Selectable Heaps and Optimal Lazy Search Trees

Bryce Sandlund, Lingyi Zhang

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)) time1, which we prove optimal for comparison-based priority queues with o(log n) time insertion. We show heap-ordered tree selection (Kaplan et al., SOSA '19) can be applied on the heap-ordered trees of the classic Fibonacci heap and Brodal queue, in O(k log(n/k)) amortized and worst-case time, respectively. We additionally show the deletion of k elements or selection without extraction can be performed on both heaps, also in O(k log(n/k)) time. Surprisingly, all operations are possible with no modifications to the original Fibonacci heap and Brodal queue data structures.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 Δi from O(log(n/|Δi|) + log log n) to O(log(n/|Δi|)) 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, all runtimes of the lazy search tree can be made worst-case.

External IDs:dblp:conf/soda/SandlundZ22