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A fast algorithm for finding dominators in a flowgraph
 ACM Transactions on Programming Languages and Systems
, 1979
"... A fast algoritbm for finding dominators in a flowgraph is presented. The algorithm uses depthfirst search and an efficient method of computing functions defined on paths in trees. A simple implementation of the algorithm runs in O(m log n) time, where m is the number of edges and n is the number o ..."
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Cited by 144 (3 self)
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A fast algoritbm for finding dominators in a flowgraph is presented. The algorithm uses depthfirst search and an efficient method of computing functions defined on paths in trees. A simple implementation of the algorithm runs in O(m log n) time, where m is the number of edges and n is the number of vertices in the problem graph. A more sophisticated implementation runs in O(ma(m, n)) time, where a(m, n) is a functional inverse of Ackermann's function. Both versions of the algorithm were implemented in Algol W, a Stanford University version of Algol, and tested on an IBM 370/168. The programs were compared with an implementation by Purdom and Moore of a straightforward O(mn)time algorithm, and with ~a bit vector algorithm described by Aho and Ullman. The fast algorithm beat the straightforward algorithm and the bit vector algorithm on all but the smallest graphs tested.
Ambivalent Data Structures For Dynamic 2EdgeConnectivity And k Smallest Spanning Trees
 SIAM J. Comput
, 1991
"... . Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log #(m, n) + min{k 3/2 ,km 1/2 }) time, where m is the number of edges and n the number of vertice ..."
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Cited by 83 (1 self)
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. Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log #(m, n) + min{k 3/2 ,km 1/2 }) time, where m is the number of edges and n the number of vertices in the graph. The techniques are extended to find the k smallest spanning trees in an embedded planar graph in O(n + k(log n) 3 ) time. Ambivalent data structures are also used to dynamically maintain 2edgeconnectivity information. Edges and vertices can be inserted or deleted in O(m 1/2 ) time, and a query as to whether two vertices are in the same 2edgeconnected component can be answered in O(log n) time, where m and n are understood to be the current number of edges and vertices, respectively. Key words. analysis of algorithms, data structures, embedded planar graph, fully persistent data structures, k smallest spanning trees, minimum spanning tree, online updating, topology tr...
A shortest path algorithm for realweighted undirected graphs
 in 13th ACMSIAM Symp. on Discrete Algs
, 1985
"... Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) ti ..."
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Cited by 12 (3 self)
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Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverseAckermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the allpairs and singlesource shortest paths problems. We solve the allpairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the singlesource problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchybased approach invented by Thorup. Key words. singlesource shortest paths, allpairs shortest paths, undirected graphs, Dijkstra’s
Approaching optimality for solving SDD linear systems ∗
, 2010
"... We present an algorithm that on input a graph G with n vertices and m + n − 1 edges and a value k, produces an incremental sparsifier ˆ G with n − 1+m/k edges, such that the condition number of G with ˆ G is bounded above by Õ(k log2 n), with probability 1 − p. The algorithm runs in time Õ((m log n ..."
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Cited by 10 (2 self)
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We present an algorithm that on input a graph G with n vertices and m + n − 1 edges and a value k, produces an incremental sparsifier ˆ G with n − 1+m/k edges, such that the condition number of G with ˆ G is bounded above by Õ(k log2 n), with probability 1 − p. The algorithm runs in time Õ((m log n + n log 2 n) log(1/p)). 1 As a result, we obtain an algorithm that on input an n × n symmetric diagonally dominant matrix A with m + n − 1 nonzero entries and a vector b, computes a vector ¯x satisfying x − A + bA <ɛA + bA, in time Õ(m log 2 n log(1/ɛ)). The solver is based on a recursive application of the incremental sparsifier that produces a hierarchy of graphs which is then used to construct a recursive preconditioned Chebyshev iteration. 1
A Few Remarks On The History Of MSTProblem
, 1997
"... On the background of Boruvka's pioneering work we present a survey of the development related to the Minimum Spanning Tree Problem. We also complement the historical paper GrahamHell [GH] by a few remarks and provide an update of the extensive literature devoted to this problem. ..."
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Cited by 1 (0 self)
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On the background of Boruvka's pioneering work we present a survey of the development related to the Minimum Spanning Tree Problem. We also complement the historical paper GrahamHell [GH] by a few remarks and provide an update of the extensive literature devoted to this problem.
Backtracking
"... Contents 1 Introduction 3 2 Models of computation 6 3 The Set Union Problem 9 4 The WorstCase Time Complexity of a Single Operation 15 5 The Set Union Problem with Deunions 18 6 Split and the Set Union Problem on Intervals 22 7 The Set Union Problem with Unlimited Backtracking 26 1 Introduction A ..."
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Contents 1 Introduction 3 2 Models of computation 6 3 The Set Union Problem 9 4 The WorstCase Time Complexity of a Single Operation 15 5 The Set Union Problem with Deunions 18 6 Split and the Set Union Problem on Intervals 22 7 The Set Union Problem with Unlimited Backtracking 26 1 Introduction An equivalence relation on a finite set S is a binary relation that is reflexive symmetric and transitive. That is, for s; t and u in S, we have that sRs, if sRt then tRs, and if sRt and tRu then sRu. Set S is partitioned by R into equivalence classes where each class cointains all and only the elements that obey R pairwise. Many computational problems involve representing, modifying and tracking the evolution of equivalenc