Sequences of Radius k: How to Fetch Many Huge Objects into Small Memory for Pairwise Computations

Jaromczyk, Jerzy W.; Lonc, Zbigniew

doi:10.1007/978-3-540-30551-4_52

Cited by 11 publications

(39 citation statements)

References 3 publications

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“…Proof of Corollary The lower bound was already shown in , so we shall prove the upper bound only. Observe first that, for

m \geq 2 k

ψ false(P_{m}^{false[k false]} false) \geq ψ false(C_{m - k}^{false[k false]} false)

, where

[k] = {1, 2, \dots, k}

.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 90%

“…Proof of Corollary 1. The lower bound was already shown in [15], so we shall prove the upper bound only.…”

Section: Proof Of Theoremmentioning

confidence: 79%

“…Formally, a sequence

(s_{1}, \dots, s_{m})

over an alphabet A is a k ‐ radius sequence if for every 2‐element subset

{a, b} \subseteq A

there is at least one pair of indices

(i, j)

, where

1 \leq i < j \leq m

, such that

j - i \leq k

and

false{s_{i}, s_{j} false} = false{a, b false}

. The study of k ‐radius sequences arose from some practical motivations in data transmission problems (see ). In these problems it is desirable to find a shortest possible k ‐radius sequence over an alphabet of size n ; the minimum possible length of such a sequence is denoted by

f_{k} false(n false)

.…”

Section: Introductionmentioning

confidence: 99%

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Achromatic and Harmonious Colorings of Circulant Graphs

Dębski

Lonc

Rzążewski

2017

Journal of Graph Theory

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Abstract:A proper vertex coloring of a graph G is achromatic (respectively harmonious) if every two colors appear together on at least one (resp. at most one) edge. The largest (resp. the smallest) number of colors in an achromatic (resp. a harmonious) coloring of G is called the achromatic (resp. harmonious chromatic) number of G and denoted by ψ (G) (resp. h(G)). For a finite set of positive integers D and a positive integer n, a circulant graph, denoted by C D n , is an undirected graph on the set of vertices {0, 1, . . . , n − 1} that has an edge i j if and only if either i − j or j − i is a member of D (where substraction is computed modulo n). For any fixed set D, we show that ψ (C D n ) is asymptotically equal to √ 2 |D| n, with the error term O(log n). We also prove that h(C D n ) is asymptotically equal to ACHROMATIC AND HARMONIOUS COLORINGS OF CIRCULANT GRAPHS 19√ 2 |D| n, with the error term O(n 1 4 log n). As corollaries, we get results that improve, for a fixed k, the previously best estimations on the lengths of a shortest k-radius sequence over an n-ary alphabet (i.e., a sequence in which any two distinct elements of the alphabet occur within distance k of each other) and a longest packing k-radius sequence over an n-ary alphabet (which is a dual counterpart of a k-radius sequence). C 2017 Wiley Periodicals, Inc.

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“…Proof of Corollary The lower bound was already shown in , so we shall prove the upper bound only. Observe first that, for

m \geq 2 k

ψ false(P_{m}^{false[k false]} false) \geq ψ false(C_{m - k}^{false[k false]} false)

, where

[k] = {1, 2, \dots, k}

.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 90%

“…Proof of Corollary 1. The lower bound was already shown in [15], so we shall prove the upper bound only.…”

Section: Proof Of Theoremmentioning

confidence: 79%

“…Formally, a sequence

(s_{1}, \dots, s_{m})

over an alphabet A is a k ‐ radius sequence if for every 2‐element subset

{a, b} \subseteq A

there is at least one pair of indices

(i, j)

, where

1 \leq i < j \leq m

, such that

j - i \leq k

and

false{s_{i}, s_{j} false} = false{a, b false}

f_{k} false(n false)

.…”

Section: Introductionmentioning

confidence: 99%

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Achromatic and Harmonious Colorings of Circulant Graphs

Dębski

Lonc

Rzążewski

2017

Journal of Graph Theory

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“…Let k and ≥ n k be positive integers. The concept of n-ary k-radius sequences was introduced by Jaromczyk and Lonc [26], when describing a First-In First-Out caching strategy for computing functions which require pairwise computations among a set of n large objects (such as medical images), where the memory allows at most k + 1 objects cached at any one time.…”

Section: Introductionmentioning

confidence: 99%

“…For exact values of f n ( ) k , only the case when k = 1 was completely determined in [21] when studied in the context of database applications. When ≥ k 2, Jaromczyk and Lonc [26] proved the following lower bound,…”

Section: Introductionmentioning

confidence: 99%

Short k‐radius sequences, k‐difference sequences and universal cycles

Zhang

2020

J of Combinatorial Designs

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Sequences of Radius k for Complete Bipartite Graphs

DăźBski

Lonc

Rzążewski

2016

Graph-Theoretic Concepts in Computer Science

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A k-radius sequence for a graph G is a sequence of vertices of G (typically with repetitions) such that for every edge uv of G vertices u and v appear at least once within distance k in the sequence. The length of a shortest k-radius sequence for G is denoted by f k (G). We give an asymptotically tight estimation on f k (G) for complete bipartite graphs which matches a lower bound, valid for all bipartite graphs. We also show that determining f k (G) for an arbitrary graph G is NP-hard for every constant k > 1.

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Sequences of Radius k: How to Fetch Many Huge Objects into Small Memory for Pairwise Computations

Cited by 11 publications

References 3 publications

Achromatic and Harmonious Colorings of Circulant Graphs

Achromatic and Harmonious Colorings of Circulant Graphs

Short k‐radius sequences, k‐difference sequences and universal cycles

Sequences of Radius k for Complete Bipartite Graphs

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