TY - JOUR

T1 - A decomposition theorem for task systems and bounds for randomized server problems

AU - Blum, Avrim

AU - Karloff, Howard

AU - Rabani, Yuval

AU - Saks, Michael

PY - 2000

Y1 - 2000

N2 - A lower bound of Ω(√log k/ log log k) is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (having at least k + 1 points) and provides a new lower bound for the metrical task system problem as well. This improves the previous best lower bound of Ω(log log k) for arbitrary metric spaces [H.J. Karloff, Y. Rabani, and Y. Ravid, SIAM J. Comput., 23 (1994), pp. 293-312] and more closely approaches the conjectured lower bound of Ω(log k). For the server problem on k + 1 equally spaced points on a line, which corresponds to a natural motion-planning problem, a lower bound of Ω(log k/ log log k) is obtained. The results are deduced from a general decomposition theorem for a simpler version of both the k-server and the metrical task system problems, called the "pursuit-evasion game." It is shown that if a metric space script M sign can be decomposed into two spaces script M signℒ and script M signscript R sign such that the distance between them is sufficiently large compared to their diameter, then the competitive ratio for this game on script M sign can be expressed nearly exactly in terms of the ratios on each of the two subspaces. This yields a divide-and-conquer approach to bounding the competitive ratio of a space.

AB - A lower bound of Ω(√log k/ log log k) is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (having at least k + 1 points) and provides a new lower bound for the metrical task system problem as well. This improves the previous best lower bound of Ω(log log k) for arbitrary metric spaces [H.J. Karloff, Y. Rabani, and Y. Ravid, SIAM J. Comput., 23 (1994), pp. 293-312] and more closely approaches the conjectured lower bound of Ω(log k). For the server problem on k + 1 equally spaced points on a line, which corresponds to a natural motion-planning problem, a lower bound of Ω(log k/ log log k) is obtained. The results are deduced from a general decomposition theorem for a simpler version of both the k-server and the metrical task system problems, called the "pursuit-evasion game." It is shown that if a metric space script M sign can be decomposed into two spaces script M signℒ and script M signscript R sign such that the distance between them is sufficiently large compared to their diameter, then the competitive ratio for this game on script M sign can be expressed nearly exactly in terms of the ratios on each of the two subspaces. This yields a divide-and-conquer approach to bounding the competitive ratio of a space.

KW - Competitive analysis

KW - Lower bounds

KW - On-line algorithms

KW - Randomized algorithms

KW - Task systems

KW - k-server

UR - http://www.scopus.com/inward/record.url?scp=0035178705&partnerID=8YFLogxK

U2 - 10.1137/S0097539799351882

DO - 10.1137/S0097539799351882

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AN - SCOPUS:0035178705

SN - 0097-5397

VL - 30

SP - 1624

EP - 1661

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 5

ER -