for weighted perfect matching

We give a generalization of the hypergreedy algorithm for minimum weight perfect matching on a complete edge weighted graph whose weights satisfy the triangle inequality. With a modified version of this algorithm we obtain a logn-approximate perfect matching heuristic for points in the Euclidean plane, inO(n log² n) time.This research was supported in part by the DIMACS Grant No. NSF-STC88-09648.This research was supported in part by the NSF under Grant No. CCR 88-07518.     

Improved bounds on weak ε-nets for convex sets

LetS be a set ofn points in ℝ ^d . A setW is aweak ε-net for (convex ranges of)S if, for anyT⊆S containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size , whereβ ₂=0,β ₃=1, andβ _d ≈0.149·2 ^d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log^1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas. Work by Bernard Chazelle has been supported by NSF Grant CCR-90-02352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Michelangelo Grigni has been supported by NSERC Operating Grants and NSF Grant DMS-9206251. Work by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sharir has been supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by NSF Grant CCR-91-22103, and by a grant from the Fund for Basic Research administered by the Israeli Academy of Sciences.     

l p n Superspaces of spans of independent random variables

We show that for 1 ≦p < ∞,p ≠ 2, ifɛ > 0 is small enough andX ≦L _p is the span ofn independent Rademacher functions orn independent Gaussian random variables, then any superspaceY ofX satisfyingd(Y,L _p ^m ) ≦ 1 +ɛ has dimension larger thanr ⁿ, wherer =r(ɛ, p) > 1. This forms part of the author’s doctoral dissertation prepared at Texas A&M University under the direction of Professor W. B. Johnson. Supported in part by NSF DMS-85 00764.     

Arrangements of segments that share endpoints: Single face results

We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement ofn line segments determined byh endpoints isO(h logh). While the previous upper bound,O(nα(n)), is tight for segments with distinct endpoints, it is far from being optimal whenn=Ω(h ²). Our results show that, in a sense, the fundamental combinatorial complexity of a face arises not as a result of the number ofsegments, but rather as a result of the number ofendpoints. The research of E. M. Arkin was partially supported by NSF Grants ECSE-8857642 and CCR-9204585. K. Kedem's research was partially supported by AFOSR Grant AFOSR-91-0328. The research of J. S. B. Mitchell was partially supported by a grant from Boeing Computer Services, Hughes Research Laboratories, AFOSR Grant AFOSR-91-0328, and by NSF Grants ECSE-8857642 and CCR-9204585.     

A random 1-011-011-01algorithm for depth first search

In this paper we present a fast parallel algorithm for constructing a depth first search tree for an undirected graph. The algorithm is anRNC algorithm, meaning that it is a probabilistic algorithm that runs in polylog time using a polynomial number of processors on aP-RAM. The run time of the algorithm isO(T _MM(n) log³ n), and the number of processors used isP _MM (n) whereT _MM(n) andP _MM(n) are the time and number of processors needed to find a minimum weight perfect matching on ann vertex graph with maximum edge weightn.This research was done while the first author was visiting the Mathematical Research Institute in Berkeley. Research supported in part by NSF grant 8120790.Supported by Air Force Grant AFOSR-85-0203A.     

A capacity-rounding algorithm for the minimum-cost circulation problem: A dual framework of the Tardos algorithm

Recently, É. Tardos gave a strongly polynomial algorithm for the minimum-cost circulation problem and solved the open problem posed in 1972 by J. Edmonds and R.M. Karp. Her algorithm runs in O(m ² T(m, n) logm) time, wherem is the number of arcs,n is the number of vertices, andT(m, n) is the time required for solving a maximum flow problem in a network withm arcs andn vertices. In the present paper, taking an approach that is a dual of Tardos's, we also give a strongly polynomial algorithm for the minimum-cost circulation problem. Our algorithm runs in O(m ² S(m, n) logm) time and reduces the computational complexity, whereS(m, n) is the time required for solving a shortest path problem with a fixed origin in a network withm arcs,n vertices, and a nonnegative arc length function. The complexity is the same as that of Orlin's algorithm, recently developed by efficiently implementing the Edmonds-Karp scaling algorithm.     

Polynomial dual network simplex algorithms

We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m ² logn) bound on the number of pivots, wheren andm denotes the number of nodes and arcs in the input network. If the demands are integral and at mostB, we also give an O(m(m+n logn) min(lognB, m logn))-time implementation of a strategy that requires somewhat more pivots.Research supported by AFOSR-88-0088 through the Air Force Office of Scientific Research, by NSF grant DOM-8921835 and by grants from Prime Computer Corporation and UPS.Research supported by NSF Research Initiation Award CCR-900-8226, by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by ONR Contract N00014-88-K-0166.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

On the complexity of a single cell in certain arrangements of surfaces related to motion planning

We obtain near-quadratic upper bounds on the maximum combinatorial complexity of a single cell in certain arrangements ofn surfaces in 3-space where the lower bound for this quantity is Ω(n ²) or slightly larger. We prove a theorem that identifies a collection of topological and combinatorial conditions for a set of surface patches in space, which make the complexity of a single cell in an arrangement induced by these surface patches near-quadratic. We apply this result to arrangements related to motion-planning problems of two types of robot systems with three degrees of freedom and also to a special type of arrangements of triangles in space. The complexity of the entire arrangement in each case that we study can be Θ(n ³) in the worst case, and our single-cell bounds are of the formO(n ² α(n)), O(n ² logn), orO(n ² α(n)logn). The only previously known similar bounds are for the considerably simpler arrangements of planes or of spheres in space, where the bounds are Θ(n) and Θ(n ²), respectively. For some of the arrangements that we study we derive near-quadratic-time algorithms to compute a single cell. A preliminary version of this paper has appeared inProc. 7th ACM Symposium on Computational Geometry, North Conway, NH, 1991, pp. 314–323.     

A new pivot selection rule for the network simplex algorithm

We present a new network simplex pivot selection rule, which we call theminimum ratio pivot rule, and analyze the worst-case complexity of the resulting network simplex algorithm. We consider networks withn nodes,m arcs, integral arc capacities and integral supplies/demands of nodes. We define a {0, 1}-valued penalty for each arc of the network. The minimum ratio pivot rule is to select that eligible arc as the entering arc whose addition to the basis creates a cycle with the minimum cost-to-penalty ratio. We show that the so-defined primal network simplex algorithm solves minimum cost flow problem within O(nΔ) pivots and in O(Δ(m + n logn)) time, whereΔ is any upper bound on the sum of all arc flows in every feasible flow. For assignment and shortest path problems, our algorithm runs in O(n ²) pivots and O(nm +n ² logn) time.     

A parallel algorithm for generating multiple ordering spanning trees in undirected weighted graphs

1.IntroductionLetG=(V,E,W)beaconnected,weightedandundirectedgraph,VeEE,w(e)(相似文献