Heapable Sequences and Subsequences
John Byers, Brent Heeringa, Michael Mitzenmacher, and Georgios Zervas

TL;DR
This paper introduces the concept of heapable sequences, explores their properties, provides algorithms for their analysis, and investigates their probabilistic behavior in random permutations, connecting to classical problems like the longest increasing subsequence.
Contribution
It defines heapable sequences, develops an efficient algorithm for testing heapability, proves NP-hardness for complete heap arrangements, and analyzes probabilistic properties of longest heapable subsequences.
Findings
Efficient algorithm for heapability testing.
NP-hardness of arranging sequences into complete heaps.
High probability of long heapable subsequences in random permutations.
Abstract
Let us call a sequence of numbers heapable if they can be sequentially inserted to form a binary tree with the heap property, where each insertion subsequent to the first occurs at a leaf of the tree, i.e. below a previously placed number. In this paper we consider a variety of problems related to heapable sequences and subsequences that do not appear to have been studied previously. Our motivation for introducing these concepts is two-fold. First, such problems correspond to natural extensions of the well-known secretary problem for hiring an organization with a hierarchical structure. Second, from a purely combinatorial perspective, our problems are interesting variations on similar longest increasing subsequence problems, a problem paradigm that has led to many deep mathematical connections. We provide several basic results. We obtain an efficient algorithm for determining the…
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Taxonomy
TopicsOptimization and Search Problems · Nanocluster Synthesis and Applications · Auction Theory and Applications
