Cascade Heap: Towards Time-Optimal Extractions

Abstract

Heaps are well-studied fundamental data structures, having myriads of applications, both theoretical and practical. We consider the problem of designing a heap with an “optimal” extract-min operation. Assuming an arbitrary linear ordering of keys, a heap with n elements typically takes $O(\log n)$ time to extract the minimum. Extracting all elements faster is impossible as this would violate the ${\Omega }(n \log n)$ bound for comparison-based sorting. It is known, however, that is takes only $O(n + k \log k)$ time to sort just k smallest elements out of n given, which prompts that there might be a faster heap, whose extract-min performance depends on the number of elements extracted so far. In this paper we show that this is indeed the case. We present a version of heap that performs insert in $O(1)$ time and takes only $O(\log ^{*} n + \log k)$ time to carry out the k-th extraction (where $\log ^{*}$ denotes the iterated logarithm). All the above bounds are worst-case.