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An optimal minimum spanning tree algorithm
 J. ACM
, 2000
"... Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is ..."
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Cited by 46 (10 self)
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Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is the minimum number of edgeweight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine. Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T ∗ are T ∗ (m, n) = �(m) and T ∗ (m, n) = O(m · α(m, n)), where α is a certain natural inverse of Ackermann’s function. Even under the assumption that T ∗ is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edgeexposure martingale for Gn,p.
A Fast, Parallel Spanning Tree Algorithm for Symmetric Multiprocessors (SMPs) (Extended Abstract)
, 2004
"... Our study in this paper focuses on implementing parallel spanning tree algorithms on SMPs. Spanning tree is an important problem in the sense that it is the building block for many other parallel graph algorithms and also because it is representative of a large class of irregular combinatorial probl ..."
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Cited by 31 (11 self)
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Our study in this paper focuses on implementing parallel spanning tree algorithms on SMPs. Spanning tree is an important problem in the sense that it is the building block for many other parallel graph algorithms and also because it is representative of a large class of irregular combinatorial problems that have simple and efficient sequential implementations and fast PRAM algorithms, but often have no known efficient parallel implementations. In this paper we present a new randomized algorithm and implementation with superior performance that for the firsttime achieves parallel speedup on arbitrary graphs (both regular and irregular topologies) when compared with the best sequential implementation for finding a spanning tree. This new algorithm uses several techniques to give an expected running time that scales linearly with the number p of processors for suitably large inputs (n> p 2). As the spanning tree problem is notoriously hard for any parallel implementation to achieve reasonable speedup, our study may shed new light on implementing PRAM algorithms for sharedmemory parallel computers. The main results of this paper are 1. A new and practical spanning tree algorithm for symmetric multiprocessors that exhibits parallel speedups on graphs with regular and irregular topologies; and 2. An experimental study of parallel spanning tree algorithms that reveals the superior performance of our new approach compared with the previous algorithms. The source code for these algorithms is freelyavailable from our web site hpc.ece.unm.edu.
Fast SharedMemory Algorithms for Computing the Minimum Spanning Forest of Sparse Graphs
, 2006
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Concurrent Threads and Optimal Parallel Minimum Spanning Trees Algorithm
 J. ACM
, 2001
"... This paper resolves a longstanding open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in O(log n) time. Specically, we present a new algorithm to so ..."
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Cited by 16 (1 self)
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This paper resolves a longstanding open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in O(log n) time. Specically, we present a new algorithm to solve these problems in O(log n) time using a linear number of processors on the exclusiveread exclusivewrite PRAM. The logarithmic time bound is actually optimal since it is well known that even computing the \OR" of n bits
On the architectural requirements for efficient execution of graph algorithms
 In Proc. 34th Int’l Conf. on Parallel Processing (ICPP
, 2005
"... Combinatorial problems such as those from graph theory pose serious challenges for parallel machines due to noncontiguous, concurrent accesses to global data structures with low degrees of locality. The hierarchical memory systems of symmetric multiprocessor (SMP) clusters optimize for local, conti ..."
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Cited by 15 (7 self)
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Combinatorial problems such as those from graph theory pose serious challenges for parallel machines due to noncontiguous, concurrent accesses to global data structures with low degrees of locality. The hierarchical memory systems of symmetric multiprocessor (SMP) clusters optimize for local, contiguous memory accesses, and so are inefficient platforms for such algorithms. Few parallel graph algorithms outperform their best sequential implementation on SMP clusters due to long memory latencies and high synchronization costs. In this paper, we consider the performance and scalability of two graph algorithms, list ranking and connected components, on two classes of sharedmemory computers: symmetric multiprocessors such as the Sun Enterprise servers and multithreaded architectures
Minimizing Randomness in Minimum Spanning Tree, Parallel Connectivity, and Set Maxima Algorithms
 In Proc. 13th Annual ACMSIAM Symposium on Discrete Algorithms (SODA'02
, 2001
"... There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing conne ..."
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Cited by 7 (4 self)
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There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing connected components and (minimum) spanning trees in parallel, and the problem of performing sensitivity analysis on shortest path trees and minimum spanning trees. However, while each of these problems has a randomized algorithm whose performance meets a known lower bound, all of these randomized algorithms use a number of random bits which is linear in the number of operations they perform. We address the issue of reducing the number of random bits used in these randomized algorithms. For each of the problems listed above, we present randomized algorithms that have optimal performance but use only a polylogarithmic number of random bits; for some of the problems our optimal algorithms use only log n random bits. Our results represent an exponential savings in the amount of randomness used to achieve the same optimal performance as in the earlier algorithms. Our techniques are general and could likely be applied to other problems.
The Euler tour technique and parallel rooted spanning tree
 In Proc. Int’l Conf. on Parallel Processing (ICPP
, 2004
"... Many parallel algorithms for graph problems start with finding a spanning tree and rooting the tree to define some structural relationship on the vertices which can be used by following problem specific computations. The generic procedure is to find an unrooted spanning tree and then root the spanni ..."
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Cited by 5 (4 self)
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Many parallel algorithms for graph problems start with finding a spanning tree and rooting the tree to define some structural relationship on the vertices which can be used by following problem specific computations. The generic procedure is to find an unrooted spanning tree and then root the spanning tree using the Euler tour technique. With a randomized worktime optimal unrooted spanning tree algorithm and worktime optimal list ranking, finding rooted spanning trees can be done worktime optimally on EREW PRAM w.h.p. Yet the Euler tour technique assumes as “given ” a circular adjacency list, it is not without implications though to construct the circular adjacency list for the spanning tree found on the fly by a spanning tree algorithm. In fact our experiments show that this “hidden ” step of constructing a circular adjacency list could take as much time as both spanning tree and list ranking combined. In this paper we present new efficient algorithms that find rooted spanning trees without using the Euler tour technique and incur little or no overhead over the underlying spanning tree algorithms. We also present two new approaches that construct Euler tours efficiently when the circular adjacency list is not given. One is a deterministic PRAM algorithm and the other is a randomized algorithm in the symmetric multiprocessor (SMP) model. The randomized algorithm takes a novel approach for the problems of constructing the Euler tour and rooting a tree. It computes a rooted spanning tree first, then constructs an Euler tour directly for the tree using depthfirst traversal. The tour constructed is cachefriendly with adjacent edges in the
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 1 (0 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,
CLUSTERING
"... Abstract—Clustering of data has numerous applications and has been studied extensively. Though most of the algorithms in the literature are sequential, many parallel algorithms have also been designed. In this paper, we present parallel algorithms with better performance than known algorithms. We co ..."
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Abstract—Clustering of data has numerous applications and has been studied extensively. Though most of the algorithms in the literature are sequential, many parallel algorithms have also been designed. In this paper, we present parallel algorithms with better performance than known algorithms. We consider algorithms that work well in the worst case as well as algorithms with good expected performance. Index Terms—Reconfigurable networks, meshconnected computers, meshes with optical buses, hierarchical clustering, PRAMs, singlelink metric. æ