2022/10/12 by Sander Borst, Borst, Sander, Daniel Dadush +5 · 1 citation
Computer Science · Mathematics · #Algorithm #Algorithms and Data Compression #Arithmetic #Binary logarithm #Binary number #Binary search tree #Binary tree #Combinatorics #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Heap (data structure) #Machine Learning and Algorithms #Mathematics #Optimization and Control (math.OC) #Optimization and Search Problems #Randomized algorithm #Running time #Time complexity #Traverse
paper · pdf · doi:10.48550/arxiv.2210.05982
openalex publication_date 2022/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Explorable heap selection is the problem of selecting the nth smallest value in a binary heap. The key values can only be accessed by traversing through the underlying infinite binary tree, and the complexity of the algorithm is measured by the total distance traveled in the tree (each edge has unit cost). This problem was originally proposed as a model to study search strategies for the branch-and-bound algorithm with storage restrictions by Karp, Saks and Widgerson (FOCS '86), who gave deterministic and randomized n⋅ exp(O(√logn)) time algorithms using O(log(n)2.5) and O(√(log n)) space respectively. We present a new randomized algorithm with running time O(nlog(n)3) using O(log n) space, substantially improving the previous best randomized running time at the expense of slightly increased space usage. We also show an Ω(log(n)n/log(log(n))) for any algorithm that solves the problem in the same amount of space, indicating that our algorithm is nearly optimal.