Distribution-Sensitive Construction of Minimum-Redundancy Prefix Codes

Optimal Prefix Free Codes with Partial Sorting

“…Our main contribution consists in the analysis of the running time of this solution, described in Section 3: the formal definition of the parameter of the analysis in Section 3.1, the upper bound in Section 3.2 and the matching lower bound in Section 3.3. We conclude with a comparison of our results with those from Belal et al [5] in Section 4.…”

“…-When the weights are given in sorted order, van Leeuwen [16] showed that an optimal code can be computed using within O(n) algebraic operations. -When the weights consist of r ∈ [1..n] distinct values and are given in a sorted, compressed form, Moffat and Turpin [21] showed how to compute an optimal code using within O(r(1 + log(n/r))) algebraic operations, which is often sublinear in n. -In the case where the weights are given unsorted, Belal et al [5,6] described several families of instances for which an optimal prefix free code can be computed in linear time, along with an algorithm claimed to perform O(kn) algebraic operations, in the worst case over instances formed by n weights such that there is an optimal binary prefix free code with k distinct code lengths 3 . This complexity was later downgraded to O(16 k n) in an extended version [4] of their article.…”

“…a doubling search [7] would find such a rank r in 2⌈log 2 r⌉ comparisons). 4 Example 1. Consider an instance of the optimal prefix free code problem formed by n sorted positive weights W [1.…”

“…Their data structure supports q queries in time within O(n(1 + lg q)), all in the comparison model. To achieve this results, it partially sorts its data in order to minimize the computational cost of future queries, but avoids sorting all of the data if the queries don't require it: the queries have 5 then become operators (they modify the data). Note that whereas the running time of each individual query depends on the state of the data, the answer to each query is independent of the state of the data.…”

“…Like the GDM algorithm, the algorithm described by Belal et al [5] for the unsorted case is based on several computations of the median of the weights within a given interval, in particular, in order to select the weights smaller than some well chosen value. The essential difference between both work is the use of deferred data structure, which simplifies both the algorithm and the analysis of its complexity.…”