Sorting and Searching Using Controlled Density Arrays

Abstract

Algorithms like insertion sort run slowly because of costly shifting of array elements when a value is inserted or deleted. The amount of shifting, however, can be reduced by leaving gaps - unused array locations into which new values can be inserted - at regular intervals in the array. The proper arrangement of gaps is maintained by periodic adjustment. We demonstrate this technique with a stable comparison sort algorithm with expected time $O(N \log N)$, worst case time $O(N \sqrt{N})$, and space requirements 2N. We conjecture that, by using an interpolation search, the expected time can be reduced to $O(N \log \log N)$. By comparison, Quicksort takes expected time $O(N \log N)$, worst case time $O(N^{2})$ and space $N + \log N$. Second, we show that for any fixed $d \geq 2$ a table management algorithm can be constructed that can process a sequence of $N$ instructions chosen from among INSERT, DELETE, SEARCH, and, MIN in worst case time $O(N^{1+1/d})$. Experiments with a version of the algorithms using $d=2$ show a marked improvement over balanced tree schemes for $N$ as large as several thousand.