A polylog competitive algorithm for the kserver problem

A polylog competitive algorithm for the k-server problem Nikhil Bansal (Eindhoven) Niv Buchbinder (Open Univ. , Israel) Aleksander Madry (MIT) Seffi Naor (Technion)

The k-server Problem • 1 2 Move Closest Sever Algorithm 3

The Paging/Caching Problem Set of pages {1, 2, …, n} , cache of size k<n. Request sequence of pages 1, 6, 4, 1, 4, 7, 6, 1, 3, … a) If requested page already in cache, no penalty. b) Else, cache miss. Need to fetch page in cache (possibly) evicting some other page. Goal: Minimize the number of cache misses. Paging: K-server on the uniform metric. (Server on location p = page p in cache) 1 . . . n

Previous Results: Paging (Deterministic) [Sleator Tarjan 85]: • Any deterministic algorithm >= k-competitive. • LRU is k-competitive (also other algorithms) Paging (Randomized): • Rand. Marking O(log k) [Fiat, Karp, Luby, Mc. Geoch, Sleator, Young 91]. • Lower bound Hk [Fiat et al. 91], tight results known.

K-server conjecture [Manasse-Mc. Geoch-Sleator ’ 88]: There exists k competitive algorithm on any metric space. Initially no f(k) guarantee. Fiat-Rababi-Ravid’ 90: exp(k log k) … Koutsoupias-Papadimitriou’ 95: 2 k-1 Chrobak-Larmore’ 91: k for trees.

Randomized k-server Conjecture There is an O(log k) competitive algorithm for any metric. Uniform Metric: log k Polylog for very special cases (uniform-like) Line: n 2/3 [Csaba-Lodha’ 06] exp(O(log n)1/2) [Bansal-Buchbinder-Naor’ 10] Depth 2 -tree: No o(k) guarantee

Our Result Thm: There is an O(log 2 k log 3 n) competitive* algorithm for k-server on any metric with n points. * Hiding some log n terms

Our Approach Hierarchically Separated Trees (HSTs) [Bartal 96]. Any Metric space O(log n) Problems on HST often reduced to Uniform metrics. [Bartal-Blum-Burch-Tomkins 97, Kleinberg-Tardos 01, …] Allocation Problem (uniform metrics): [Cote-Meyerson-Poplawski’ 08] (We work with a weaker “fractional” allocation problem)

Outline • • Introduction HSTs + Allocation Problem Fractional view of Randomized Algorithms Fractional Allocation Problem

Designing Algorithm on HST d+1 level HST

Allocation Problem •

Allocation to k-server •

Outline • • Introduction Allocation Problem Fractional view of Randomized Algorithms Fractional Allocation Algorithm

Fractional View of Randomized Algorithms To specify a randomized algorithm: i) Prob. distribution on states at time t. ii) How it changes at time t+1. Fractional view: Just specify some marginals. Eg. Paging, actual algorithm = distribution over k-tuples but, Fractional: p 1, …, pn s. t. p 1 + …+ pn = k Cost: If p 1, …, pn changes to q 1, …, qn , pay i |pi – qi| Suffices: Fractional Paging -> Randomized Paging (2 x loss)

Fractional Allocation Problem •

A gap example Allocation Problem on 2 points Left Right Requests alternate on locations. Left: (1, 1, …, 1, 0) Right: (1, 0, …, 0, 0) Any integral solution must pay (T) over T steps. Claim: Fractional Algorithm pays only T/(2 k). Left: 0 servers w/p 1/k, and k servers w/p 1 -1/k Right: has 1 server w/p 1. No move cost. Hit cost of 1/k on left requests.

Main Steps •

A word about Fractional Allocation •

Concluding Remarks Removing dependence of depth on aspect ratio. Thm: HST -> Weighted HST with O(log n) depth. 1 2 4 8 Extend Allocation to weighted star. Main question: Can we remove dependence on n. 1. Metric -> HST 2. But even on HST (lose depth of HST)

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