A polylog competitive algorithm for the k-server problem

1. A polylog competitive algorithm for the k-server problem

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

3. . . . n 1 The Paging/Caching Problem Set of pages {1,2,…,n} , cache of size k<n. Request sequence of pages1, 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)

4. Previous Results: Paging 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, McGeoch, Sleator, Young 91]. • Lower bound Hk[Fiat et al. 91], tight results known.

5. K-server conjecture [Manasse-McGeoch-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’95:2k-1 Chrobak-Larmore’91: k for trees.

6. 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: n2/3[Csaba-Lodha’06] exp(O(log n)1/2) [Bansal-Buchbinder-Naor’10] Depth 2-tree: No o(k) guarantee

7. Our Result Thm: There is an O(log2 k log3 n) competitive* algorithm for k-server on any metric with n points. * Hiding some log log n terms

8. Our Approach Hierarchically Separated Trees (HSTs)[Bartal 96]. Any Metric space 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) O(log n)

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

10. Designing Algorithm on HST d+1 level HST

11. Allocation Problem

12. Allocation to k-server

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

14. 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: p1,…,pn s.t. p1 + …+ pn = k Cost: If p1,…,pn changes to q1,…,qn , pay i |pi – qi| Suffices: Fractional Paging -> Randomized Paging (2x loss)

15. Fractional Allocation Problem

16. 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/(2k) . 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.

17. Main Steps

18. A word about Fractional Allocation

19. Concluding Remarks Removing dependence of depth on aspect ratio. Thm: HST -> Weighted HST with O(log n) depth. 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) 1 2 4 8

20. Thank you