Sukumar Mondal

AN OPTIMAL ALGORITHM FOR SOLVING ALL-PAIRS SHORTEST PATHS ON TRAPEZOID GRAPHS

The shortest-paths problem is an important problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The Shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one node to another often gives the best way to route message between the nodes. This paper presents an O(n2) time algorithm for solving all pairs shortest path problems on trapezoid graphs which are extensions of interval graphs and permutation graphs. The space complexity of this algorithm is of O(n2). This problem has been solved by constructing n breadth-first search (BFS) trees with each of the n vertices as root. As the lower bound of time complexity for computing the all pairs shortest paths is known to be of O(n2), this proposed algorithm is optimal.

An optimal parallel algorithm for computing cut vertices and blocks on interval graphs

In this paper, a parallel algorithm is presented to find all cut-vertices and blocks of an interval graph. If the list of sorted end points of the intervals of an interval graph is given then the proposed algorithm takes O(log n) time and O(n/log n) processors on an EREW PRAM, if the sorted list is not given then the time and processors complexities are respectively O(log n) and O(n).

An Optimal Algorithm to Solve the All-Pairs Shortest Paths Problem on Permutation Graphs

In this paper we present an optimal algorithm to solve the all-pairs shortest path problem on permutation graphs with n vertices and m edges which runs in O(n2) time. Using this algorithm, the average distance of a permutation graph can also be computed in O(n2) time.

An Optimal Algorithm to Solve 2-Neighbourhood Covering Problem on Interval Graphs

Let G =( V , E ) be a simple graph and k be a fixed positive integer. A vertex w is said to be a k -neighbourhood-cover of an edge ( u , v ) if d ( u , w ) h k and d ( v , w ) h k . A set C ³ V is called a k -neighbourhood-covering set if every edge in E is k -neighbourhood-covered by some vertices of C . This problem is NP-complete for general graphs even it remains NP-complete for chordal graphs. Using dynamic programming technique, an O ( n ) time algorithm is designed to solve minimum 2-neighbourhood-covering problem on interval graphs. A data structure called interval tree is used to solve this problem.

An optimal algorithm for finding depth-first spanning tree on permutation graphs

LetG be a connected graph ofn vertices. The problem of finding a depth-first spanning tree ofG is to find a connected subgraph ofG with then vertices andn − 1 edges by depth-first-search. In this paper, we propose anO(n) time algorithm to solve this problem on permutation graphs.

Optimal Sequential And Parallel Algorithms To Compute A Steiner Tree On Permutation Graphs

This paper presents an optimal sequential and an optimal parallel algorithm to compute a minimum cardinality Steiner set and a Steiner tree. The sequential algorithm takes O ( n ) time and parallel algorithm takes O (log n ) time and O ( n /log n ) processors on an EREW PRAM model.

Interval-valued fuzzy \(\phi\) -tolerance competition graphs

This paper develops an interval-valued fuzzy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}

A linear time algorithm to construct a tree 4-spanner on trapezoid graphs

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