Dynamic Storage Allocation and On-Line Colouring Interval Graphs

Narayanaswamy, N. S.

doi:10.1007/978-3-540-27798-9_36

Cited by 16 publications

(19 citation statements)

References 5 publications

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“…These are not necessarily the best algorithms for the problem. Narayanaswamy [Narayanaswamy 2004;Azar et al 2006] presented an algorithm that is 10-competitive for this problem, and this is the best bound known.…”

Section: Online Coloring Of Intervals With Bandwidthmentioning

confidence: 99%

Max-coloring and online coloring with bandwidths on interval graphs

Pemmaraju

Raman

Varadarajan

2011

ACM Trans. Algorithms

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Given a graph G = (V, E) and positive integral vertex weights w : V → N, the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C 1 , C 2 , . . . , C k , minimizemax v∈C i w(v). This problem, restricted to interval graphs, arises whenever there is a need to design dedicated memory managers that provide better performance than the general purpose memory management of the operating system.Though this problem seems similar to the dynamic storage allocation problem, there are fundamental differences. We make a connection between max-coloring and on-line graph coloring and use this to devise a simple 2-approximation algorithm for max-coloring on interval graphs. We also show that a simple first-fit strategy, that is a natural choice for this problem, yields an 8-approximation algorithm. We show this result by proving that the first-fit algorithm for on-line coloring an interval graph G uses no more than 8 · χ(G) colors, significantly improving the bound of 26·χ(G) by Kierstead and Qin (Discrete Math., 144, 1995). We also show that the max-coloring problem is NP-hard.The problem of online coloring of intervals with bandwidths is a simultaneous generalization of online interval coloring and online bin packing. The input is a set I of intervals, each interval i ∈ I having an associated bandwidth b(i) ∈ (0, 1]. We seek an online algorithm that produces a coloring of the intervals such that for any color c and any real r, the sum of the bandwidths of intervals containing r and colored c is at most 1. Motivated by resource allocation problems, Adamy and Erlebach (Proceedings of the First International Workshop on Online and Approximation Algorithms, 2003, LNCS 2909, pp 1-12 ) consider this problem and present an algorithm that uses at most 195 times the number of colors used by an optimal off-line algorithm. Using the new analysis of first-fit coloring of interval graphs, we show that the Adamy-Erlebach algorithm is 35-competitive. Finally, we generalize the Adamy-Erlebach algorithm to a class of algorithms and show that a different instance from this class is 30-competitive.

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Section: Online Coloring Of Intervals With Bandwidthmentioning

confidence: 99%

Max-coloring and online coloring with bandwidths on interval graphs

Pemmaraju

Raman

Varadarajan

2011

ACM Trans. Algorithms

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“…The following KT b algorithm for the online interval coloring with bandwidth problem was studied by Epstein and Levy [EL05a,EL05b] (see also [Nar04]). We are given an upper bound b on the maximum bandwidth request.…”

Section: Preliminariesmentioning

confidence: 99%

“…Our algorithm for this case is the following adaptation of the algorithm of Narayanaswamy [Nar04] for the online interval coloring problem with bandwidth. We partition the requests into three groups.…”

Section: The Bounded Modelmentioning

confidence: 99%

“…We use disjoint colors for coloring requests of distinct groups. Our algorithm is different from the algorithm of [Nar04] mainly in the procedure for coloring the small requests.…”

Section: The Bounded Modelmentioning

confidence: 99%

“…The intervals are to be colored so that at each point, the sum of bandwidths of intervals colored by a certain color does not exceed 1. This problem was studied also in [Nar04], giving an improved competitive ratio of 10, and in [EL05a], showing a lower bound of 3.2609 for this model.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Variable Sized Online Interval Coloring with Bandwidth

2007

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We consider online coloring of intervals with bandwidth in a setting where colors have variable capacities. Whenever the algorithm opens a new color, it must choose the capacity for that color and cannot change it later. The goal is to minimize the total capacity of all the colors used. We consider the bounded model, where all capacities must be chosen in the range (0, 1], and the unbounded model, where the algorithm may use colors of any positive capacity. For the absolute competitive ratio, we give an upper bound of 14 and a lower bound of 4.59 for the bounded model, and an upper bound of 4 and a matching lower bound of 4 for the unbounded model. We also consider the offline version of these problems and show that whereas the unbounded model is polynomially solvable, the bounded model is NP-hard in the strong sense and admits a 3.6-approximation algorithm.

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Online Interval Coloring and Variants

Epstein

Levy

2005

Lecture Notes in Computer Science

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We study interval coloring problems and present new upper and lower bounds for several variants. We are interested in four problems, online coloring of intervals with and without bandwidth and a new problem called lazy online coloring again with and without bandwidth. We consider both general interval graphs and unit interval graphs. Specifically, we establish the difference between the two main problems which are interval coloring with and without bandwidth. We present the first nontrivial lower bound of 3.2609 for the problem with bandwidth. This improves the lower bound of 3 that follows from the tight results for interval coloring without bandwidth presented in [9].

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Dynamic Storage Allocation and On-Line Colouring Interval Graphs

Cited by 16 publications

References 5 publications

Max-coloring and online coloring with bandwidths on interval graphs

Max-coloring and online coloring with bandwidths on interval graphs

Variable Sized Online Interval Coloring with Bandwidth

Online Interval Coloring and Variants

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