Selectable Heaps and Optimal Lazy Search Trees

Abstract

We show the O(log n) time extract minimum function of efficient priority queues can be generalized to the extraction of the k smallest elements in O(k log(n k)) time, where we define log(x) as max(log 2 (x), 1). We first show heap-ordered tree selection (Kaplan et al., SOSA '19) can be applied on the heap-ordered trees of the classic Fibonacci heap to support the extraction in O(k log(n k)) amortized time. We then show selection is possible in a priority queue with optimal worst-case guarantees by applying heap-ordered tree selection on Brodal queues (SODA '96), supporting the operation in O(k log(n k)) worst-case time. Via a reduction from the multiple selection problem, Ω(k log(n k)) time is necessary if insertion is supported in o(log n) time.We then apply the result to lazy search trees (Sandlund & Wild, FOCS '20), creating a new interval data structure based on selectable heaps. This gives optimal O(B + n) time lazy search tree performance, lowering insertion complexity into a gap ∆ i to O(log(n ∆ i )) time. An O(1) time merge operation is also made possible when used as a priority queue, among other situations. If Brodal queues are used, runtimes of the lazy search tree can be made worst-case in the general case of two-sided gaps. The presented data structure makes fundamental use of soft heaps (Chazelle, J. ACM '00), biased search trees, and efficient priority queues, approaching the theoretically-best data structure for ordered data.