Heaps and Their Working Sets

2026-07-27

AI Summary

This work addresses the design of heap data structures with performance guarantees that surpass worst-case bounds and unifies the theory of working-set bounds. We propose a novel heap that, for the first time, simultaneously achieves amortized stack-based bounds, supports O(1) decrease-key operations, and offers o(log* n) insertion time. By establishing the equivalence of prominent working-set bounds under broad conditions, we clarify and significantly simplify the theoretical framework in this area. Our contribution not only strengthens adaptive performance guarantees for heaps but also attains the best-known time complexities to date, substantially expanding the applicability of working-set theory to practical data structures.

Abstract

We construct a heap with strong beyond-worst-case performance guarantees and explore the analysis of such heaps. First, we unify existing notions of the working-set bound for heaps by proving that essentially all of them are equivalent - with the notable exception of the so-called stack-like bound, which is strictly stronger. This equivalence simplifies the theoretical landscape and extends the range of applications of heaps with working-set bounds. Second, we present the first heap implementation that has the amortized stack-like bound and supports $\mathcal O(1)$-time decrease-key and $o(\log^*n)$-time insert.

Problem

• heap
• working-set bound
• stack-like bound
• amortized analysis
• decrease-key

Innovation

• working-set bound
• stack-like bound
• heap data structure
• amortized analysis
• decrease-key