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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 146 (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.
A Randomized LinearTime Algorithm to Find Minimum Spanning Trees
, 1994
"... We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost ra ..."
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Cited by 120 (7 self)
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We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost randomaccess machine with the restriction that the only operations allowed on edge weights are binary comparisons.
Ambivalent data structures for dynamic 2edgeconnectivity and k smallest spanning trees
 SIAM J. Comput
, 1997
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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 11 (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.
An InverseAckermann Type Lower Bound for Online Minimum Spanning Tree Verification
 Combinatorica
"... Given a spanning tree T of some graph G, the problem of minimum spanning tree verication is to decide whether T = MST(G). A celebrated result of Komlos shows that this problem can be solved in linear time. Somewhat unexpectedly, MST verication turns out to be useful in actually computing minimum spa ..."
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Cited by 2 (1 self)
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Given a spanning tree T of some graph G, the problem of minimum spanning tree verication is to decide whether T = MST(G). A celebrated result of Komlos shows that this problem can be solved in linear time. Somewhat unexpectedly, MST verication turns out to be useful in actually computing minimum spanning trees from scratch. It is this application that has led some to wonder whether a more flexible version of MST Verification could be used to derive a faster deterministic minimum spanning tree algorithm.
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.
Re intcd from JOURNAL OP COMPUTER AND Smmm SCIENCES AIKights Reserved by Academic Press, New York and London A Data Structure for Dynamic Trees
, 1983
"... A data structure is proposed to maintain a collection of vertexdisjoint trees under a sequence of two kinds of operations: a link operation that combines two trees into one by adding an edge, and a cut operation that divides one tree into two by deleting an edge. Each operation requires O(log n) ti ..."
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A data structure is proposed to maintain a collection of vertexdisjoint trees under a sequence of two kinds of operations: a link operation that combines two trees into one by adding an edge, and a cut operation that divides one tree into two by deleting an edge. Each operation requires O(log n) time. Using this data structure, new fast algorithms are obtained for the following problems: (1) Computing nearest common ancestors. (2) Solving various network flow problems including finding maximum flows, blocking flows, and acyclic flows. (3) Computing certain kinds of constrained minimum spanning trees. (4) Implementing the network simplex algorithm for minimumcost flows. The most significant application is (2); an O(mn log n)time algorithm is obtained to find a maximum flow in a network of n vertices and m edges, beating by a factor of log n the fastest algorithm previously known for sparse graphs. 1.
Published In Approaching Optimality For Solving SDD Linear Systems
"... Abstract—We present an algorithm that on input of an nvertex medge weighted graph G and a value k, produces an incremental sparsifier G ̂ with n−1+m/k edges, such that the condition number of G with G ̂ is bounded above by Õ(k log2 n), with probability 1 − p. The algorithm runs in time Õ((m logn ..."
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Abstract—We present an algorithm that on input of an nvertex medge weighted graph G and a value k, produces an incremental sparsifier G ̂ with n−1+m/k edges, such that the condition number of G with G ̂ is bounded above by Õ(k log2 n), with probability 1 − p. The algorithm runs in time Õ((m logn+ n log2 n) log(1/p)). As a result, we obtain an algorithm that on input of an n × n symmetric diagonally dominant matrix A with m nonzero entries and a vector b, computes a vector x satisfying x − A+bA < A+bA, in expected time Õ(m log2 n log(1/)). The solver is based on repeated applications of the incremental sparsifier that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration. Keywordsalgorithms; spectral graph theory; linear systems; combinatorial preconditioning I.