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154
Greedy optimal homotopy and homology generators
 Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms
, 2005
"... Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops t ..."
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Cited by 102 (11 self)
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Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdi`ere and Lazarus.
Minimum cuts in nearlinear time
 Proc. of the 28th STOC
, 1996
"... Abstract. We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carl ..."
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Cited by 95 (12 self)
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Abstract. We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an medge, nvertex graph with high probability in O(m log 3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n 2 log n) time. This variant has an optimal ᏺᏯ parallelization. Both variants improve on the previous best time bound of O(n 2 log 3 n). Other applications of the treepacking approach are new, nearly tight bounds on the number of nearminimum cuts a graph may have and a new data structure for representing them in a spaceefficient manner.
Cp(graph): Introducing a graph computation domain in constraint programming
 In CP2005 Proceedings
, 2005
"... Abstract. In an increasing number of domains such as bioinformatics, combinatorial graph problems arise. We propose a novel way to solve these problems, mainly those that can be translated to constrained subgraph finding. Our approach extends constraint programming by introducing CP(Graph), a new co ..."
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Cited by 52 (12 self)
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Abstract. In an increasing number of domains such as bioinformatics, combinatorial graph problems arise. We propose a novel way to solve these problems, mainly those that can be translated to constrained subgraph finding. Our approach extends constraint programming by introducing CP(Graph), a new computation domain focused on graphs including a new type of variable: graph domain variables as well as constraints over these variables and their propagators. These constraints are subdivided into kernel constraints and additional constraints formulated as networks of kernel constraints. For some of these constraints a dedicated global constraint and its associated propagator are sketched. CP(Graph) is integrated with finite domain and finite sets computation domains, allowing the combining of constraints of these domains with graph constraints. A prototype of CP(Graph) built over finite domains and finite sets in Oz is presented. And we show that a problem of biochemical network analysis can be very simply described and solved within CP(Graph). 1
Dynamic Generators of Topologically Embedded Graphs
, 2003
"... We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g ..."
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Cited by 49 (1 self)
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We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log 4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for lowgenus graphs, and to find in linear time a treedecomposition of lowgenus lowdiameter graphs.
Efficient and simple generation of random simple connected graphs with prescribed degree sequence
 in The Eleventh International Computing and Combinatorics Conference, Aug. 2005, kumming
, 2005
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A Fully Dynamic Algorithm for Maintaining the Transitive Closure
 In Proc. 31st ACM Symposium on Theory of Computing (STOC'99
, 1999
"... This paper presents an efficient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form "Is there a directed path fro ..."
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Cited by 49 (1 self)
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This paper presents an efficient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form "Is there a directed path from i to j?" can be answered in O(1) time. The algorithm is randomized; it is correct when answering yes, but has O(1/n^c) probability of error when answering no, for any constant c. In acyclic graphs, worst case update time is O(n^2). In general graphs, update time is O(n^(2+alpha)), where alpha = min {.26, maximum size of a strongly connected component}. The space complexity of the algorithm is O(n^2).
Logarithmic lower bounds in the cellprobe model
 SIAM Journal on Computing
"... Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the ..."
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Cited by 47 (3 self)
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Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the data structure. We use our technique to prove better bounds for the partialsums problem, dynamic connectivity and (by reductions) other dynamic graph problems. Our proofs are surprisingly simple and clean. The bounds we obtain are often optimal, and lead to a nearly complete understanding of the problems. We also present new matching upper bounds for the partialsums problem. Key words. cellprobe complexity, lower bounds, data structures, dynamic graph problems, partialsums problem AMS subject classification. 68Q17
Analyzing Graph Structure via Linear Measurements
"... We initiate the study of graph sketching, i.e., algorithms that use a limited number of linear measurements of a graph to determine the properties of the graph. While a graph on n nodes is essentially O(n 2)dimensional, we show the existence of a distribution over random projections into ddimensio ..."
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Cited by 41 (11 self)
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We initiate the study of graph sketching, i.e., algorithms that use a limited number of linear measurements of a graph to determine the properties of the graph. While a graph on n nodes is essentially O(n 2)dimensional, we show the existence of a distribution over random projections into ddimensional “sketch ” space (d ≪ n 2) such that the relevant properties of the original graph can be inferred from the sketch with high probability. Specifically, we show that: 1. d = O(n · polylog n) suffices to evaluate properties including connectivity, kconnectivity, bipartiteness, and to return any constant approximation of the weight of the minimum spanning tree. 2. d = O(n 1+γ) suffices to compute graph sparsifiers, the exact MST, and approximate the maximum weighted matchings if we permit O(1/γ)round adaptive sketches, i.e., a sequence of projections where each projection may be chosen dependent on the outcome of earlier sketches. Our results have two main applications, both of which have the potential to give rise to fruitful lines of further research. First, our results can be thought of as giving the first compressedsensing style algorithms for graph data. Secondly, our work initiates the study of dynamic graph streams. There is already extensive literature on processing massive graphs in the datastream model. However, the existing work focuses on graphs defined by a sequence of inserted edges and does not consider edge deletions. We think this is a curious omission given the existing work on both dynamic graphs in the nonstreaming setting and dynamic geometric streaming. Our results include the first dynamic graph semistreaming algorithms for connectivity, spanning trees, sparsification, and matching problems.
Improved Dynamic Reachability Algorithms for Directed Graphs
, 2002
"... We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the ..."
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Cited by 35 (3 self)
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We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the transitive closure of a directed graph, through an arbitrary sequence of edge deletions, in O(mn) total expected time, essentially the time needed for computing the transitive closure of the initial graph. Such a result was previously known only for acyclic graphs.
Cell probe complexity  a survey
 In 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 1999. Advances in Data Structures Workshop
"... The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1 ..."
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Cited by 32 (0 self)
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The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1