Recursively Divided Pancake Graphs with a Small Network Cost

Jung-Hyun Seo,Hyeong-Ok Lee

doi:10.3390/sym13050844

Abstract

Graphs are often used as models to solve problems in computer science, mathematics, and biology. A pancake sorting problem is modeled using a pancake graph whose classes include burnt pancake graphs, signed permutation graphs, and restricted pancake graphs. The network cost is degree × diameter. Finding a graph with a small network cost is like finding a good sorting algorithm. We propose a novel recursively divided pancake (RDP) graph that has a smaller network cost than other pancake-like graphs. In the pancake graph Pn, the number of nodes is n!, the degree is n − 1, and the network cost is O(n2). In an RDPn, the number of nodes is n!, the degree is 2log2n − 1, and the network cost is O(n(log2n)3). Because O(n(log2n)3) < O(n2), the RDP is superior to other pancake-like graphs. In this paper, we propose an RDPn and analyze its basic topological properties. Second, we show that the RDPn is recursive and symmetric. Third, a sorting algorithm is proposed, and the degree and diameter are derived. Finally, the network cost is compared between the RDP graph and other classes of pancake graphs.

Highlights

We propose an recursively divided pancake (RDP) that reduces the degree to O(log2 n) and the network cost to O(log2 n3 ) by using recursively divided edges
Pancake-like graphs are used in various fields wherein network cost is an important evaluation measure
We have provided an RDP that has a smaller network cost than existing pancake-like graphs

Summary

Introduction

The pancake graphs originated with pancake sorting problems. An n-pancake sorting problem refers to the task of sorting n pancakes of varied sizes piled on a stack. The number of flips in order to complete the sort in the worst time complexity is 2n − 3. The n-pancake sorting problem can be represented by the pancake graph Pn. The permutations are mapped to the nodes and flips to the edges. When a sorting problem is expressed as a graph, the number of main operations is mapped to the degree and the worst time complexity to the diameter. The degree of a pancake graph is n − 1, and the network cost is O(n2 ). W. Gates proved that the lower bound for the number of flips is 1.5n and the upper bound is 2n + 3 in the sorting of n burnt pancakes [2].

Preliminaries and Definition of Recursively Divided Pancake

Topological Properties

Sorting Algorithm and Network Cost Comparison

Conclusions