On the History of the Minimum Spanning Tree Problem

Graham, Ronald L.; Hell, Pavol

doi:10.1109/mahc.1985.10011

Cited by 661 publications

(344 citation statements)

References 72 publications

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“…It has important applications in computer and communication networks, playing a major role in network reliability, classification and routing [1]. In many application domains, such as…”

Section: Introductionmentioning

confidence: 99%

Hardware and Software Implementations of Prim’s Algorithm for Efficient Minimum Spanning Tree Computation

Mariano

Lee

Gerstlauer

et al. 2013

IFIP Advances in Information and Communication Technology

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Abstract. Minimum spanning tree (MST) problems play an important role in many networking applications, such as routing and network planning. In many cases, such as wireless ad-hoc networks, this requires efficient high-performance and low-power implementations that can run at regular intervals in real time on embedded platforms. In this paper, we study custom software and hardware realizations of one common algorithm for MST computations, Prim's algorithm. We specifically investigate a performance-optimized realization of this algorithm on reconfigurable hardware, which is increasingly present in such platforms. Prim's algorithm is based on graph traversals, which are inherently hard to parallelize. We study two algorithmic variants and compare their performance against implementations on desktop-class and embedded CPUs. Results show that the raw execution time of an optimized implementation of Prim's algorithm on a Spartan-class Xilinx FPGA running at 24 MHz and utilizing less than 2.5% of its logic resources is 20% faster than an embedded ARM9 processor. When scaled to moderate clock frequencies of 150 and 250 MHz in more advanced FPGA technology, speedups of 7x and 12x are possible (at 56% and 94% of the ARM9 clock frequency, respectively).

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“…It has important applications in computer and communication networks, playing a major role in network reliability, classification and routing [1]. In many application domains, such as…”

Section: Introductionmentioning

confidence: 99%

Hardware and Software Implementations of Prim’s Algorithm for Efficient Minimum Spanning Tree Computation

Mariano

Lee

Gerstlauer

et al. 2013

IFIP Advances in Information and Communication Technology

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“…Maybe the two most widely known algorithms to compute the MST are Prim's and Kruskal's [1]. There is a third classical algorithm by Boruvka [1] that mysteriously remained shadowed by the other two.…”

Section: Introductionmentioning

confidence: 99%

“…There is a third classical algorithm by Boruvka [1] that mysteriously remained shadowed by the other two. This fact is emphasized by the fact that Boruvka's algorithm is also known as Sollin's algorithm, despite the fact that Sollin re-discovered it independently years later.…”

Section: Introductionmentioning

confidence: 99%

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Boruvka Meets Nearest Neighbors

Tepper

Musé

Almansa

et al. 2013

Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications

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Abstract.Computing the minimum spanning tree (MST) is a common task in the pattern recognition and the computer vision fields. However, little work has been done on efficient general methods for solving the problem on large datasets where graphs are complete and edge weights are given implicitly by a distance between vertex attributes. In this work we propose a generic algorithm that extends the classical Boruvka's algorithm by using nearest neighbors search structures to significantly reduce time and memory consumption. The algorithm can also compute in a straightforward way approximate MSTs thus further improving speed. Experiments show that the proposed method outperforms classical algorithms on large low-dimensional datasets by several orders of magnitude.

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“…Finding the MST of a graph has a long, distinguished history [3,10,12]. Currently the best known deterministic algorithm of Chazelle [2] runs in O(mα(m, n)) time, where n (resp.…”

Section: Introductionmentioning

confidence: 99%

Approximating the Minimum Spanning Tree Weight in Sublinear Time

Chazelle

Rubinfeld²,

Trevisan

2001

Automata, Languages and Programming

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We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1, . . . , w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε −2 log dw ε ) the weight of the minimum spanning tree of G with a relative error of at most ε. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dwε −2 ) on the probe and time complexity of any approximation algorithm for MST weight.The essential component of our algorithm is a procedure for estimating in time O(dε −2 log d ε ) the number of connected components of an unweighted graph to within an additive error of εn. (This becomes O(ε −2 log 1 ε ) for d = O(1).) The time bound is shown to be tight up to within the log d ε factor. Our connected-components algorithm picks O(1/ε 2 ) vertices in the graph and then grows "local spanning trees" whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST.

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On the History of the Minimum Spanning Tree Problem

Cited by 661 publications

References 72 publications

Hardware and Software Implementations of Prim’s Algorithm for Efficient Minimum Spanning Tree Computation

Hardware and Software Implementations of Prim’s Algorithm for Efficient Minimum Spanning Tree Computation

Boruvka Meets Nearest Neighbors

Approximating the Minimum Spanning Tree Weight in Sublinear Time

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