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A Polylogarithmic-Competitive Algorithm for the k-Server Problem

Nikhil Bansal, Niv Buchbinder, Aleksander Madry, Joseph, Naor

arXiv:1110.1580v1cs.DS

TL;DR

The paper addresses the lack of nontrivial randomized upper bounds for the k-server problem on arbitrary finite metrics. It uses fractional allocation, HST reductions, weighted-HST embeddings, and online rounding to obtain the first polylogarithmic guarantee, while noting limitations of the fractional allocation relaxation and algorithm analysis.

  • Problem

    Randomized k-server algorithms had no better general-metric upper bound than the deterministic 2k −1 guarantee, despite an Ω(log k) lower bound.

  • Method

    The approach reduces k-server to fractional allocation, builds fractional solutions on HSTs, embeds metrics into shallow weighted HSTs, and rounds online to randomized solutions.

  • Results

    O(log^2 k log^3 n log log n) competitive ratio is achieved for randomized k-server on any metric space with n points.

  • Takeaways & Limitations

    A weak fractional allocation relaxation, despite being Ω(k) from integral allocation in some instances, can still yield a strong randomized k-server algorithm.

  • Takeaways & Limitations

    The fractional allocation view is too weak to approximate the integral allocation problem, with an Ω(k) integral-to-fractional cost gap on some instances.

Abstract

from arXiv · show

We give the first polylogarithmic-competitive randomized online algorithm for the $k$-server problem on an arbitrary finite metric space. In particular, our algorithm achieves a competitive ratio of O(log^3 n log^2 k log log n) for any metric space on n points. Our algorithm improves upon the deterministic (2k-1)-competitive algorithm of Koutsoupias and Papadimitriou [J.ACM'95] whenever n is sub-exponential in k.

1 Introduction

The randomized k-server problem has a large gap between known lower and upper competitive bounds on general metrics. This paper closes part of that gap with the first polylogarithmic-competitive randomized algorithm for finite metrics, using fractional allocation, HST reductions, and weighted-HST depth control.

  • 1 Introduction: Randomized k-server algorithms previously had only an Ω(log k) lower bound and the deterministic 2k −1 upper bound on general metrics, leaving an exponential gap.The gap motivates seeking a polylogarithmic guarantee for arbitrary metric spaces.
  • 1.1 Our Result: The algorithm reduces k-server to a fractional allocation problem, constructs a fractional k-server solution on HSTs, and rounds it online with only an O(1) competitive-loss factor.The fractional relaxation is easier to design than a randomized allocation algorithm, yet still yields an integral randomized k-server algorithm.
  • 1.2 Preliminaries: The fractional allocation relaxation can be Ω(k) cheaper than every integral allocation solution, yet it still suffices to obtain a good integral randomized k-server solution.The paper exhibits an instance with integral cost Ω(T) and fractional cost O(T/k).
  • 1.1 Our Result: For a sufficiently well-separated HST of depth ℓ, the resulting algorithm is roughly O(ℓ log(kℓ))-competitive.The HST construction recursively distributes servers among child subtrees through allocation instances.
  • 1.1 Our Result: Weighted HST embeddings reduce depth from O(log ∆) to O(log n) with O(1) leaf-distance distortion, eliminating dependence on potentially exponential diameter.This depth reduction replaces a log ∆ factor by log n in the competitive bound.

2 Overview of Our Approach

The approach combines a fractional allocation algorithm on weighted stars with reductions to fractional k-server algorithms on HSTs, rounding, and metric embeddings. Composing these ingredients yields the stated polylogarithmic competitive ratio for general finite metrics.

  • Fractional allocation algorithm: A fractional allocation algorithm on weighted stars provides the core building block for the k-server construction.Theorem 5 gives a (1 + ε, O(log(k/ε)))-competitive fractional allocation algorithm.
  • From allocation to k-server problem: O(ℓlog(kℓ))-competitive fractional k-server algorithms on depth-ℓ weighted HSTs follow when the HST separation satisfies σ = Ω(ℓlog(kℓ)).This reduction uses the fractional allocation algorithm as its building block.
  • Putting it all together: Fractional k-server solutions on HSTs can be converted into randomized integral algorithms with only an O(1) competitive-ratio loss.The stated rounding result applies to HSTs with σ > 5.
  • Putting it all together: The proof embeds an arbitrary metric into σ-HSTs, compresses tree depth to O(log n), and transfers the resulting solution back with bounded distortion.The embedding uses σ = Θ(log n log(k log n)); the depth reduction distorts leaf distances by at most 2σ/(σ −1).
  • Putting it all together: O(log^2 k log^3 n log log n) competitive ratio follows by composing the HST algorithm, embedding distortion, and final reduction.The proof expresses the expected cost as the product of the HST competitiveness and the embedding distortion.

3 The Fractional Allocation Problem

The fractional allocation algorithm models server distributions on weighted stars and evolves them through quota-fixing and hit-response stages. Its analysis establishes feasibility, well-defined dynamics, and competitive cost bounds against an integral optimum.

  • 3.1 Description of the Algorithm: Hit costs are represented by non-increasing vectors, whose successive differences encode marginal costs for serving with fewer servers.The allocation state uses variables describing probabilities of server counts at locations, with transformed variables y satisfying quota and consistency constraints.
  • 3.1 Description of the Algorithm: The algorithm processes each quota and hit-cost update through a fix stage followed by a hit stage governed by continuous differential equations.The fix stage enforces the current server quota; the hit stage modifies the fractional configuration in response to the arriving cost vector.
  • 3.1 Description of the Algorithm: The hit-stage dynamics maintain blocks of indices with equal y-values, averaging cost coefficients within blocks and merging adjacent blocks when required.Blocks never split once formed, and their structure preserves the required prefix-average property.
  • 3.2 Correctness: Lemma 11 proves that the algorithm’s differential equations have a unique solution throughout both the fix and hit stages.This establishes that the continuous evolution remains well-defined despite potentially discontinuous changes during the hit stage.
  • 3.3 Competitive Analysis: The potential analysis also bounds each step’s cost by (1 + ε)α · (wmax · |κ(t) − κ(t −1)| + M*t + H*t).Quota changes contribute through the wmax term, while M*t and H*t denote the corresponding optimum costs.

4 Fractional k-server on Weighted HSTs

The paper recursively combines fractional allocation algorithms across weighted HST nodes to construct a fractional k-server algorithm. For sufficiently separated trees, this yields an O(ℓlog(kℓ))-competitive guarantee while using a weaker fractional allocation primitive.

  • 4.1 Construction: A (1 + ε, log(k/ε))-competitive fractional allocation algorithm on weighted stars yields an O(ℓlog(kℓ))-competitive fractional k-server algorithm on weighted σ-HSTs.The guarantee requires σ = Ω(ℓlog(kℓ)).
  • 4.1 Construction: Each internal HST node runs allocation instances with different quota patterns, whose convex combination recursively determines server distributions among child subtrees.The root starts with the total server quota, and child allocations inherit consistency constraints from their parent instances.
  • 4.1 Construction: The construction uses a weaker fractional allocation problem instead of requiring a randomized integral allocation algorithm.This is the main methodological difference from the earlier recursive approach based on integral allocation algorithms.
  • 4.1 Construction: Weighted HSTs allow non-uniform child-edge lengths while preserving exponential separation along every root-to-leaf path.This avoids a dependence on HST depth that could otherwise become polynomial in the metric’s number of points.
  • 4.3 Performance Analysis: Theorem 23 gives the allocation-based movement bound for weighted σ-HSTs, leading to the O(ℓlog(kℓ)) guarantee when σ = Ω(ℓlog(kℓ)).The bound applies to request sequences on the HST under the recursively maintained quota pattern.

5 Weighted HSTs and Online Rounding

This section reduces arbitrary-depth HSTs to shallow weighted HSTs and rounds fractional k-server states online with constant-factor additional cost.

  • 5.2 Rounding the Fractional k-server Solution Online: The section establishes an online randomized rounding procedure that converts fractional k-server states into randomized integral states on σ-HSTs.The procedure maintains states consistent with the fractional solution and targets an O(1)-factor maintenance cost.
  • 5.1 Embedding σ-HSTs into Weighted σ-HSTs: A σ-HST with n leaves can be transformed into a weighted σ-HST of depth O(log n), preserving leaves and distorting leaf distances by at most 2σ/(σ −1).The transformation contracts selected edges while retaining enough of each leaf-to-leaf path's longest-edge structure.
  • 5.2 Rounding the Fractional k-server Solution Online: The rounding proof modifies configurations locally, using swaps and movements whose cost is O(δw(p)), within a constant factor of the corresponding fractional change.This local repair is combined with the balancing lemma to obtain the desired consistent and balanced successor state.
  • 5.2 Rounding the Fractional k-server Solution Online: At every time, the rounded state remains consistent with the fractional state, and a fractional movement cost c_t can be matched by an O(c_t) integral-state modification cost.The construction assumes σ > 5 and uses balanced configurations to control deviations from fractional marginals.
  • 5.2 Rounding the Fractional k-server Solution Online: A configuration is balanced when each subtree contains either the floor or ceiling of its fractional server count, and any consistent state can be balanced at cost O(G(S,x)).The balance gap measures the state’s weighted distance from satisfying these floor-and-ceiling constraints.

A Proof of Lemma 11

This proof analyzes a continuous evolution of allocation variables, showing that normalization is locally monotone and changes only at discrete horizons.

  • A Proof of Lemma 11: The fix-stage evolution preserves feasibility while tracking active coordinates, hit-cost blocks, monotonicity, and the server-quota constraint.The proof extends the process over a maximal interval on which these conditions remain valid.
  • A Proof of Lemma 11: During the hit stage, blocks never split because merged coordinates retain identical y-values and identical hit-cost parameters.Once merged, the coordinates evolve identically thereafter.
  • A Proof of Lemma 11: The weighted normalization N(η) does not increase during the local evolution, because weights shift from above-average to at-most-average coordinates.This yields the monotonicity inequality N(η+) ≤ N(η).
  • A Proof of Lemma 11: N(η) can increase only at a horizon, where a block merge, quota hit, or coordinate inactivation occurs.Between consecutive horizons, the normalization remains non-increasing.

B Proof of Lemma 22

This proof establishes the quota-pattern comparison lemma by induction on quota changes, relating variable-server optima to fixed-quota optima and controlling transition costs.

  • B Proof of Lemma 22: Lemma 30 supplies a configuration Y maximally overlapping X while bounding the excess fixed-quota optimum by the original variable-quota excess cost.The overlap property bounds the transition movement by D|κ(τ + 1) −κ(τ)|.
  • B Proof of Lemma 22: The proof inducts on g(κ), the total variation of the quota pattern, with constant quota as the base case.When g(κ)=0, the relevant hit-cost sum telescopes directly.
  • B Proof of Lemma 22: At the earliest quota change, the proof replaces the original pattern with one having a smaller g(κ), then transfers the argument inductively.The modified pattern changes only the quota at the transition time and satisfies g(κ′) < g(κ).
  • B Proof of Lemma 22: Constructed solutions splice the inductive solution before the quota change with the original solution afterward, yielding the required lower and upper bounds on Optcost(κ).The transition cost and variation term are combined so that c(Y,X) + D·g(κ′) ≤ D·g(κ).
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