Pancake problems with restricted prefix reversals and some corresponding Cayley networks

Bass, Douglas W.; Sudborough, Ivan Hal

doi:10.1109/icpp.1998.708457

Cited by 8 publications

(26 citation statements)

References 26 publications

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“…Pancake and burnt pancake networks have better diameter and better vertex degree than the popular hypercubes [17]. There exist some other variations of pancake flipping, giving different efficient interconnection networks [3].…”

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 [17,18]. A well studied variation of pancake flipping problem is the burnt pancake flipping problem [3,17,18] 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 [17].…”

Section: Introductionmentioning

confidence: 99%

“…The pancake flipping problem [3,8,14,17,18] 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 Dweighter [8] 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%

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Pancake flipping and sorting permutations

Hasan

Rahman

et al. 2015

Journal of Discrete Algorithms

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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. 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 from practical point of view 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%

Section: Introductionmentioning

confidence: 99%

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Pancake flipping and sorting permutations

Hasan

Rahman

et al. 2015

Journal of Discrete Algorithms

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“…Let P(n) (alternatively referred to as an n-pancake graph) denotes the structure of a graph in which each vertex represents a permutation of n elements from 1 to n; also know as an n-dimensional pancake graph. Moreover, 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, then 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). The cardinality of the vertex set in P(n) is n!, and the n-pancake graph is n − 1 regular.…”

Section: Introductionmentioning

confidence: 99%

Constructing Independent Spanning Trees on Pancake Networks

2020

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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.INDEX TERMS Interconnection networks, independent spanning trees, pancake networks.

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“…The augmented cubeconnected cycles networks ACCC 1 n and ACCC 2 n [3] are derived from adding links to CCC n . The subcube network Subcube n [2] 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 [5] is the first regular spanning subnetwork of Q n with degree less than n and diameter n. Q n + lg n , n = 2 k 6 3n / 2 Subcube n Q n -1 , n = 2 k lg n (3n / 2) -2 Q n, 2, 1 Q n , n is even n / 2 + 1 n However, none of the networks of Table 2 are part of any k-factorization of the hypercubes they span, as they use all of the links for one or more dimensions of Q n . (n / 2)-factorizations of Q 8 and Q 12 , where the factors have the same diameter as their respective hypercubes, were identified by computer search in [4].…”

Section: Progress In Finding Factors With Small Diametermentioning

confidence: 99%

Symmetric k-factorizations of hypercubes with factors of small diameter

Bass

Sudborough

Proceedings International Symposium on Parallel Architectures, Algorithms and Networks. I-Span'02

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The links of the hypercube Q n can be partitioned into multiple link-disjoint spanning subnetworks, or factors. Each of these factors could simulate Q n . We therefore identify k-factorizations, or partitions of the links of Q n into factors of degree k, where 1) the factorization exists for all values of n such that n mod k = 0, 2) k is as small as possible, 3) the n/k factors have a similar structure, 4) the factors have as small a diameter as possible, and 5) the factors host Q n with as small a dilation as possible.In this paper, we give an (n/2)-factorization of Q n , where n is even, generated by variations on reduced and thin hypercubes. The two factors are isomorphic, and both of the factors have diameter n+2. The diameter is an improvement over the best result known. Both of the factors also host Q n with Θ(1) dilation.

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Pancake problems with restricted prefix reversals and some corresponding Cayley networks

Cited by 8 publications

References 26 publications

Pancake flipping and sorting permutations

Pancake flipping and sorting permutations

Constructing Independent Spanning Trees on Pancake Networks

Symmetric k-factorizations of hypercubes with factors of small diameter

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