Verification of minimum-redundancy prefix codes

Optimal Prefix Free Codes with Partial Sorting

“…-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.…”

“…Such a deferred data structure is sufficient to simply execute van Leeuwen's algorithm [16] on an unsorted array of positive integers, but would not result in an improvement in the computational complexity: 6 van Leeuwen's algorithm [16] is simply performing n select operations on the input, effectively sorting the unsorted array.…”

Optimal prefix codes with fewer distinct codeword lengths are faster to construct

“…We have shown in [1] that the verification of a given prefix code for optimality requires Ω(n log n) in the algebraic decision-tree model. That lower bound was illustrated through an example of a prefix code with k = Θ(log n) distinct codeword lengths.…”

“…In [1], we called this second property the exclusion property. In general, building a Huffman tree T that has the sibling property (both the first and second properties) for a list of n weights by evaluating all its internal nodes requires Ω(n log n).…”

“…The running time of our algorithm is O(k • n). Following our previous work in [1], no algorithm can possibly construct optimal prefix codes in o(k • n) time. When the given weights are presorted our algorithm performs O(9 k • log 2k n) comparisons.…”