Storing a Sparse Table with 0(1) Worst Case Access Time

Reviewer: William Fennell Smyth

This important little paper describes methods which almost achieve for searching n items what heapsort achieves for sorting: worst case execution time at the theoretical minimum order of magnitude using only n storage locations. The main idea is to store the items in a two-tier data structure, where the first tier consists of pointers to the second; the second tier consists of blocks, each of which typically contains one or two of the n items arranged so as to be directly accessible by a perfect hashing function. The authors first describe the basic method, which guarantees 0(1) worst case execution time using 0( n) storage locations. At the cost of multiplying storage requirements by a constant factor, the basic data structure may be updated in “expected” time 0( n). Applying an auxiliary data structure attributed to Tarjan and Yao [1], the authors then show how the basic method may be modified to reduce storage requirement to n+ o( n) locations, while at the same time maintaining 0(1) execution time. Other possible variations on the basic method are briefly described. Although the data structure is certainly of more than theoretical interest, no implementation of update/retrieval algorithms is given or discussed. The use of the term “sparse” in the title really disguises the importance and generality of the methods described since, as the authors point out, the methods apply independent of the sparseness of the n items in their universe of items.