155 research outputs found
More Efficient Algorithms and Analyses for Unequal Letter Cost Prefix-Free Coding
There is a large literature devoted to the problem of finding an optimal
(min-cost) prefix-free code with an unequal letter-cost encoding alphabet of
size. While there is no known polynomial time algorithm for solving it
optimally there are many good heuristics that all provide additive errors to
optimal. The additive error in these algorithms usually depends linearly upon
the largest encoding letter size.
This paper was motivated by the problem of finding optimal codes when the
encoding alphabet is infinite. Because the largest letter cost is infinite, the
previous analyses could give infinite error bounds. We provide a new algorithm
that works with infinite encoding alphabets. When restricted to the finite
alphabet case, our algorithm often provides better error bounds than the best
previous ones known.Comment: 29 pages;9 figures
The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions
The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis.
This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case.
More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0.
For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors
Prefix Codes: Equiprobable Words, Unequal Letter Costs
Describes a near-linear-time algorithm for a variant of Huffman coding, in
which the letters may have non-uniform lengths (as in Morse code), but with the
restriction that each word to be encoded has equal probability. [See also
``Huffman Coding with Unequal Letter Costs'' (2002).]Comment: proceedings version in ICALP (1994
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