37 research outputs found
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
Two new Probability inequalities and Concentration Results
Concentration results and probabilistic analysis for combinatorial problems
like the TSP, MWST, graph coloring have received much attention, but generally,
for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example).
Here, we prove two probability inequalities which generalize and strengthen
Martingale inequalities. The inequalities provide the tools to deal with more
general heavy-tailed and inhomogeneous distributions for combinatorial
problems. We prove a wide range of applications - in addition to the TSP, MWST,
graph coloring, we also prove more general results than known previously for
concentration in bin-packing, sub-graph counts, Johnson-Lindenstrauss random
projection theorem. It is hoped that the strength of the inequalities will
serve many more purposes.Comment: 3