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

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.Comment: ACM-SIAM Symposium on Discrete Algorithms (SODA) 201

Topics in Approximation Algorithms

This thesis focuses on approximation and online algorithms for a few different problems.1) There have been continued improvements in the approximation algorithms for packing and covering problems. Some recent algorithms, including Fastpc, have provable good worst-case running times, but it is not known how they perform in practice compared to the simplex and interior-point algorithms that are widely used to solve these problems. We present an empirical comparison of these algorithms, using our own implementation of Fastpc and CPLEX implementations of simplex and interior-point methods. We use a variety of inputs for this experimental study. We find that Fastpc is slower for small problem instances, but its performance, relative to CPLEX algorithms, improves as the instances get bigger. Our experiments show that for reasonably large inputs Fastpc performs better than the CPLEX algorithms.2) We give deterministic algorithms for some variants of online file caching by reducing the problems to online covering. The variants considered in this study include one or both of the following features: (i) a rental cost for each slot occupied in the cache, and (ii) zapping a file by paying a cost so that the zapped file does not occupy any space in the cache and does not incur any retrieval cost. The rental cost is motivated by the idea of energy efficient caching where caching systems can save power by turning off slots not being used to store files. Our approach is based on the online covering algorithm by Koufogiannakis and Young. We give deterministic lower bounds for these variants, which are tight within con-stant factors of the upper bounds for most of the cases. We also give randomized algorithms and lower bounds for the variants with rental cost.3) We introduce online Huffman coding. In Huffman coding, the symbols are drawn from a probability distribution and revealed one by one, and the goal is to find a minimum cost prefix-code for the symbols. In the online version, the algorithm has to assign a codeword to a symbol when it is revealed for the first time. We propose an online greedy algorithm and show that it is constant-competitive for online Huffman coding. We also show a lower bound of 10/9 on the competitive ratio of any deterministic online algorithm