Minimal spanning trees and partial sorting

Brennan, J. J.

doi:10.1016/0167-6377(82)90010-4

Cited by 9 publications

(3 citation statements)

References 10 publications

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“…J. Brennan (1982) presented a modification to Kruskal's classic minimum spanning tree (MST) algorithm (J.B. Kruskal, 1956) that operated similar in a manner to quicksort; splitting an edge set into 'light' and 'heavy' subsets. Recently, Osipov (Vitaly, 2009) further expanded this idea by adding a multi-core friendly filtering step designed to eliminate edges that were obviously not in the MST (Filter-Kruskal).…”

Section: Literature Reviewmentioning

confidence: 99%

Development of Gis Tool for the Solution of Minimum Spanning Tree Problem using Prim's Algorithm

Dutta¹,

Patra²,

Shankar

et al. 2014

Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.

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ABSTRACT:A minimum spanning tree (MST) of a connected, undirected and weighted network is a tree of that network consisting of all its nodes and the sum of weights of all its edges is minimum among all such possible spanning trees of the same network. In this study, we have developed a new GIS tool using most commonly known rudimentary algorithm called Prim's algorithm to construct the minimum spanning tree of a connected, undirected and weighted road network. This algorithm is based on the weight (adjacency) matrix of a weighted network and helps to solve complex network MST problem easily, efficiently and effectively. The selection of the appropriate algorithm is very essential otherwise it will be very hard to get an optimal result. In case of Road Transportation Network, it is very essential to find the optimal results by considering all the necessary points based on cost factor (time or distance). This paper is based on solving the Minimum Spanning Tree (MST) problem of a road network by finding it's minimum span by considering all the important network junction point. GIS technology is usually used to solve the network related problems like the optimal path problem, travelling salesman problem, vehicle routing problems, location-allocation problems etc. Therefore, in this study we have developed a customized GIS tool using Python script in ArcGIS software for the solution of MST problem for a Road Transportation Network of Dehradun city by considering distance and time as the impedance (cost) factors. It has a number of advantages like the users do not need a greater knowledge of the subject as the tool is user-friendly and that allows to access information varied and adapted the needs of the users. This GIS tool for MST can be applied for a nationwide plan called Prime Minister Gram Sadak Yojana in India to provide optimal all weather road connectivity to unconnected villages (points). This tool is also useful for constructing highways or railways spanning several cities optimally or connecting all cities with minimum total road length.

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Section: Literature Reviewmentioning

confidence: 99%

Development of Gis Tool for the Solution of Minimum Spanning Tree Problem using Prim's Algorithm

Dutta¹,

Patra²,

Shankar

et al. 2014

Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.

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“…(If b is small relative to |FS(i)|, then we could instead perform a partial sorting to find the (b + 1)-smallest values of v * (k−m, j) + c ij , ∀j ∈ FS(i). This method could be performed in O(|FS(i)| + b log b) time [6,17].) The overall complexity is given by…”

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confidence: 99%

Dynamic shortest‐path interdiction

Sefair

Smith²

2016

Networks

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We study a dynamic network game between an attacker and a user. The user wishes to find a shortest path between a pair of nodes in a directed network, and the attacker seeks to interdict a subset of arcs to maximize the user's shortest-path cost. In contrast to most previous studies, the attacker can interdict arcs any time the user reaches a node in the network, and the user can respond by dynamically altering its chosen path. We assume that the attacker can interdict a limited number of arcs, and that an interdicted arc can still be traversed by the user at an increased cost. The challenge is therefore to find an optimal path (possibly repeating arcs in the network), coupled with the attacker's optimal interdiction strategy (i.e., which arcs to interdict and when to interdict them). We propose an exact exponential-state dynamic-programming algorithm for this problem, which can be reduced to a polynomial-time algorithm in the case of acyclic networks. We also develop lower and upper bounds on the optimal objective function value based on classical interdiction and robust optimization models, or based on an exact solution to variations of this problem. We examine the efficiency of our algorithms and the quality of our bounds on a set of randomly generated instances.

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“…Brennan [7] presented a modification to Kruskal's classic minimum spanning tree (MST) algorithm [24] that operated in a manner similar to quicksort; splitting an edge set into "light" and "heavy" subsets. Recently, Osipov et al [32] further expanded this idea by adding a multi-core friendly filtering step designed to eliminate edges that were obviously not in the MST (Filter-Kruskal).…”

Section: Introductionmentioning

confidence: 99%

Geometric Minimum Spanning Trees with GeoFilterKruskal

Chatterjee

Connor

Kumar

2010

Experimental Algorithms

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Abstract. Let P be a set of points in R d . We propose GEOFILTERKRUSKAL, an algorithm that computes the minimum spanning tree of P using well separated pair decomposition in combination with a simple modification of Kruskal's algorithm. When P is sampled from uniform random distribution, we show that our algorithm takes one parallel sort plus a linear number of additional steps, with high probability, to compute the minimum spanning tree. Experiments show that our algorithm works better in practice for most data distributions compared to the current state of the art [31]. Our algorithm is easy to parallelize and to our knowledge, is currently the best practical algorithm on multi-core machines for d > 2.

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Minimal spanning trees and partial sorting

Cited by 9 publications

References 10 publications

Development of Gis Tool for the Solution of Minimum Spanning Tree Problem using Prim's Algorithm

Development of Gis Tool for the Solution of Minimum Spanning Tree Problem using Prim's Algorithm

Dynamic shortest‐path interdiction

Geometric Minimum Spanning Trees with GeoFilterKruskal

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