H.A. Kierstead. The linearity of first-fit coloring of interval graphs. SIAM J. Discrete Math, 1:526--530, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which case it is easy to see that without loss of generality s(i) is integral too. Dynamic Storage Allocation was proven NP Complete in 1976 by Stockmeyer. See problem SR2 in Garey and Johnson [2] The rst polynomial time, constant factor approximation algorithm was given by Kierstead in 1988 [5], using a reduction from Dynamic Storage Allocation to on line coloring of interval graphs discovered by Woodall in 1973 [7] and independently by Chrobak and Slusarek in 1988 [1] Kierstead s algorithm is an 80 approximation algorithm. Kierstead [6] later exhibited a 6 approximation algorithm for ....

H. A. Kierstead, \The Linearity of First-Fit Coloring of Interval Graphs," SIAM J. Discrete Math 1 (1988), 526-530.

....behavior on others. The survey paper of Kierstead and Trotter [KT] mentions a number of problems related to the First Fit coloring algorithm; for example, Woodall asked in 1974 whether the number of colors used by First Fit was within a constant factor of (G) when G is an interval graph (see also [Kier]) In the new language, this is simply the question of whether First Fit achieves a constant competitive ratio for this type of graph. In the last five years, there have been a number of papers on on line graph coloring in the competitive framework; these will be discussed briefly in Section ....

H. Kierstead, "The linearity of First-Fit for coloring interval graphs," SIAM J. Discrete Math, 1(1988), pp. 526--530.

....D ae G and v 2 V (G) D v and D Gamma v denotes the subgraph of G induced by V (D) fvg and V (D) n fvg, respectively. For further definitions and notations see in subsequent sections. The concept of on line chromatic number of graphs was introduced in [GL1] GL2] and investigated in [GL3] [K], KPT] and [LST] A survey of recent results can be found in [KT] GKL2] and [GKL3] What we know about the power of on line algorithms used for particular graph classes is the following. In [GL1] it is proved that FF is perfect for P 4 free graphs, i.e. FF colors any P 4 free graph G ....

H.A. Kierstead, The linearity of First-Fit coloring of interval graphs, SIAM Journal on Discrete Math. 1 (1988) 526-530.

....remarks There are many online problems related to graphs that we have not addressed in this survey. A classical problem is online coloring : The nodes of a graph arrive online and we wish to color them using as few colors as possible. There is a significant body of work on this problem, see e.g. [37,45,48,52]. In online matching , a newly arriving node can be matched to a node already present. Unweighted and weighted versions of this problem have been considered [43,41,44] The generalized Steiner tree problem is an extensively studied problem where we have to construct a minimum weight tree in a ....

H.A. Kierstead. The linearity of First-Fit coloring of interval graphs. SIAM Journal on Discrete Mathematics, 1:526--530, 1988.

....connections. Raghavan and Upfal [RU94] considered the off line version of the problem, which was further studied in [AR95, KP96, MKR95, Ra96] For a line topology, the problem is the well studied interval graph coloring problem. On line algorithms for this problem have been analyzed in [KT81, Ki88] We consider trees, trees of rings, and meshes topologies, previously studied in the off line case. We give on line algorithms with competitive ratio O(log n) for all these topologies. We give a matching Omega Gammaing n) lower bound for meshes. We also prove that any algorithm for trees ....

....algorithm for meshes and certain nearly Eulerian planar graphs . Rabani [Ra96] improves the bound for meshes to O(poly(log log n) The on line path coloring problem has been studied in the case of a line topology in the context of interval graph coloring by Kierstead and Trotter [KT81, Ki88] They give an optimal 3 competitive algorithm for the line ( KT81] 1 The path coloring problem is closely related to the virtual circuit routing problem, motivated by its application to ATM networks. The load version of this problem is where every requested pair must be assigned a path as to ....

H.A. Kierstead. The Linearity of First-Fit Coloring of Interval Graphs. In Siam Journal on Discrete Math. 1(4), pages 526--530, 1988.

....graphs. Consider the undirected graph, call it Gamma, whose nodes are the messages in the chatting problem M and whose edges comprise all pairs of messages fM 0 ; M 00 g from M for which the line segments (s M 0 # x) and (s M 00 # x) overlap. Note that Gamma is the interval graph [6] that is induced by the set of line segments f(s M #x) M 2 Mg. If two segments (s M 0#x) and (s M 00#x) cannot be allocated to the same track, then the pair fM 0 ; M 00 g is an edge of Gamma. If we associate tracks with colors, then the compaction problem is equivalent to finding a ....

....to the same track. The minimum node coloring problem for interval graphs admits a simple (low degree) polynomial time solution. One orders the nodes according to the left endpoints of their corresponding intervals and then applies a first fit on line coloring strategy to the sorted sequence [6]. The number of colors used is the maximum clique size of Gamma which is clearly equal, in our case, to the message congestion C(M) 2 By applying Theorem 3.2 to the virtual chatting schedule obtained by solving the rectangle compaction problem in the proof of Theorem 4.4 we obtain: Corollary ....

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H.A. Kierstead (1988): The linearity of first-fit coloring of interval graphs. SIAM J. Discr. Math. 1, 526-530.

.... results show that on line algorithms on certain families of graphs are competitive in a stronger sense (compared with off line ones) A very interesting example is the family of interval graphs where on line algorithm is known with performance ratio 3 ( KT1] and FF has constant performance ratio ([K1]) Another important recent result ( KPT1] is that on the family of cocomparability graphs on line algorithms are competitive in this stronger sense. Our approach in this paper was to define on line competitive algorithms to compete within the family of on line coloring algorithms. ....

H. A. Kierstead, The Linearity of First Fit for Coloring Interval Graphs, SIAM J. Discr. Math. 1 (1988) 526-530.

....axis, and an edge exists between each pair of nodes whose intervals intersect. The simplest on line coloring algorithm assigns each node u the color ff which is the least positive integer not already assigned to a neighbor of u. This algorithm is called First Fit. For interval graphs, Kierstead [Kie88] showed that First Fit is 40 competitive with respect to the number of different colors used, and later Kierstead and Qin [KQ92] improved this bound to 25:8. Chrobak and Slusarek [CS88] give an example that proves that First Fit cannot be better than 4:4 competitive. Theorem 3.8 Given a linear ....

H.A. Kierstead. The linearity of first-fit for coloring interval graphs. SIAM Journal of Discrete Mathematics, 1(4):526--530, 1988.

....problem for linear array networks. 5.1 GREEDY for linear array networks The simplest on line coloring algorithm assigns each node u the color ff which is the least positive integer not already assigned to a neighbor of u. This algorithm is called First Fit. For interval graphs, Kierstead [Kie88] showed that First Fit is 40 competitive with respect to the number of different colors used, and later Kierstead and Qin [KQ92] improved this bound to 25:8. Chrobak and Slusarek [CS88] give an example that proves that First Fit cannot be better than 4:4 competitive. Theorem 5.1 Given a linear ....

H.A. Kierstead. The linearity of first-fit for coloring interval graphs. SIAM Journal of Discrete Mathematics, 1(4):526--530, 1988.

....interval graphs. Consider the undirected graph, call it Gamma, whose nodes are the messages in the chatting problem M and whose edges comprise all pairs of messages fM 0 ; M 00 g from M for which the line segments (s M 0 #x) and (s M 00 #x) overlap. Note that Gamma is the interval graph [9] that is induced by the set of line segments f(s M #x) M 2 Mg. If two segments (s M 0 #x) and (s M 00 #x) cannot be allocated to the same track, then the pair fM 0 ; M 00 g is an edge of Gamma. If we associate tracks with colors, then the compaction problem is equivalent to finding a ....

....to the same track. The minimum node coloring problem for interval graphs admits a simple (low degree) polynomial time solution. One orders the nodes according to the left endpoints of their corresponding intervals and then applies a first fit on line coloring strategy to the sorted 24 sequence [9]. The number of colors used is the maximum clique size of Gamma which is clearly equal, in our case, to the message congestion C(M) By applying Theorem 2 to the virtual chatting schedule obtained by solving the rectangle compaction problem in the proof of Theorem 4 we obtain: Corollary 2: In ....

[Article contains additional citation context not shown here]

H.A. Kierstead (1988): The linearity of first-fit coloring of interval graphs. SIAM J. Discr. Math. 1, 526-530.

....can be done to improve the performance. Bartal et al. 11] gave an O( p n) competitive deterministic algorithm and prove that no randomized algorithm can do beter than Omega Gamma n 1 Gammalog 4 3 ) Also, on line coloring algorithms have been developed for special families of graphs [38, 40, 50]. ....

H. A. Kierstead. The linearity of first-fit coloring of interval graphs. SIAM Journal on Discrete Mathematics, 1(4):526--530, ???? 1988.

....algorithm. We note that the problem of determining an optimal weighted coloring for a given set of intervals, i.e. the off line version) is NP complete in the strong sense [9, page 226] Recently, polynomial time approximation algorithms for the off line version were obtained. Kierstead [10] showed that if the intervals are sorted by their weights, then the competitive ratio of First Fit is 80. Later, Kierstead [11] gave a different approximation algorithm which achieves a competitive factor of 6. Both algorithms use (G) as a benchmark for measuring the approximation factor. The ....

....the competitive ratio of First Fit is also O(log(Wmax ) It is interesting to note that Robson [17] also showed that the competitive ratio of Best Fit can be as bad as Wmax . As previously mentioned, algorithms for the off line case with constant approximation factors have been recently obtained [10, 11]. 1.2. Our contribution. 1.2.1. First Fit. Our first contribution is to provide tighter bounds for First Fit by considering a new parameter for analyzing the competitive ratio, i.e. the maximum number of concurrent active processes in memory. Let the chromatic number of G be denoted by (G) ....

H. A. Kierstead, The linearity of first-fit coloring of interval graphs, SIAM J. Disc. Math, 1, (1988), pp. 526-530.

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H.A. Kierstead. The linearity of first-fit coloring of interval graphs. SIAM J. Discrete Math, 1:526--530, 1988.

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H.A. Kierstead, The linearity of first-fit coloring of interval graphs, SIAM J. on Disc. Math., vol. 1, pp. 526-530, 1988.

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H.A. Kierstead. The Linearity of First-Fit Coloring of Interval Graphs. In Siam Journal on Discrete Math. 1(4), pp. 526--530, 1988.

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H.A. Kierstead. The Linearity of First-Fit Coloring of Interval Graphs. In Siam Journal on Discrete Math. 1(4), pp. 526--530, 1988.

No context found.

H. A. Kierstead, The linearity of first-fit coloring of interval graphs, Siam J. Disc. Math, 1, pp. 526-? (1988).

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