Sorting Permutations: Games, Genomes, and Cycles

Adamyk, Katharine; Holmes, Edward C.; Mayfield, Georgia; Moritz, Dominik; Scheepers, Marion; Tenner, Bridget Eileen; Wauck, Helen

doi:10.48550/arxiv.1410.2353

Cited by 2 publications

(9 citation statements)

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“…As an example we perform cds (3,4), (6,7) on β = [3,6,5,2,4,8,1,7]: Label β with the relevant pointers: β = [3 (3,4) , 6 (6,7) , 5, 2, (3,4) 4, 8, 1, (6,7) 7]. Performing the swap yields δ = [3,4,8,1,5,2,6,7]. Both 3 and 4 and 6 and 7 are now adjacent in δ and pointers (3,4) and (6, 7) no longer interlock.…”

Section: Notation and Terminologymentioning

confidence: 99%

“…When these two edges belong to different cycles, then the strategic pile of α is empty. The notion of the strategic pile of a permutation was introduced and studied in [1].…”

Section: Cycle Graphs and The Strategic Pilementioning

confidence: 99%

“…(1) (cds Sortability Theorem, [1], Theorem 2.5) α is cds-sortable if, and only if, the strategic pile of α is empty.…”

Section: Cycle Graphs and The Strategic Pilementioning

confidence: 99%

“…(2) (Strategic Pile Removal Theorem, [1], Theorem 2.15 ) Suppose that α has more than one element in the strategic pile. For each pointer p in the strategic pile there exists a pointer q such that cds applied to the pointer pair p, q results in removal of p from the strategic pile.…”

Section: Cycle Graphs and The Strategic Pilementioning

confidence: 99%

“…Christie [3] used the cycle graph to compute the minimum number of arbitrary swaps required to sort a permutation. The cycle graph is also instrumental in the (linear time) characterization of cds-sortability of a permutation, derived in Theorem 2.5 of [1]. This characterization shows that for a cds sortable permutation π only the identity permutation is a fixed point for applications of cds to π.A new phenomenon emerges when π is not cds sortable: Several fixed points may result from applications of cds to π.…”

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confidence: 99%

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Permutation sorting and a game on graphs

Jansen,

Scheepers,

Simon

et al. 2014

Preprint

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We introduce a game on graphs. By a theorem of Zermelo, each instance of the game on a finite graph is determined. While the general decision problem on which player has a winning strategy in a given instance of the game is unsolved, we solve the decision problem for a specific class of finite graphs. This result is then applied to a permutation sorting game to prove the optimality of a proportional bound under which TWO has a winning strategy.Permutation sorting by block interchanges, also called swaps, has been studied before [3]. A constrained version of the block interchange sorting operation on permutations has been postulated to perform steps in decrypting the ciliate micronuclear genome to establish a new macronuclear genome [6]. We shall refer to this constrained block interchange operation, defined later, by the name cds (for context directed swap). Though any permutation is sortable by arbitrary block interchanges, not all finite permutations are sortable by cds. For any permutation repeated applications of cds produce a permutation to which cds no longer applies, called a cds fixed point. For a positive integer n there are n cds fixed points in S n , the set of permutations of {1, 2, . . . , n}. From this point on we consider only permutations in S n for some n.Two fundamental graph structures associated with a permutation encode the permutation's properties in connection with the cds sorting operation. One is the cycle graph introduced in [2], and the other is the overlap graph that was introduced in [5] -see also [7].The cycle graph keeps track of the status of a permutation in terms of being sorted by block interchanges. Christie [3] used the cycle graph to compute the minimum number of arbitrary swaps required to sort a permutation. The cycle graph is also instrumental in the (linear time) characterization of cds-sortability of a permutation, derived in Theorem 2.5 of [1]. This characterization shows that for a cds sortable permutation π only the identity permutation is a fixed point for applications of cds to π.A new phenomenon emerges when π is not cds sortable: Several fixed points may result from applications of cds to π. In [1] the strategic pile 1 of π, a structure encoded by the cycle graph of π, was discovered. The strategic pile determines the various fixed points that are achievable as results of cds operations.For cds-non-sortable permutations variability in achievable fixed points suggests combinatorial games. The following game, played by players ONE and TWO, was introduced in [1]: A cds non-sortable permutation π ∈ S n and a subset A of its strategic pile P are given. The game CDS(π, A) proceeds as follows: π is the initial permutation. Starting with ONE the players alternate applying a cds operation to the current permutation. The game ends when a cds fixed point is reached. If the code for this fixed point, necessarily a member of the strategic pile P, is a member of A, then ONE wins. Else, TWO wins. Note that the length of the game is bounded by n. Zermelo's Theorem [8] implies tha...

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Section: Notation and Terminologymentioning

confidence: 99%

“…When these two edges belong to different cycles, then the strategic pile of α is empty. The notion of the strategic pile of a permutation was introduced and studied in [1].…”

Section: Cycle Graphs and The Strategic Pilementioning

confidence: 99%

“…(1) (cds Sortability Theorem, [1], Theorem 2.5) α is cds-sortable if, and only if, the strategic pile of α is empty.…”

Section: Cycle Graphs and The Strategic Pilementioning

confidence: 99%

Section: Cycle Graphs and The Strategic Pilementioning

confidence: 99%

mentioning

confidence: 99%

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Permutation sorting and a game on graphs

Jansen,

Scheepers,

Simon

et al. 2014

Preprint

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Context Directed Reversals and the Ciliate Decryptome

Jansen¹,

Scheepers²,

Simon³

et al. 2016

Preprint

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Prior studies of the efficiency of the block interchange (swap) and the reversal sorting operations on (signed) permutations identified specialized versions of the these operations. These specialized operations are here called context directed reversal, abbreviated cdr, and context directed swap, abbreviated cds. Prior works have also characterized which (signed) permutations are sortable by cdr or by cds.It is now known that when a permutation π is cds sortable in n steps, then any n consecutive applicable cds operations will sort π. Examples show that this is not the case for cdr. This phenomenon is the focus of this paper. It is proven that if a signed permutation is cdr sortable, then any cdr fixed point of it is cds sortable (the cds Rescue Theorem). The cds Rescue Theorem is discussed in the context of a mathematical model for ciliate micronuclear decryption.It is also known that if applications of cds to a permutation π reaches a cds fixed point in n steps, then any n consecutive applicable cds operations will terminate in a cds fixed point of π. This is not the case for cdr: It is proven that though for a given signed permutation the number of cdr operations leading to different cdr fixed points may be different from each other, the parity of the number of operations is the same (the cdr Parity Theorem). This result provides a solution to two previously formulated decision problems regarding certain combinatorial games.

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Sorting Permutations: Games, Genomes, and Cycles

Cited by 2 publications

References 16 publications

Permutation sorting and a game on graphs

Permutation sorting and a game on graphs

Context Directed Reversals and the Ciliate Decryptome

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