A survey of the all-pairs shortest paths problem and its variants in graphs

Reddy, K. Rasool

doi:10.1515/ausi-2016-0002

Cited by 7 publications

(3 citation statements)

References 88 publications

(132 reference statements)

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“…In many cases, distances can be used to define cost functions. For instance, the problem of finding a minimum average distance (MAD) tree is one of the well-known problems in computer science (see the survey [32]). A MAD tree of a graph is defined as a spanning tree with minimum average distance or, equivalently, with the minimum Wiener index.…”

Section: Discussionmentioning

confidence: 99%

Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

Nadjafi-Arani¹,

Mirzargar²,

Emmert‐Streib

et al. 2020

Symmetry

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If G is a graph and P is a partition of V(G), then the partition distance of G is the sum of the distances between all pairs of vertices that lie in the same part of P. A colored distance is the dual concept of the partition distance. These notions are motivated by a problem in the facility location network and applied to several well-known distance-based graph invariants. In this paper, we apply an extended cut method to induce the partition and color distances to some subsets of vertices which are not necessary a partition of V(G). Then, we define a two-dimensional weighted graph and an operator to prove that the induced partition and colored distances of a graph can be obtained from the weighted Wiener index of a two-dimensional weighted quotient graph induced by the transitive closure of the Djoković–Winkler relation as well as by any partition that is coarser. Finally, we utilize our main results to find some upper bounds for the modified Wiener index and the number of orbits of partial cube graphs under the action of automorphism group of graphs.

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Section: Discussionmentioning

confidence: 99%

Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

Nadjafi-Arani¹,

Mirzargar²,

Emmert‐Streib

et al. 2020

Symmetry

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“…Another approach is to use new-generation hardware, such as the GPU, to parallelize existing algorithms and reduce runtime [48]. According to Reddy (2016) [49], most algorithms are based on the following two types of computational models:…”

Section: All Pairs Shortest Path Problem (Apsp)mentioning

confidence: 99%

Algorithm for the Accelerated Calculation of Conceptual Distances in Large Knowledge Graphs

Quintero,

Mendiola,

Guzmán

et al. 2023

Mathematics

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Conceptual distance refers to the degree of proximity between two concepts within a conceptualization. It is closely related to semantic similarity and relationships, but its measurement strongly depends on the context of the given concepts. DIS-C represents an advancement in the computation of semantic similarity/relationships that is independent of the type of knowledge structure and semantic relations when generating a graph from a knowledge base (ontologies, semantic networks, and hierarchies, among others). This approach determines the semantic similarity between two indirectly connected concepts in an ontology by propagating local distances by applying an algorithm based on the All Pairs Shortest Path (APSP) problem. This process is implemented for each pair of concepts to establish the most effective and efficient paths to connect these concepts. The algorithm identifies the shortest path between concepts, which allows for an inference of the most relevant relationships between them. However, one of the critical issues with this process is computational complexity, combined with the design of APSP algorithms, such as Dijkstra, which is 𝒪n3. This paper studies different alternatives to improve the DIS-C approach by adapting approximation algorithms, focusing on Dijkstra, pruned Dijkstra, and sketch-based methods, to compute the conceptual distance according to the need to scale DIS-C to analyze very large graphs; therefore, reducing the related computational complexity is critical. Tests were performed using different datasets to calculate the conceptual distance when using the original version of DIS-C and when using the influence area of nodes. In situations where time optimization is necessary for generating results, using the original DIS-C model is not the optimal method. Therefore, we propose a simplified version of DIS-C to calculate conceptual distances based on centrality estimation. The obtained results for the simple version of DIS-C indicated that the processing time decreased 2.381 times when compared to the original DIS-C version. Additionally, for both versions of DIS-C (normal and simple), the APSP algorithm decreased the computational cost when using a two-hop coverage-based approach.

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“…The survey by [12] reviews the shortest path algorithms on static graphs that produce exact results for the APSP problems for both weighted and unweighted graphs as well as dense and sparse graphs. He also represented some studies on APSP for restricted families of graphs, such as interval graphs that determine an interval for each vertex with its neighbours.…”

Section: Introductionmentioning

confidence: 99%

A New Approach to Measuring Distances in Dense Graphs

Almulhim

Thwaites

Taylor

2019

Machine Learning, Optimization, and Data Science

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The problem of computing distances and shortest paths between vertices in graphs is one of the fundamental issues in graph theory. It is of great importance in many different applications, for example, transportation, and social network analysis. However, efficient shortest distance algorithms are still desired in many disciplines. Basically, the majority of dense graphs have ties between the shortest distances. Therefore, we consider a different approach and introduce a new measure to solve all-pairs shortest paths for undirected and unweighted graphs. This measures the shortest distance between any two vertices by considering the length and the number of all possible paths between them. The main aim of this new approach is to break the ties between equal shortest paths SP, which can be obtained by the Breadth-first search algorithm (BFS), and distinguish meaningfully between these equal distances. Moreover, using the new measure in clustering produces higher quality results compared with SP. In our study, we apply two different clustering techniques: hierarchical clustering and K-means clustering, with four different graph models, and for a various number of clusters. We compare the results using a modularity function to check the quality of our clustering results.

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A survey of the all-pairs shortest paths problem and its variants in graphs

Cited by 7 publications

References 88 publications

Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

Algorithm for the Accelerated Calculation of Conceptual Distances in Large Knowledge Graphs

A New Approach to Measuring Distances in Dense Graphs

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