The Greedy Gray Code Algorithm

Williams, Aaron

doi:10.1007/978-3-642-40104-6_46

Cited by 39 publications

(26 citation statements)

References 7 publications

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“…In this paper, we are not interested in shortest paths in pancake networks, but rather Hamilton cycles. There are myriad ways that researchers attempt to build Hamilton cycles in highly-symmetric graphs, and the greedy approach is perhaps the simplest (see Williams [25]). This approach initializes a path at a specific vertex, then repeatedly extends the path by a single edge.…”

Section: (Greedy) Hamilton Cyclesmentioning

confidence: 99%

A Hamilton Cycle in the $k$-Sided Pancake Network

Cameron

Sawada

Williams

2021

Preprint

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We present a Hamilton cycle in the k-sided pancake network and four combinatorial algorithms to traverse the cycle. The network's vertices are coloured permutations π = p1p2 • • • pn, where each pi has an associated colour in {0, 1, . . . , k−1}. There is a directed edge (π1, π2) if π2 can be obtained from π1 by a "flip" of length j, which reverses the first j elements and increments their colour modulo k. Our particular cycle is created using a greedy min-flip strategy, and the average flip length of the edges we use is bounded by a constant. By reinterpreting the order recursively, we can generate successive coloured permutations in O(1)-amortized time, or each successive flip by a loop-free algorithm. We also show how to compute the successor of any coloured permutation in O(n)-time. Our greedy min-flip construction generalizes known Hamilton cycles for the pancake network (where k = 1) and the burnt pancake network (where k = 2). Interestingly, a greedy max-flip strategy works on the pancake and burnt pancake networks, but it does not work on the k-sided network when k > 2.

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Section: (Greedy) Hamilton Cyclesmentioning

confidence: 99%

A Hamilton Cycle in the $k$-Sided Pancake Network

Cameron

Sawada

Williams

2021

Preprint

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“…The girths of the burnt Pancake graphs over the hyperoctahedral group was considered in [2]. The (burnt) Pancake graphs are commonly used in computer science to represent interconnection networks [1,10,11].…”

Section: Introductionmentioning

confidence: 99%

The girths of the cubic Pancake graphs

Konstantinova,

Gun

2022

Preprint

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The Pancake graphs P n , n 2, are Cayley graphs over the symmetric group Sym n generated by prefix-reversals. There are six generating sets of prefixreversals of cardinality three which give connected Cayley graphs over the symmetric group known as cubic Pancake graphs. In this paper we study the girth of the cubic Pancake graphs. It is proved that considered cubic Pancake graphs have the girths at most twelve.

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“…1. the ECO framework developed by Bacchelli, Barcucci, Grazzini, and Pergola [2] that generates Gray codes for a variety of combinatorial objects such as Dyck words in constant amortized time per instance; 2. the twisted lexico computation tree by Takaoka [22] that generates Gray codes for multiple combinatorial objects in constant amortized time per instance; 3. loopless algorithms developed by Walsh [25] to generate Gray codes for multiple combinatorial objects, which extend algorithms initially given by Ehrlich in [8]; 4. greedy algorithms observed by Williams [28] that provide a uniform understanding for many previous published results; 5. the reflectable language framework by Li and Sawada [13] for generating Gray codes of k-ary strings, restricted growth strings, and k-ary trees with n nodes; 6. the bubble language framework developed by Ruskey, Sawada and Williams [17] that provides algorithms to generate shift Gray codes for fixed-weight necklaces and Lyndon words, k-ary Dyck words, and representations of interval graphs; 7. the permutation language framework developed by Hartung, Hoang, Mütze and Williams [11] that provides algorithms to generate Gray codes for a variety of combinatorial objects based on encoding them as permutations.…”

Section: Introductionmentioning

confidence: 99%

Inside the Binary Reflected Gray Code: Flip-Swap Languages in 2-Gray Code Order

Sawada¹,

Williams²,

Wong³

2021

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A flip-swap language is a set S of binary strings of length n such that S∪{0 n } is closed under two operations (when applicable): (1) Flip the leftmost 1; and (2) Swap the leftmost 1 with the bit to its right. Flip-swap languages model many combinatorial objects including necklaces, Lyndon words, prefix normal words, left factors of k-ary Dyck words, and feasible solutions to 0-1 knapsack problems. We prove that any flip-swap language forms a cyclic 2-Gray code when listed in binary reflected Gray code (BRGC) order. Furthermore, a generic successor rule computes the next string when provided with a membership tester. The rule generates each string in the aforementioned flip-swap languages in O(n)amortized per string, except for prefix normal words of length n which require O(n 1.864 )-amortized per string. Our work generalizes results on necklaces and Lyndon words by Vajnovski [Inf.

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The Greedy Gray Code Algorithm

Cited by 39 publications

References 7 publications

A Hamilton Cycle in the $k$-Sided Pancake Network

A Hamilton Cycle in the $k$-Sided Pancake Network

The girths of the cubic Pancake graphs

Inside the Binary Reflected Gray Code: Flip-Swap Languages in 2-Gray Code Order

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