Most algorithms of computational geometry are designed for the Real-RAM and non-degenerate inputs... more Most algorithms of computational geometry are designed for the Real-RAM and non-degenerate inputs. We call such algorithms idealistic. Executing an idealistic algorithm with floating point arithmetic may fail. Controlled perturbation replaces an input x by a random nearby x in the δ-neighborhood of x and then runs the floating point version of the idealistic algorithm on x. The hope is that this will produce the correct result for x with constant probability provided that δ is small and the precision L of the floating point system is large enough. We turn this hope into a theorem for a large class of geometric algorithms and describe a general methodology for deriving a relation between δ and L. We exemplify the usefulness of the methodology by examples.
In this paper, we present two results on slime mold computations. The first one treats a biologic... more In this paper, we present two results on slime mold computations. The first one treats a biologically-grounded model, originally proposed by biologists analyzing the behavior of the slime mold Physarum polycephalum. This primitive organism was empirically shown by Nakagaki et al. to solve shortest path problems in wet-lab experiments (Nature'00). We show that the proposed simple mathematical model actually generalizes to a much wider class of problems, namely undirected linear programs with a non-negative cost vector. For our second result, we consider the discretization of a biologically-inspired model. This model is a directed variant of the biologically-grounded one and was never claimed to describe the behavior of a biological system. Straszak and Vishnoi showed that it can ε-approximately solve flow problems (SODA'16) and even general linear programs with positive cost vector (ITCS'16) within a finite number of steps. We give a refined convergence analysis that improves the dependence on ε from polynomial to logarithmic and simultaneously allows to choose a step size that is independent of ε. Furthermore, we show that the dynamics can be initialized with a more general set of (infeasible) starting points.
We introduce the slime mold graph repository or SMGR, a novel data collection promoting the visib... more We introduce the slime mold graph repository or SMGR, a novel data collection promoting the visibility, accessibility and reuse of experimental data revolving around network-forming slime molds. By making data readily available to researchers across multiple disciplines, the SMGR promotes novel research as well as the reproduction of original results. While SMGR data may take various forms, we stress the importance of graph representations of slime mold networks due to their ease of handling and their large potential for reuse. Data added to the SMGR stands to gain impact beyond initial publications or even beyond its domain of origin. We initiate the SMGR with the comprehensive Kist Europe data set focusing on the slime mold Physarum polycephalum, which we obtained in the course of our original research. It contains sequences of images documenting growth and network formation of the organism under constant conditions. Suitable image sequences depicting the typical P. polycephalum network structures are used to compute sequences of graphs faithfully capturing them. Given such sequences, node identities are computed, tracking the development of nodes over time. Based on this information we demonstrate two out of many possible ways to begin exploring the data. The entire data set is well-documented, self-contained and ready for inspection at http://smgr.mpi-inf.mpg.de.
R. Lavi and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-... more R. Lavi and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-in-expectation mechanisms from linear programming based approximation algorithms. Due to the use of the Ellipsoid method, a direct implementation of the method is unlikely to be efficient in practice. We propose to use the much simpler and usually faster multiplicative weights update method instead. The simplification comes at the cost of slightly weaker approximation and truthfulness guarantees.
A certifying algorithm is an algorithm that produces with each output, a certificate or witness (... more A certifying algorithm is an algorithm that produces with each output, a certificate or witness (easy-to-verify proof) that the particular output has not been compromised by a bug. A user of a certifying program P (= the implementation of a certifying algorithm) inputs x, receives an output y and a certificate w, and then checks, either manually or by use of a checking program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust P. We refer the reader to the recent survey paper [9] for a detailed discussion of certifying algorithms.
The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simult... more The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously work-e cient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random edge weights uniformly distributed in 0;1]. We use the G(n;d=n) model: the graph consists of n nodes and each edge is chosen with probability d=n. Our PRAM algorithm needs O(n 1=3 log n) time and O(n log n+dn) work with high probability (whp). We also give extensions to external memory computation. Simulations show the applicability of our approach even on non-random graphs. 1 Introduction Computing shortest paths is an important combinatorial optimization problem with numerous applications. Let G = (V; E) be a directed graph, jEj = m, jV j = n, let s be a distinguished vertex of the graph, and c be a function assigning a non-negative real-valued weight to each edge of G. The single source shortest path problem (SSSP) is that of computing, for each vertex v reachable from s, the weight dist(v) of a minimum-weight path from s to v; the weight of a path is the sum of the weights of its edges. The theoretically most e cient sequential algorithm on digraphs with nonnegative edge weights is Dijkstra's algorithm 8]. Using Fibonacci heaps its running time is O(n logn + m) 1. Dijkstra's algorithm maintains a partition of V into settled, queued and unreached nodes and for each node v a tentative distance tent(v); tent(v) is always the weight of some path from s to v and hence an upper bound on dist(v). For unreached nodes, tent(v) = 1. Initially, s is queued, tent(s) = 0, and all other nodes are unreached. In each iteration, the queued node v with smallest tentative distance is selected and declared settled and all edges (v; w) are relaxed, i.e., tent(w) is set to minftent(w); tent(v) + c(v; w)g. 1 There is also an O(n + m) time algorithm for undirected graphs 20], but it requires the RAM model instead of the comparison model which is used in this work.
We present a tight tradeoff between the expected communication complexity [`</font &am... more We present a tight tradeoff between the expected communication complexity [`</font >(C)]\bar C and the number R of random bits used by any Las Vegas protocol (for a two-processor system) for the list-disjointness function of two lists of n numbers of n bits each. This function evaluates to 1 if and only if the two lists correspond in at least
We introduce a general framework for processing a set of curves defined on a continuous two-dimen... more We introduce a general framework for processing a set of curves defined on a continuous two-dimensional parametric surface, while sweeping the parameter space. A major goal of our work is to maximize code reuse in implementing algorithms that employ the prevalent sweep-line paradigm, and consequently to minimize the effort needed to extend the implementation of the paradigm to various surfaces and families of curves embedded on them. We show how the sweep-line paradigm is used to construct an arrangement of curves embedded on an orientable parametric surface, and explain how the arrangement package of Cgal, which previously handled only arrangements of bounded planar curves, is extended to handle curves embedded on a general surface. To the best of our knowledge, this is the first software implementation of generic algorithms that can handle arrangements on general parametric surfaces.
We show that a maximum flow in a network with n vertices can be computed deterministically in O(n... more We show that a maximum flow in a network with n vertices can be computed deterministically in O(n3/logn) time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of O(n3). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is O(nS/3(log n)4/3), in contrast with f~(nm) flow operations for all previous algorithms, where m denotes the number of edges in the network. A randomized version of our algorithm executes O(nal2rn 1/2 (log n) 312 + n 2 (log n) 2) flow operations with high probability. Specializing to the case in which all capacities are integers bounded by U, we show that a maximum flow can be computed using O(n3/2ml/2 + n2{log U) ~/2) flow operations. Finally, we argue that several of our results yield optimal parallel algorithms.
Given two bounded convex sets X ⊆ R m and Y ⊆ R n , specified by membership oracles, and a contin... more Given two bounded convex sets X ⊆ R m and Y ⊆ R n , specified by membership oracles, and a continuous convex-concave function F : X × Y → R, we consider the problem of computing an ε-approximate saddle point, that is, a pair (x * , y *) ∈ X × Y such that sup y∈Y F (x * , y) ≤ inf x∈X F (x, y *) + ε. Grigoriadis and Khachiyan (1995) gave a simple randomized variant of fictitious play for computing an ε-approximate saddle point for matrix games, that is, when F is bilinear and the sets X and Y are simplices. In this paper, we extend their method to the general case. In particular, we show that, for functions of constant "width", an ε-approximate saddle point can be computed using O * ((n+m) ε 2 ln R) random samples from log-concave distributions over the convex sets X and Y. It is assumed that X and Y have inscribed balls of radius 1/R and circumscribing balls of radius R. As a consequence, we obtain a simple randomized polynomial-time algorithm that computes such an approximation faster than known methods for problems with bounded width and when ε ∈ (0, 1) is a fixed, but arbitrarily small constant. Our main tool for achieving this result is the combination of the randomized fictitious play with the recently developed results on sampling from convex sets.
We present an improved average case analysis of the maximum cardinality matching problem. We show... more We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every non-maximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like the Hopcroft-Karp algorithm for bipartite graphs and the Micali-Vazirani algorithm for general graphs, which have a worst case running time of O(m √ n), run in time O(m log n) with high probability, where m is the number of edges in the graph. Motwani proved these results for random graphs when the average degree is at least ln(n) [Average Case Analysis of Algorithms for Matchings and Related Problems, Journal of the ACM, 41(6):1329-1356, 1994]. Our results hold if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.
Das diesem Bericht zugrunde liegende Vorhaben wurde mit Mitteln des Bundesministers für Forschung... more Das diesem Bericht zugrunde liegende Vorhaben wurde mit Mitteln des Bundesministers für Forschung und Technologie (Betreuungskennzeichen ITS 9103) gefördert. Die Verantwortung für den Inhalt dieser Veröffentlichung liegt beim Autor." An o(n 3)-Time Maximum-Flow Algorithm 1 ,2
ABSTRACT We consider the problem of matching a set of applicants to a set of posts, where each ap... more ABSTRACT We consider the problem of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a non-empty subset of posts in order of preference, possibly involving ties. We say that a matching M is popular if there is no matching M' such that the number of applicants preferring M' to M exceeds the number of applicants preferring M to M'. In this paper, we give the first polynomial-time algorithms to determine if an instance admits a popular matching, and to find a largest such matching, if one exists. For the special case in which every preference list is strictly ordered (i.e. contains no ties), we give an O(n+m) time algorithm, where n is the total number of applicants and posts, and m is the total length of all the preference lists. For the general case in which preference lists may contain ties, we give an O(√nm) time algorithm, and show that the problem has equivalent time complexity to the maximum-cardinality bipartite matching problem.
We present an algorithm for isolating all roots of an arbitrary complex polynomial p that also wo... more We present an algorithm for isolating all roots of an arbitrary complex polynomial p that also works in the presence of multiple roots provided that (1) the number of distinct roots is given as part of the input and (2) the algorithm can ask for arbitrarily good approximations of the coefficients of p. The algorithm outputs pairwise disjoint disks each containing one of the distinct roots of p and the multiplicity of the root contained in the disk. The algorithm uses approximate factorization as a subroutine. For the case where Pan's algorithm [36] is used for the factorization, we derive complexity bounds for the problems of isolating and refining all roots, which are stated in terms of the geometric locations of the roots only. Specializing the latter bounds to a polynomial of degree d and with integer coefficients of bitsize less than τ, we show thatÕ(d 3 + d 2 τ + dκ) bit operations are sufficient to compute isolating disks of size less than 2 −κ for all roots of p, where κ is an arbitrary positive integer. In addition, we apply our root isolation algorithm to a recent algorithm for computing the topology of a real planar algebraic curve specified as the zero set of a bivariate integer polynomial and for isolating the real solutions of a bivariate polynomial system. For polynomials of degree n and bitsize τ, we improve the currently best running time fromÕ(n 9 τ + n 8 τ 2) (deterministic) toÕ(n 6 + n 5 τ) (randomized) for topology computation and fromÕ(n 8 + n 7 τ) (deterministic) toÕ(n 6 + n 5 τ) (randomized) for solving bivariate systems.
We present an algorithm for the routing problem for two-tenninal nets in generalized switchboxes.... more We present an algorithm for the routing problem for two-tenninal nets in generalized switchboxes. A generalized s\~itchbox is any subset R of the planar rectangular grid with no non-trivial holes, i.e. every finite face has exactly four incident vertices. A net is a pair of nodes of non-maximal degree on the boundary of R. A solution is a set of edge-disjoint paths, one for each net. Our algorithm solves standard generalized switchbox routing problems in time 0(n(109 n)2) where n is the number of vertices of R, i.e. it either finds a solution or indicates that there is none. A problem is standard if deg(v) + ter(v) is even for all vertices v where deg(v) is the degree of v and ter(v) is the number of nets which have v as a tenninal. For nonstandard problems we can find a solution in time O(n(log n)2 + IUIZ) where U is the set of vertices v with deg(v) + ter(v) is odd.
The dynamic planar point location problem is the task of maintaining a dynamic set S of n non-int... more The dynamic planar point location problem is the task of maintaining a dynamic set S of n non-intersecting, except possibly at endpoints, line segments in the plane under the following operations: Locate(q: point): Report the segment immediately above q, i.e., the rst segment i n tersected by a n u p ward vertical ray starting at q Insert(s: s e g m e n t): Add segment s to the collection S of segments Delete(s: s e g m e n t): Remove segment s from the collection S of segments. We present a solution which requires space O(n), has query and insertion time O(log n loglog n) and deletion time O(log 2 n). A query time below O(log 2 n) w as previously only known for monotone subdivisions and horizontal segments and required non-linear space.
Devdatt Dubhashi § We establish a lower bound on the efficiency of area-universal circuits. The a... more Devdatt Dubhashi § We establish a lower bound on the efficiency of area-universal circuits. The area Au of every graph H that can host any graph G of area (at most) A with dilation d, and congestion c ~ ../AI log log A satisfies the tradeoff Au = O(AlogAI(c 2 log(2d»),. In particular, ifAu = 0 (A) then max(c, d) = O(Jlog AI log log A).
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1990
In this paper we introduce a general framework for compaction on a torus. This problem comes up w... more In this paper we introduce a general framework for compaction on a torus. This problem comes up whenever an array of identical cells has to be compacted. We instantiate our framework with several specific compaction algorithms: one-dimensional compaction without and with automatic jog insertion and two-dimensional compaction. I. INTRODUCTION COMPACTER takes as input a VLSI layout and pro-A duces as output an equivalent layout of smaller area. An effective compaction system frees the designer from the details of the design rules, and hence, increases his productivity and on the other hand produces high quality layouts. For these reasons, compaction algorithms have gained widespread attention in the VLSI literature [4], [7], [9], [lo], [12], [15], [17], and are the basis for several computer-aided circuit design systems [2], [4], [ 1 11, [ 191, Regular layouts composed of one-or two-dimensional arrays of identical cells arise frequently in practice, e.g., bit slice architecture or systolic arrays. Let S be the cell to be replicated. We address the following problem. Compact S into a cell S' such that cell S' still can be used to tile the plane (or an infinite strip). This problem is called the compaction problem on the torus, because a layout S can be used to tile the plane iff its left and right, and top and bottom boundaries are compatible, i.e., if the cell S can be drawn on a torus, cf. Fig.
We consider the problem of computing geometric transformations (rotation, translation, reflexion)... more We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point set A exactly or approximately into a point set B. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).
Most algorithms of computational geometry are designed for the Real-RAM and non-degenerate inputs... more Most algorithms of computational geometry are designed for the Real-RAM and non-degenerate inputs. We call such algorithms idealistic. Executing an idealistic algorithm with floating point arithmetic may fail. Controlled perturbation replaces an input x by a random nearby x in the δ-neighborhood of x and then runs the floating point version of the idealistic algorithm on x. The hope is that this will produce the correct result for x with constant probability provided that δ is small and the precision L of the floating point system is large enough. We turn this hope into a theorem for a large class of geometric algorithms and describe a general methodology for deriving a relation between δ and L. We exemplify the usefulness of the methodology by examples.
In this paper, we present two results on slime mold computations. The first one treats a biologic... more In this paper, we present two results on slime mold computations. The first one treats a biologically-grounded model, originally proposed by biologists analyzing the behavior of the slime mold Physarum polycephalum. This primitive organism was empirically shown by Nakagaki et al. to solve shortest path problems in wet-lab experiments (Nature'00). We show that the proposed simple mathematical model actually generalizes to a much wider class of problems, namely undirected linear programs with a non-negative cost vector. For our second result, we consider the discretization of a biologically-inspired model. This model is a directed variant of the biologically-grounded one and was never claimed to describe the behavior of a biological system. Straszak and Vishnoi showed that it can ε-approximately solve flow problems (SODA'16) and even general linear programs with positive cost vector (ITCS'16) within a finite number of steps. We give a refined convergence analysis that improves the dependence on ε from polynomial to logarithmic and simultaneously allows to choose a step size that is independent of ε. Furthermore, we show that the dynamics can be initialized with a more general set of (infeasible) starting points.
We introduce the slime mold graph repository or SMGR, a novel data collection promoting the visib... more We introduce the slime mold graph repository or SMGR, a novel data collection promoting the visibility, accessibility and reuse of experimental data revolving around network-forming slime molds. By making data readily available to researchers across multiple disciplines, the SMGR promotes novel research as well as the reproduction of original results. While SMGR data may take various forms, we stress the importance of graph representations of slime mold networks due to their ease of handling and their large potential for reuse. Data added to the SMGR stands to gain impact beyond initial publications or even beyond its domain of origin. We initiate the SMGR with the comprehensive Kist Europe data set focusing on the slime mold Physarum polycephalum, which we obtained in the course of our original research. It contains sequences of images documenting growth and network formation of the organism under constant conditions. Suitable image sequences depicting the typical P. polycephalum network structures are used to compute sequences of graphs faithfully capturing them. Given such sequences, node identities are computed, tracking the development of nodes over time. Based on this information we demonstrate two out of many possible ways to begin exploring the data. The entire data set is well-documented, self-contained and ready for inspection at http://smgr.mpi-inf.mpg.de.
R. Lavi and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-... more R. Lavi and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-in-expectation mechanisms from linear programming based approximation algorithms. Due to the use of the Ellipsoid method, a direct implementation of the method is unlikely to be efficient in practice. We propose to use the much simpler and usually faster multiplicative weights update method instead. The simplification comes at the cost of slightly weaker approximation and truthfulness guarantees.
A certifying algorithm is an algorithm that produces with each output, a certificate or witness (... more A certifying algorithm is an algorithm that produces with each output, a certificate or witness (easy-to-verify proof) that the particular output has not been compromised by a bug. A user of a certifying program P (= the implementation of a certifying algorithm) inputs x, receives an output y and a certificate w, and then checks, either manually or by use of a checking program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust P. We refer the reader to the recent survey paper [9] for a detailed discussion of certifying algorithms.
The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simult... more The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously work-e cient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random edge weights uniformly distributed in 0;1]. We use the G(n;d=n) model: the graph consists of n nodes and each edge is chosen with probability d=n. Our PRAM algorithm needs O(n 1=3 log n) time and O(n log n+dn) work with high probability (whp). We also give extensions to external memory computation. Simulations show the applicability of our approach even on non-random graphs. 1 Introduction Computing shortest paths is an important combinatorial optimization problem with numerous applications. Let G = (V; E) be a directed graph, jEj = m, jV j = n, let s be a distinguished vertex of the graph, and c be a function assigning a non-negative real-valued weight to each edge of G. The single source shortest path problem (SSSP) is that of computing, for each vertex v reachable from s, the weight dist(v) of a minimum-weight path from s to v; the weight of a path is the sum of the weights of its edges. The theoretically most e cient sequential algorithm on digraphs with nonnegative edge weights is Dijkstra's algorithm 8]. Using Fibonacci heaps its running time is O(n logn + m) 1. Dijkstra's algorithm maintains a partition of V into settled, queued and unreached nodes and for each node v a tentative distance tent(v); tent(v) is always the weight of some path from s to v and hence an upper bound on dist(v). For unreached nodes, tent(v) = 1. Initially, s is queued, tent(s) = 0, and all other nodes are unreached. In each iteration, the queued node v with smallest tentative distance is selected and declared settled and all edges (v; w) are relaxed, i.e., tent(w) is set to minftent(w); tent(v) + c(v; w)g. 1 There is also an O(n + m) time algorithm for undirected graphs 20], but it requires the RAM model instead of the comparison model which is used in this work.
We present a tight tradeoff between the expected communication complexity [`</font &am... more We present a tight tradeoff between the expected communication complexity [`</font >(C)]\bar C and the number R of random bits used by any Las Vegas protocol (for a two-processor system) for the list-disjointness function of two lists of n numbers of n bits each. This function evaluates to 1 if and only if the two lists correspond in at least
We introduce a general framework for processing a set of curves defined on a continuous two-dimen... more We introduce a general framework for processing a set of curves defined on a continuous two-dimensional parametric surface, while sweeping the parameter space. A major goal of our work is to maximize code reuse in implementing algorithms that employ the prevalent sweep-line paradigm, and consequently to minimize the effort needed to extend the implementation of the paradigm to various surfaces and families of curves embedded on them. We show how the sweep-line paradigm is used to construct an arrangement of curves embedded on an orientable parametric surface, and explain how the arrangement package of Cgal, which previously handled only arrangements of bounded planar curves, is extended to handle curves embedded on a general surface. To the best of our knowledge, this is the first software implementation of generic algorithms that can handle arrangements on general parametric surfaces.
We show that a maximum flow in a network with n vertices can be computed deterministically in O(n... more We show that a maximum flow in a network with n vertices can be computed deterministically in O(n3/logn) time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of O(n3). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is O(nS/3(log n)4/3), in contrast with f~(nm) flow operations for all previous algorithms, where m denotes the number of edges in the network. A randomized version of our algorithm executes O(nal2rn 1/2 (log n) 312 + n 2 (log n) 2) flow operations with high probability. Specializing to the case in which all capacities are integers bounded by U, we show that a maximum flow can be computed using O(n3/2ml/2 + n2{log U) ~/2) flow operations. Finally, we argue that several of our results yield optimal parallel algorithms.
Given two bounded convex sets X ⊆ R m and Y ⊆ R n , specified by membership oracles, and a contin... more Given two bounded convex sets X ⊆ R m and Y ⊆ R n , specified by membership oracles, and a continuous convex-concave function F : X × Y → R, we consider the problem of computing an ε-approximate saddle point, that is, a pair (x * , y *) ∈ X × Y such that sup y∈Y F (x * , y) ≤ inf x∈X F (x, y *) + ε. Grigoriadis and Khachiyan (1995) gave a simple randomized variant of fictitious play for computing an ε-approximate saddle point for matrix games, that is, when F is bilinear and the sets X and Y are simplices. In this paper, we extend their method to the general case. In particular, we show that, for functions of constant "width", an ε-approximate saddle point can be computed using O * ((n+m) ε 2 ln R) random samples from log-concave distributions over the convex sets X and Y. It is assumed that X and Y have inscribed balls of radius 1/R and circumscribing balls of radius R. As a consequence, we obtain a simple randomized polynomial-time algorithm that computes such an approximation faster than known methods for problems with bounded width and when ε ∈ (0, 1) is a fixed, but arbitrarily small constant. Our main tool for achieving this result is the combination of the randomized fictitious play with the recently developed results on sampling from convex sets.
We present an improved average case analysis of the maximum cardinality matching problem. We show... more We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every non-maximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like the Hopcroft-Karp algorithm for bipartite graphs and the Micali-Vazirani algorithm for general graphs, which have a worst case running time of O(m √ n), run in time O(m log n) with high probability, where m is the number of edges in the graph. Motwani proved these results for random graphs when the average degree is at least ln(n) [Average Case Analysis of Algorithms for Matchings and Related Problems, Journal of the ACM, 41(6):1329-1356, 1994]. Our results hold if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.
Das diesem Bericht zugrunde liegende Vorhaben wurde mit Mitteln des Bundesministers für Forschung... more Das diesem Bericht zugrunde liegende Vorhaben wurde mit Mitteln des Bundesministers für Forschung und Technologie (Betreuungskennzeichen ITS 9103) gefördert. Die Verantwortung für den Inhalt dieser Veröffentlichung liegt beim Autor." An o(n 3)-Time Maximum-Flow Algorithm 1 ,2
ABSTRACT We consider the problem of matching a set of applicants to a set of posts, where each ap... more ABSTRACT We consider the problem of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a non-empty subset of posts in order of preference, possibly involving ties. We say that a matching M is popular if there is no matching M' such that the number of applicants preferring M' to M exceeds the number of applicants preferring M to M'. In this paper, we give the first polynomial-time algorithms to determine if an instance admits a popular matching, and to find a largest such matching, if one exists. For the special case in which every preference list is strictly ordered (i.e. contains no ties), we give an O(n+m) time algorithm, where n is the total number of applicants and posts, and m is the total length of all the preference lists. For the general case in which preference lists may contain ties, we give an O(√nm) time algorithm, and show that the problem has equivalent time complexity to the maximum-cardinality bipartite matching problem.
We present an algorithm for isolating all roots of an arbitrary complex polynomial p that also wo... more We present an algorithm for isolating all roots of an arbitrary complex polynomial p that also works in the presence of multiple roots provided that (1) the number of distinct roots is given as part of the input and (2) the algorithm can ask for arbitrarily good approximations of the coefficients of p. The algorithm outputs pairwise disjoint disks each containing one of the distinct roots of p and the multiplicity of the root contained in the disk. The algorithm uses approximate factorization as a subroutine. For the case where Pan's algorithm [36] is used for the factorization, we derive complexity bounds for the problems of isolating and refining all roots, which are stated in terms of the geometric locations of the roots only. Specializing the latter bounds to a polynomial of degree d and with integer coefficients of bitsize less than τ, we show thatÕ(d 3 + d 2 τ + dκ) bit operations are sufficient to compute isolating disks of size less than 2 −κ for all roots of p, where κ is an arbitrary positive integer. In addition, we apply our root isolation algorithm to a recent algorithm for computing the topology of a real planar algebraic curve specified as the zero set of a bivariate integer polynomial and for isolating the real solutions of a bivariate polynomial system. For polynomials of degree n and bitsize τ, we improve the currently best running time fromÕ(n 9 τ + n 8 τ 2) (deterministic) toÕ(n 6 + n 5 τ) (randomized) for topology computation and fromÕ(n 8 + n 7 τ) (deterministic) toÕ(n 6 + n 5 τ) (randomized) for solving bivariate systems.
We present an algorithm for the routing problem for two-tenninal nets in generalized switchboxes.... more We present an algorithm for the routing problem for two-tenninal nets in generalized switchboxes. A generalized s\~itchbox is any subset R of the planar rectangular grid with no non-trivial holes, i.e. every finite face has exactly four incident vertices. A net is a pair of nodes of non-maximal degree on the boundary of R. A solution is a set of edge-disjoint paths, one for each net. Our algorithm solves standard generalized switchbox routing problems in time 0(n(109 n)2) where n is the number of vertices of R, i.e. it either finds a solution or indicates that there is none. A problem is standard if deg(v) + ter(v) is even for all vertices v where deg(v) is the degree of v and ter(v) is the number of nets which have v as a tenninal. For nonstandard problems we can find a solution in time O(n(log n)2 + IUIZ) where U is the set of vertices v with deg(v) + ter(v) is odd.
The dynamic planar point location problem is the task of maintaining a dynamic set S of n non-int... more The dynamic planar point location problem is the task of maintaining a dynamic set S of n non-intersecting, except possibly at endpoints, line segments in the plane under the following operations: Locate(q: point): Report the segment immediately above q, i.e., the rst segment i n tersected by a n u p ward vertical ray starting at q Insert(s: s e g m e n t): Add segment s to the collection S of segments Delete(s: s e g m e n t): Remove segment s from the collection S of segments. We present a solution which requires space O(n), has query and insertion time O(log n loglog n) and deletion time O(log 2 n). A query time below O(log 2 n) w as previously only known for monotone subdivisions and horizontal segments and required non-linear space.
Devdatt Dubhashi § We establish a lower bound on the efficiency of area-universal circuits. The a... more Devdatt Dubhashi § We establish a lower bound on the efficiency of area-universal circuits. The area Au of every graph H that can host any graph G of area (at most) A with dilation d, and congestion c ~ ../AI log log A satisfies the tradeoff Au = O(AlogAI(c 2 log(2d»),. In particular, ifAu = 0 (A) then max(c, d) = O(Jlog AI log log A).
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1990
In this paper we introduce a general framework for compaction on a torus. This problem comes up w... more In this paper we introduce a general framework for compaction on a torus. This problem comes up whenever an array of identical cells has to be compacted. We instantiate our framework with several specific compaction algorithms: one-dimensional compaction without and with automatic jog insertion and two-dimensional compaction. I. INTRODUCTION COMPACTER takes as input a VLSI layout and pro-A duces as output an equivalent layout of smaller area. An effective compaction system frees the designer from the details of the design rules, and hence, increases his productivity and on the other hand produces high quality layouts. For these reasons, compaction algorithms have gained widespread attention in the VLSI literature [4], [7], [9], [lo], [12], [15], [17], and are the basis for several computer-aided circuit design systems [2], [4], [ 1 11, [ 191, Regular layouts composed of one-or two-dimensional arrays of identical cells arise frequently in practice, e.g., bit slice architecture or systolic arrays. Let S be the cell to be replicated. We address the following problem. Compact S into a cell S' such that cell S' still can be used to tile the plane (or an infinite strip). This problem is called the compaction problem on the torus, because a layout S can be used to tile the plane iff its left and right, and top and bottom boundaries are compatible, i.e., if the cell S can be drawn on a torus, cf. Fig.
We consider the problem of computing geometric transformations (rotation, translation, reflexion)... more We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point set A exactly or approximately into a point set B. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).
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Papers by K. Mehlhorn