NC ALGORITHMS FOR THE SINGLE MOST VITAL EDGE PROBLEM WITH RESPECT TO ALL PAIRS SHORTEST PATHS

For a weighted, undirected graph G=(V, E) where |V|=n and |E|=m, we examine the single most vital edge with respect to all-pairs shortest paths (APSP) under two different measurements. The first measurement considers only the impact of the removal of a single edge from the APSP on the shortest distance between each vertex pair. The second considers the total weight of all the edges which make up the APSP, that is, calculate the sum of the distance between each vertex pair after the deletion of any edge belonging to a shortest path. We give a sequential algorithm for this problem, and show how to obtain an NC algorithm running in O( log n) time using mn2 processors and O(mn2) space on the MINIMUM CRCW PRAM. Given the shortest distance between each pair of vertices u and v, the diameter of the graph is defined as the longest of these distances. The Most vital edge with respect to the diameter is the edge lying on such a u–v shortest path which when removed causes the greatest increase in the diameter. We show how to modify the above algorithm to solve this problem using the same time and number of processors. Both algorithms compare favourably with the straightforward solution which simply recalculates the all pairs shortest path information.

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Resilience of Urban Street Network Configurations under Low Demands

Transportation Research Record Journal of the Transportation Research Board ◽

10.1177/0361198120933269 ◽

2020 ◽

Vol 2674 (9) ◽

pp. 982-994

Author(s):

Zhengyao Yu ◽

Vikash V. Gayah

Keyword(s):

Routing Algorithm ◽

Shortest Paths ◽

Left Turn ◽

Urban Street ◽

Traffic Conditions ◽

Street Networks ◽

Shortest Distance ◽

Left Turns ◽

The Impact ◽

Travel Distances

Urban street networks are subject to a variety of random disruptions. The impact of movement restrictions (e.g., one-way or left-turn restrictions) on the ability of a network to overcome these disruptions—that is, its resilience—has not been thoroughly studied. To address this gap, this paper investigates the resilience of one-way and two-way square grid street networks with and without left turns under light traffic conditions. Networks are studied using a simplified routing algorithm that can be examined analytically and a microsimulation that describes detailed vehicle dynamics. In the simplified method, routing choices are enumerated for all possible origin–destination (OD) combinations to identify how the removal of a link affects operations, both when knowledge of the disruption is and is not available at the vehicle’s origin. Disruptions on two-way networks that allow left turns tend to have little impact on travel distances because of the availability of multiple shortest paths between OD pairs and the flexibility in route modification. Two-way networks that restrict left turns at intersections only have a single shortest-distance path between any OD pair and thus experience larger increases in travel distance, even when the disruption is known ahead of time. One-way networks sometimes have multiple shortest-distance routes and thus travel distances increase less than two-way network without left turns when links are disrupted. These results reveal a clear tradeoff between improved efficiency and reduced resilience for networks that have movement restrictions, and can be used as a basis to study network resilience under more congested scenarios and in more realistic network structures.

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PENENTUAN LINTASAN TERPENDEK DARI FMIPA KE REKTORAT DAN FAKULTAS LAIN DI UNSRAT MANADO MENGGUNAKAN ALGORITMA DJIKSTRA

JURNAL ILMIAH SAINS ◽

10.35799/jis.11.1.2011.46 ◽

2011 ◽

Vol 11 (1) ◽

pp. 73

Author(s):

Deiby T. Salaki

Keyword(s):

Shortest Path ◽

Local Minimum ◽

Shortest Paths ◽

Weighted Graph ◽

Initial Vertex ◽

North Sulawesi ◽

Shortest Distance ◽

Highway Facilities

Universitas Sam Ratulangi Manado adalah salah satu perguruan tinggi di Sulawesi Utara yang terdiri atas 11 fakultas dan satu gedung rektorat. Setiap fakultas dan rektorat terhubung dengan fasilitas jalan raya. Secara matematis kondisi seperti ini dapat direpresentasikan sebagai sebuah graf yang bisa diterapkan untuk mencari lintasan terpendek. Pada penelitian ini akan dicari lintasan terpendek dari FMIPA ke rektorat dan fakultas lainnya. Dengan menggunakan algoritma Djikstra, lintasan terpendek dari FMIPA diperoleh dengan memilih minimum lokal atau akses dengan jarak terdekat dari setiap lokasi yang kemudian digabungkan menjadi sebuah kumpulan lintasan dari satu lokasi ke lokasi lainnya dengan jarak terpendek. DETERMINATION OF SHORTEST PATH FROM FMIPA TO RECTORATE AND OTHER FACULTIES AT SAM RATULANGI UNIVERSITY USING DJIKSTRA ALGORITHMABSTRACTSam Ratulangi University is one of the colleges in North Sulawesi consisting of 11 faculties and one rectorate building. Every faculty and rectorate connected by highway facilities. Mathemathically this condition can be represented as an undirected weighted graph that can be applied to find the shortest path. By using the Djikstra algorithm, the shortest paths are obtained by setting the FMIPA as the initial vertex and then select the local minimum or access to the closest distance from each location, then combined the collection of path from one location to another with the shortest distance.

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An all-pairs shortest path algorithm for bipartite graphs

Open Computer Science ◽

10.2478/s13537-013-0110-4 ◽

2013 ◽

Vol 3 (4) ◽

Author(s):

Svetlana Torgasin ◽

Karl-Heinz Zimmermann

Keyword(s):

Shortest Path ◽

Time Complexity ◽

Shortest Paths ◽

Distance Matrix ◽

Bipartite Graphs ◽

Matrix Product ◽

Complex Structures ◽

Shortest Path Algorithm ◽

Order Of Magnitude ◽

All Pairs Shortest Path

AbstractBipartite graphs are widely used for modeling of complex structures in biology, engineering, and computer science. The search for shortest paths in such structures is a highly demanded procedure that requires optimization. This paper presents a variant of the all-pairs shortest path algorithm for bipartite graphs. The method is based on the distance matrix product and improves the general algorithm by exploiting the graph topology. The space complexity is reduced by a factor of at least four and the time complexity decreased by almost an order of magnitude when compared with the basic APSP algorithm.

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A novel approach for solving all-pairs shortest path problem in an interval-valued fuzzy network: Suggested modifications

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-210182 ◽

2021 ◽

pp. 1-18

Author(s):

Tanveen Kaur Bhatia ◽

Amit Kumar ◽

S.S. Appadoo

Keyword(s):

Shortest Path ◽

Fuzzy Number ◽

Fuzzy Systems ◽

Shortest Path Problem ◽

Shortest Path Problems ◽

Novel Approach ◽

Shortest Distance ◽

All Pairs Shortest Path ◽

Fuzzy Network ◽

Interval Valued

Enayattabr et al. (Journal of Intelligent and Fuzzy Systems 37 (2019) 6865– 6877) claimed that till now no one has proposed an approach to solve interval-valued trapezoidal fuzzy all-pairs shortest path problems (all-pairs shortest path problems in which distance between every two nodes is represented by an interval-valued trapezoidal fuzzy number). Also, to fill this gap, Enayattabr et al. proposed an approach to solve interval-valued trapezoidal fuzzy all-pairs shortest path problems. In this paper, an interval-valued trapezoidal fuzzy shortest path problem is considered to point out that Enayattabr et al.’s approach fails to find correct shortest distance between two fixed nodes. Hence, it is inappropriate to use Enayattabr et al.’s approach in its present from. Also, the required modifications are suggested to resolve this inappropriateness of Enayattabr et al.’s approach.

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Computing nearest neighbour interchange distances between ranked phylogenetic trees

Journal of Mathematical Biology ◽

10.1007/s00285-021-01567-5 ◽

2021 ◽

Vol 82 (1-2) ◽

Author(s):

Lena Collienne ◽

Alex Gavryushkin

Keyword(s):

Cancer Research ◽

Computational Complexity ◽

Phylogenetic Tree ◽

Shortest Path ◽

Phylogenetic Trees ◽

Shortest Paths ◽

Nearest Neighbour ◽

Tree Inference ◽

Subtree Prune And Regraft ◽

Comparison Algorithms

AbstractMany popular algorithms for searching the space of leaf-labelled (phylogenetic) trees are based on tree rearrangement operations. Under any such operation, the problem is reduced to searching a graph where vertices are trees and (undirected) edges are given by pairs of trees connected by one rearrangement operation (sometimes called a move). Most popular are the classical nearest neighbour interchange, subtree prune and regraft, and tree bisection and reconnection moves. The problem of computing distances, however, is $${\mathbf {N}}{\mathbf {P}}$$ N P -hard in each of these graphs, making tree inference and comparison algorithms challenging to design in practice. Although anked phylogenetic trees are one of the central objects of interest in applications such as cancer research, immunology, and epidemiology, the computational complexity of the shortest path problem for these trees remained unsolved for decades. In this paper, we settle this problem for the ranked nearest neighbour interchange operation by establishing that the complexity depends on the weight difference between the two types of tree rearrangements (rank moves and edge moves), and varies from quadratic, which is the lowest possible complexity for this problem, to $${\mathbf {N}}{\mathbf {P}}$$ N P -hard, which is the highest. In particular, our result provides the first example of a phylogenetic tree rearrangement operation for which shortest paths, and hence the distance, can be computed efficiently. Specifically, our algorithm scales to trees with tens of thousands of leaves (and likely hundreds of thousands if implemented efficiently).

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Dynamic Shortest Paths Methods for the Time-Dependent TSP

Algorithms ◽

10.3390/a14010021 ◽

2021 ◽

Vol 14 (1) ◽

pp. 21

Author(s):

Christoph Hansknecht ◽

Imke Joormann ◽

Sebastian Stiller

Keyword(s):

Column Generation ◽

Traveling Salesman Problem ◽

Shortest Path ◽

Valid Inequalities ◽

Shortest Paths ◽

Traveling Salesman ◽

Time Dependent ◽

Full Generality ◽

Branching Rule ◽

The Traveling Salesman Problem

The time-dependent traveling salesman problem (TDTSP) asks for a shortest Hamiltonian tour in a directed graph where (asymmetric) arc-costs depend on the time the arc is entered. With traffic data abundantly available, methods to optimize routes with respect to time-dependent travel times are widely desired. This holds in particular for the traveling salesman problem, which is a corner stone of logistic planning. In this paper, we devise column-generation-based IP methods to solve the TDTSP in full generality, both for arc- and path-based formulations. The algorithmic key is a time-dependent shortest path problem, which arises from the pricing problem of the column generation and is of independent interest—namely, to find paths in a time-expanded graph that are acyclic in the underlying (non-expanded) graph. As this problem is computationally too costly, we price over the set of paths that contain no cycles of length k. In addition, we devise—tailored for the TDTSP—several families of valid inequalities, primal heuristics, a propagation method, and a branching rule. Combining these with the time-dependent shortest path pricing we provide—to our knowledge—the first elaborate method to solve the TDTSP in general and with fully general time-dependence. We also provide for results on complexity and approximability of the TDTSP. In computational experiments on randomly generated instances, we are able to solve the large majority of small instances (20 nodes) to optimality, while closing about two thirds of the remaining gap of the large instances (40 nodes) after one hour of computation.

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