Pancake problems with restricted prefix reversals and some corresponding Cayley networks

Bass, Douglas W.; Sudborough, Ivan Hal

doi:10.1016/s0743-7315(03)00033-9

Cited by 14 publications

(5 citation statements)

References 15 publications

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“…The pancake flipping problem [1][2][3][4][5] deals with finding the minimum number of prefix reversals (i.e., flips) required to sort a given permutation. This problem was first introduced in 1975 by [1] which describes the motivation of a chef to rearrange a stack of pancakes from the smallest pancake on the top to the largest one on the bottom by grabbing several pancakes from the top with his spatula and flipping them over, repeating them as many times as necessary.…”

Section: Introductionmentioning

confidence: 99%

“…The diameter of a network is the maximum distance between any pair of nodes in the network and corresponds to the worst communication delay for broadcasting messages in the network [4,5]. A well studied variation of pancake flipping problem is the burnt pancake flipping problem [2,4,5] where each element in the permutation has a sign, and the sign of an element changes with reversals. Pancake and burnt pancake networks have better diameter and better vertex degree than the popular hypercubes [4].…”

Section: Introductionmentioning

confidence: 99%

“…Pancake and burnt pancake networks have better diameter and better vertex degree than the popular hypercubes [4]. There exists some other variations of pancake flipping, giving different efficient interconnection networks [2].…”

Section: Introductionmentioning

confidence: 99%

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Pancake Flipping with Two Spatulas

Sharmin

Yeasmin

Hasan

et al. 2010

Electronic Notes in Discrete Mathematics

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Abstract. In this paper we study several variations of the pancake flipping problem, which is also well known as the problem of sorting by prefix reversals. We consider the variations in the sorting process by adding with prefix reversals other similar operations such as prefix transpositions and prefix transreversals. These type of sorting problems have applications in interconnection networks and computational biology. We first study the problem of sorting unsigned permutations by prefix reversals and prefix transpositions and present a 3-approximation algorithm for this problem. Then we give a 2-approximation algorithm for sorting by prefix reversals and prefix transreversals. We also provide a 3-approximation algorithm for sorting by prefix reversals and prefix transpositions where the operations are always applied at the unsorted suffix of the permutation. We further analyze the problem in more practical way and show quantitatively how approximation ratios of our algorithms improve with the increase of number of prefix reversals applied by optimal algorithms. Finally, we present experimental results to support our analysis.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pancake Flipping with Two Spatulas

Sharmin

Yeasmin

Hasan

et al. 2010

Electronic Notes in Discrete Mathematics

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“…Furthermore, Savage [Sav97] mentions X = {ρ n , ρ n−1 , (n − 1 n)} as an interesting open problem. Lastly, Bass and Sudborough [BS03] and Sawada and Williams [SW16b] suggest the instances…”

Section: (K) Compton and Williamson [Cw93]mentioning

confidence: 96%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

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“…Regular spanning subnetworks of Q n are known to exist for specific values of n. These spanning subnetworks are described in Table 2. For example, the cube-connected cycles network of dimension n CCC n [23] is known to be a spanning subnetwork of Q n+lgn , where [6] is both a spanning subnetwork of Q n−1 and a subnetwork of the pancake network of dimension n. The spanning subnetwork Q n,2,1 [7] contains all the links for dimensions 0 and 1, and uses the value of the first two bits of the label of each node to determine the dimensions of links incident to that node. Q n,2,1 is the first regular spanning subnetwork of Q n with degree less than n and diameter n.…”

Section: Progress In Finding Factors Of Small Diametermentioning

confidence: 99%

Hamilton Decompositions and (n/2)-Factorizations of Hypercubes

Bass¹,

Sudborough²

2003

JGAA

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Since Qn, the hypercube of dimension n, is known to have n linkdisjoint paths between any two nodes, the links of Qn can be partitioned into multiple link-disjoint spanning subnetworks, or factors. We seek to identify factors which efficiently simulate Qn, while using only a portion of the links of Qn. We seek to identify (n/2)-factorizations, of Qn where 1) the factors have as small a diameter as possible, and 2) mappings (embeddings) of Qn to each of the factors exist, such that the maximum number of links in a factor corresponding to one link in Qn (dilation), is as small as possible. In this paper we consider two algorithms for generating Hamilton decompositions of Qn, and three methods for constructing (n/2)-factorizations of Qn for specific values of n. The most notable (n/2)-factorization of Qn results in two mutually isomorphic factors, each with diameter n + 2, where an embedding exists which maps Qn to each of the factors with constant dilation.

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Pancake problems with restricted prefix reversals and some corresponding Cayley networks

Cited by 14 publications

References 15 publications

Pancake Flipping with Two Spatulas

Pancake Flipping with Two Spatulas

Combinatorial Gray Codes — an Updated Survey

Hamilton Decompositions and (n/2)-Factorizations of Hypercubes

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