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Distributed Verification of Minimum Spanning Trees
 Proc. 25th Annual Symposium on Principles of Distributed Computing
, 2006
"... The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in ..."
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Cited by 32 (23 self)
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The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows ” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (except when W ≤ log n). Both our bounds improve previously known bounds for the problem. Our techniques (both for the lower bound and for the upper bound) may indicate a strong relation between the fields of proof labeling schemes and implicit labeling schemes. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings.
A randomized timework optimal parallel algorithm for finding a minimum spanning forest
 SIAM J. COMPUT
, 1999
"... We present a randomized algorithm to find a minimum spanning forest (MSF) in an undirected graph. With high probability, the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write (EREW) PRAM. This result is optimal w.r.t. both work and parallel time, and is the fi ..."
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Cited by 20 (3 self)
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We present a randomized algorithm to find a minimum spanning forest (MSF) in an undirected graph. With high probability, the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write (EREW) PRAM. This result is optimal w.r.t. both work and parallel time, and is the first provably optimal parallel algorithm for this problem under both measures. We also give a simple, general processor allocation scheme for treelike computations.
Fast Minimum Spanning Tree for Large Graphs on the GPU
"... Graphics Processor Units are used for many general purpose processing due to high compute power available on them. Regular, dataparallel algorithms map well to the SIMD architecture of currentGPU.Irregularalgorithmsondiscretestructureslikegraphsare harder to map to them. Efficient datamapping prim ..."
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Cited by 14 (4 self)
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Graphics Processor Units are used for many general purpose processing due to high compute power available on them. Regular, dataparallel algorithms map well to the SIMD architecture of currentGPU.Irregularalgorithmsondiscretestructureslikegraphsare harder to map to them. Efficient datamapping primitives can play crucialroleinmappingsuchalgorithmsontotheGPU.Inthispaper, we present a minimum spanning tree algorithm on Nvidia GPUs underCUDA,asarecursiveformulationofBor˚uvka’sapproachfor undirected graphs. We implement it using scalable primitives such as scan, segmented scan and split. The irregular steps of supervertexformationandrecursivegraphconstructionaremappedtoprimitives like split to categories involving vertex ids and edge weights. We obtain 30 to 50 times speedup over the CPU implementation on most graphs and 3 to 10 times speedup over our previous GPU implementation. We construct the minimum spanning tree on a 5 million node and 30 million edge graph in under 1 second on one quarter of the TeslaS1070GPU.
Optimal randomized EREW PRAM algorithms for finding spanning forests
 J. Algorithms
, 2000
"... We present the first randomized O(log n) time and O(m+n) work EREW PRAM algorithm for finding a spanning forest of an undirected graph G = (V; E) with n vertices and m edges. Our algorithm is optimal with respect to time, work and space. As a consequence we get optimal randomized EREW PRAM algori ..."
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Cited by 14 (1 self)
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We present the first randomized O(log n) time and O(m+n) work EREW PRAM algorithm for finding a spanning forest of an undirected graph G = (V; E) with n vertices and m edges. Our algorithm is optimal with respect to time, work and space. As a consequence we get optimal randomized EREW PRAM algorithms for other basic connectivity problems such as finding a bipartite partition, finding bridges and biconnected components, finding Euler tours in Eulerian graphs, finding an ear decomposition, finding an open ear decomposition, finding a strong orientation, and finding an stnumbering.
CommunicationOptimal Parallel Minimum Spanning Tree Algorithms
, 1998
"... Lower and upper bounds for finding a minimum spanning tree (MST) in a weighted undirected graph on the BSP model are presented. We provide the first nontrivial lower bounds on the communication volume required to solve the MST problem. Let p denote the number of processors, n the number of nodes of ..."
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Cited by 13 (1 self)
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Lower and upper bounds for finding a minimum spanning tree (MST) in a weighted undirected graph on the BSP model are presented. We provide the first nontrivial lower bounds on the communication volume required to solve the MST problem. Let p denote the number of processors, n the number of nodes of the input graph, and m the number of edges of the input graph. We show that in the worst case, a total of \Omega\Gamma \Delta min(m; pn)) bits need to be communicated in order to solve the MST problem, where is the number of bits required to represent a single edge weight. This implies that if each message communicates at most bits, any BSP algorithm for finding an MST requires communication time \Omega\Gamma g \Delta min(m=p; n)), where g is the gap parameter of the BSP model. In addition, we present two algorithms with communication requirements that match our lower bound in different situations. Both algorithms perform linear work for appropriate values of n, m and p, and use a numbe...
A Randomized Linear Work EREW PRAM Algorithm to Find a Minimum Spanning Forest
, 1997
"... We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an nvertex graph the algorithm runs in o((log n) 1+ffl ) expected time for any ffl ? 0 and performs linear expected work. This is the first linear work, polylog time algorithm on th ..."
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Cited by 13 (2 self)
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We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an nvertex graph the algorithm runs in o((log n) 1+ffl ) expected time for any ffl ? 0 and performs linear expected work. This is the first linear work, polylog time algorithm on the EREW PRAM for this problem. This also gives parallel algorithms that perform expected linear work on two more realistic models of parallel computation, the QSM and the BSP. 1 Introduction The design of efficient algorithms to find a minimum spanning forest (MSF) in a weighted undirected graph is a fundamental problem that has received much attention. There have been many algorithms designed for the MSF problem that run in close to linear time (see, e.g., [CLR91]). Recently a randomized lineartime algorithm for this problem was presented in [KKT95]. Based on this work [CKT94] presented a randomized parallel algorithm on the CRCW PRAM which runs in O(2 log n log n) expected time whil...
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 ..."
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Cited by 12 (7 self)
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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.
Randomized Minimum Spanning Tree Algorithms Using Exponentially Fewer Random Bits
"... For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel mi ..."
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Cited by 5 (1 self)
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For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set maxima problems. (For the first two problems there are provably optimal deterministic algorithms with unknown, and possibly superlinear running times.) One downside of the randomized methods for solving these problems is that they use a number of random bits linear in the size of the input. In this paper we develop some general methods for reducing exponentially the consumption of random bits in comparison based algorithms. In some cases we are able to reduce the number of random bits from linear to nearly constant without affecting the expected running time. Most of our results are obtained by adjusting or reorganizing existing randomized algorithms to work well with a pairwise or O(1)wise independent sampler. The prominent exception — and the main focus of this paper — is a lineartime randomized minimum spanning tree algorithm that is not derived from the well known KargerKleinTarjan algorithm. In many ways it resembles more closely the deterministic minimum spanning tree algorithms based on Soft Heaps. Further,