First-Come-First-Served for Online Slot Allocation and Huffman Coding

TL;DR

This paper demonstrates that a simple First-Come-First-Served strategy in online slot allocation and Huffman coding achieves near-optimal results without prior knowledge of item probabilities, with proven competitive ratios.

Contribution

It introduces and analyzes the FCFS algorithm for online slot allocation and Huffman coding, showing it attains asymptotically optimal costs in an online setting.

Findings

FCFS achieves cost at most opt + 2 log(1+opt) + 2 for Huffman coding.

Optimal competitive ratios are 1+H(n-1) for general costs.

FCFS is effective without prior distribution knowledge.

Abstract

Can one choose a good Huffman code on the fly, without knowing the underlying distribution? Online Slot Allocation (OSA) models this and similar problems: There are n slots, each with a known cost. There are n items. Requests for items are drawn i.i.d. from a fixed but hidden probability distribution p. After each request, if the item, i, was not previously requested, then the algorithm (knowing the slot costs and the requests so far, but not p) must place the item in some vacant slot j(i). The goal is to minimize the sum, over the items, of the probability of the item times the cost of its assigned slot. The optimal offline algorithm is trivial: put the most probable item in the cheapest slot, the second most probable item in the second cheapest slot, etc. The optimal online algorithm is First Come First Served (FCFS): put the first requested item in the cheapest slot, the second (distinct) requested item in the second cheapest slot, etc. The optimal competitive ratios for any online algorithm are 1+H(n-1) ~ ln n for general costs and 2 for concave costs. For logarithmic costs, the ratio is, asymptotically, 1: FCFS gives cost opt + O(log opt). For Huffman coding, FCFS yields an online algorithm (one that allocates codewords on demand, without knowing the underlying probability distribution) that guarantees asymptotically optimal cost: at most opt + 2 log(1+opt) + 2.