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86
Randomized routing and sorting on fixedconnection networks
 Journal of Algorithms
, 1994
"... This paper presents a general paradigm for the design of packet routing algorithms for fixedconnection networks. Its basis is a randomized online algorithm for scheduling any set of N packets whose paths have congestion c on any boundeddegree leveled network with depth L in O(c + L + log N) steps ..."
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Cited by 88 (13 self)
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This paper presents a general paradigm for the design of packet routing algorithms for fixedconnection networks. Its basis is a randomized online algorithm for scheduling any set of N packets whose paths have congestion c on any boundeddegree leveled network with depth L in O(c + L + log N) steps, using constantsize queues. In this paradigm, the design of a routing algorithm is broken into three parts: (1) showing that the underlying network can emulate a leveled network, (2) designing a path selection strategy for the leveled network, and (3) applying the scheduling algorithm. This strategy yields randomized algorithms for routing and sorting in time proportional to the diameter for meshes, butterflies, shuffleexchange graphs, multidimensional arrays, and hypercubes. It also leads to the construction of an areauniversal network: an Nnode network with area Θ(N) that can simulate any other network of area O(N) with slowdown O(log N).
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 78 (22 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
VLSI cell placement techniques
 ACM Computing Surveys
, 1991
"... VLSI cell placement problem is known to be NP complete. A wide repertoire of heuristic algorithms exists in the literature for efficiently arranging the logic cells on a VLSI chip. The objective of this paper is to present a comprehensive survey of the various cell placement techniques, with emphasi ..."
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Cited by 75 (0 self)
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VLSI cell placement problem is known to be NP complete. A wide repertoire of heuristic algorithms exists in the literature for efficiently arranging the logic cells on a VLSI chip. The objective of this paper is to present a comprehensive survey of the various cell placement techniques, with emphasis on standard ce11and macro
Proofs from the Book
, 1998
"... Paul Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdős also said that you need not believe in God but, as a mathematician, you should believe in The ..."
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Cited by 57 (1 self)
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Paul Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdős also said that you need not believe in God but, as a mathematician, you should believe in The Book. A few years ago, we suggested to him to write up a first (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, filling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdős ’ 85th birthday. With Paul’s unfortunate death in the summer of 1997, he is not listed as a coauthor. Instead this book is dedicated to his memory. We have no definition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations. We also hope that our readers will
Randomized Routing on FatTrees
 Advances in Computing Research
, 1996
"... Fattrees are a class of routing networks for hardwareefficient parallel computation. This paper presents a randomized algorithm for routing messages on a fattree. The quality of the algorithm is measured in terms of the load factor of a set of messages to be routed, which is a lower bound on the ..."
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Cited by 51 (11 self)
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Fattrees are a class of routing networks for hardwareefficient parallel computation. This paper presents a randomized algorithm for routing messages on a fattree. The quality of the algorithm is measured in terms of the load factor of a set of messages to be routed, which is a lower bound on the time required to deliver the messages. We show that if a set of messages has load factor on a fattree with n processors, the number of delivery cycles (routing attempts) that the algorithm requires is O(+lg n lg lg n) with probability 1 \Gamma O(1=n). The best previous bound was O( lg n) for the offline problem in which the set of messages is known in advance. In the context of a VLSI model that equates hardware cost with physical volume, the routing algorithm can be used to demonstrate that fattrees are universal routing networks. Specifically, we prove that any routing network can be efficiently simulated by a fattree of comparable hardware cost. 1 Introduction Fattrees constitute...
Embedding Planar Graphs at Fixed Vertex Locations
, 1999
"... Let G be a planar graph of n vertices, v1 ; : : : ; vn , and let fp1 ; : : : ; png be a set of n points in the plane. We present an algorithm for constructing in O(n ) time a planar embedding of G, where vertex v i is represented by point p i and each edge is represented by a polygonal curve ..."
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Cited by 30 (0 self)
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Let G be a planar graph of n vertices, v1 ; : : : ; vn , and let fp1 ; : : : ; png be a set of n points in the plane. We present an algorithm for constructing in O(n ) time a planar embedding of G, where vertex v i is represented by point p i and each edge is represented by a polygonal curve with O(n) bends (internal vertices.) This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m=20 edges represented by curves with at least m=40 bends.
On VLSI Layouts Of The Star Graph And Related Networks
, 1994
"... . We prove that the minimal VLSI layout of the arrangement graph A(n; k) occupies \Theta(n!=(n \Gamma k \Gamma 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area \Theta(n!) 2 : This answers an open problem posed by Akers, Harel and Krishnamurthy [1]. ..."
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Cited by 18 (3 self)
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. We prove that the minimal VLSI layout of the arrangement graph A(n; k) occupies \Theta(n!=(n \Gamma k \Gamma 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area \Theta(n!) 2 : This answers an open problem posed by Akers, Harel and Krishnamurthy [1]. The method is also applied to the pancake graph. The results provide optimal upper and lower bounds for crossing numbers of the above graphs. Key Words: area, arrangement graph, congestion, crossing number, embedding, layout, pancake graph, star graph, VLSI 1
On Bipartite Drawings and the Linear Arrangement Problem
"... The bipartite crossing number problem is studied, and a connection between this problemand the linear arrangement problem is established. It is shown that when the arboricity is close ..."
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Cited by 17 (0 self)
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The bipartite crossing number problem is studied, and a connection between this problemand the linear arrangement problem is established. It is shown that when the arboricity is close
Distinct Distances in the Plane
, 2001
"... It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1. ..."
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Cited by 16 (0 self)
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It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1.