Constructing Independent Spanning Trees on Pancake Networks

Dun-Wei Cheng,Sun-Yuan Hsieh,Chih-Te Chan

doi:10.1109/access.2019.2962549

Dun-Wei Cheng, Sun-Yuan Hsieh + Show 1 more

Open Access

https://doi.org/10.1109/access.2019.2962549

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Abstract

For any graph G, the set of independent spanning trees (ISTs) is defined as the set of spanning trees in G. All ISTs have the same root, paths from the root to another vertex between distinct trees are vertex-disjoint and edge-disjoint. The construction of multiple independent trees on a graph has numerous applications, such as fault-tolerant broadcasting and secure message distribution. The pancake graph is a subclass of Cayley graphs and since Cayley graphs are crucial for designing interconnection networks, constructing ISTs on these graphs is necessary for many practical applications. In this paper, we propose algorithms for constructing ISTs on pancake graph. Examine the use of our algorithm for constructing ISTs on pancake graph in different dimensions. We also present proofs about the construction of ISTs on pancake graph to verify that the correctness of these algorithms.

Highlights

In graph theory, the pancake graph is a type of Cayley graphs that encodes a certain abstract structure of an element group into a graph
The graph’s edges are given between permutations transitive by prefix reversals; this means that when two permutations of n elements are generated through prefix reversals, there exists an edge that is connected between these two vertices (e.g., an edge exists between (1, 2, 3, 4) and (3, 2, 1, 4) because the prefix of three elements is a reversal)
We prove some of the rules of the aforementioned algorithms to ensure that we can construct independent spanning trees (ISTs) on the n-pancake graph (P(n))

Summary

INTRODUCTION

The pancake graph is a type of Cayley graphs that encodes a certain abstract structure of an element group into a graph. The pancake graph can be regarded as a model of interconnecting networks for a parallel computing structure [1], [3], [4], [14]. In a given interconnecting network, the ISTs are vertex-disjoint and edge-disjoint. Cheng et al.: Constructing ISTs on Pancake Networks are no longer operating properly, the system can use a different spanning tree structure to broadcast and continue to complete jobs. The conjecture has yet to be confirmed for k greater than 5 Researchers have shifted their attention to methods for constructing ISTs on restricted interconnected networks [5], [6]. The greatest number of ISTs in P(n) is n − 1

PRELIMINARIES

PROOF REGARDING ALGORITHMS

CONCLUSION