Results 11  20
of
57
A Linear Algorithm for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs
 Algorithmica
, 1992
"... We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a s ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
(Show Context)
We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heapordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a realvalued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimumcost path from r to every...
Checking mergeable priority queues
 In Digest of the 24th Symposium on FaultTolerant Computing
, 1994
"... ..."
(Show Context)
Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
, 2011
"... We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately t ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.
Setting Parameters by Example
, 1999
"... We introduce a class of “inverse parametric optimization” problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We describe algorithms for solving such problems for minimum spa ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
We introduce a class of “inverse parametric optimization” problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We describe algorithms for solving such problems for minimum spanning trees, shortest paths, and other “optimal subgraph” problems, and discuss applications in multicast routing, vehicle path planning, resource allocation, and board game programming.
Random Sampling in Matroids, with Applications to Graph Connectivity and Minimum Spanning Trees
, 1993
"... Random sampling is a powerful way to gather information about a group by considering only a small part of it. We give a paradigm for applying this technique to optimization problems, and demonstrate its effectiveness on matroids. Matroids abstractly model many optimization problems that can be solve ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
Random sampling is a powerful way to gather information about a group by considering only a small part of it. We give a paradigm for applying this technique to optimization problems, and demonstrate its effectiveness on matroids. Matroids abstractly model many optimization problems that can be solved by greedy methods, such as the minimum spanning tree (MST) problem. Our results have several applications. We give an algorithm that uses simple data structures to construct an MST in O(m+n logn) time (Klein and Tarjan [21] have recently shown that a better choice of parameters makes this algorithm run in O(m + n) time). We give bounds on the connectivity (minimum cut) of a graph suffering random edge failures. We give fast algorithms for packing matroid bases, with particular attention to packing spanning trees in graphs.
Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures
 Journal of Graph Algorithms and Applications
, 1998
"... Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
(Show Context)
Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes the diameter of the new spanning tree. Such an optimal replacement is called the best swap. As a natural extension, the allbestswaps (ABS) problem is the problem of finding the best swap for every edge of the MDST. Given a weighted graph G =(V, E), where V  = n and E  = m,wesolvetheABSprobleminO(n √ m)time and O(m + n) space, thus improving previous bounds for m = o(n 2). 1
Data structures for range minimum queries in multidimensional arrays
 In Proceedings of the 20th Annual ACMSIAM Symposium on Discrete Algorithms
, 2010
"... Given a ddimensional array A with N entries, the Range Minimum Query (RMQ) asks for the minimum element within a contiguous subarray of A. The 1D RMQ problem has been studied intensively because of its relevance to the Nearest Common Ancestor problem and its important use in stringology. If constan ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
(Show Context)
Given a ddimensional array A with N entries, the Range Minimum Query (RMQ) asks for the minimum element within a contiguous subarray of A. The 1D RMQ problem has been studied intensively because of its relevance to the Nearest Common Ancestor problem and its important use in stringology. If constanttime query answering is required, linear time and space preprocessing algorithms were known for the 1D case, but not for the higher dimensional cases. In this paper, we give the first lineartime preprocessing algorithm for arrays with fixed dimension, such that any range minimum query can be answered in constant time. 1
Tight bounds for distributed MST verification
 In Proc. 28th Symposium on Theoretical Aspects of Computer Science (STACS 2011), volume 9 of LIPIcs
, 2011
"... This paper establishes tight bounds for the Minimumweight Spanning Tree (MST) verification problem in the distributed setting. Specifically, we provide an MST verification algorithm that achieves simultaneously Õ(E) messages and Õ(√n+D) time, where E  is the number of edges in the given graph ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
This paper establishes tight bounds for the Minimumweight Spanning Tree (MST) verification problem in the distributed setting. Specifically, we provide an MST verification algorithm that achieves simultaneously Õ(E) messages and Õ(√n+D) time, where E  is the number of edges in the given graph G and D is G’s diameter. On the negative side, we show that any MST verification algorithm must send Ω(E) messages and incur Ω̃(√n+D) time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of Ω(E) messages and Ω( n+D) time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously Õ(E) messages and Õ(√n+D) time. Specifically, the best known timeoptimal algorithm (using Õ( n + D) time) requires O(E  + n3/2) messages, and the best known messageoptimal algorithm (using Õ(E) messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.
Dominator Tree Verification and VertexDisjoint Paths
, 2005
"... We present a lineartime algorithm that given a flowgraph G = (V, A, r) and a tree T, checks whether T is the dominator tree of G. Also we prove that there exist two spanning trees of G, T1 and T2, such that for any vertex v the paths from r to v in T1 and T2 intersect only at the vertices that domi ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
We present a lineartime algorithm that given a flowgraph G = (V, A, r) and a tree T, checks whether T is the dominator tree of G. Also we prove that there exist two spanning trees of G, T1 and T2, such that for any vertex v the paths from r to v in T1 and T2 intersect only at the vertices that dominate v. The proof is constructive and our algorithm can build the two spanning trees in linear time. Simpler versions of our two algorithms run in O(mα(m, n))time, where n is the number of vertices and m is the number of arcs in G. The existence of such two spanning trees implies that we can order the calculations of the iterative algorithm for finding dominators, proposed by Allen and Cocke [2], so that it builds the dominator tree in a single iteration.
An Optimal EREW PRAM Algorithm For Minimum Spanning Tree Verification
, 1997
"... We present a deterministic parallel algorithm on the EREW PRAM model to verify a minimum spanning tree of a graph. The algorithm runs on a graph with n vertices and m edges in O(log n) time and O(m + n) work. The algorithm is a parallelization of King's linear time sequential algorithm for the ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
We present a deterministic parallel algorithm on the EREW PRAM model to verify a minimum spanning tree of a graph. The algorithm runs on a graph with n vertices and m edges in O(log n) time and O(m + n) work. The algorithm is a parallelization of King's linear time sequential algorithm for the problem. 1 Introduction The problem of verifying if a given spanning tree in an edgeweighted graph is a minimum spanning tree for the graph is an important problem which is closely related to the problem of finding a minimum spanning tree of a graph. Recently Dixon, Rauch and Tarjan ([DRT92]) and King ([Kin95]) have developed sequential linear time algorithms for the problem. Dixon and Tarjan have also given an optimal CREW algorithm ([DT94]). In this paper, we present a parallel algorithm which runs in optimal time and work bounds on the weaker EREW PRAM model. This resolves an open question posed in [DT94]. The highlevel structure of the algorithm has been adapted from [DT94] and [Kin95]. W...