1 research outputs found
Heapable Sequences and Subsequences
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 heapability of a sequence, and also prove that the question of
whether a sequence can be arranged in a complete binary heap is NP-hard.
Regarding subsequences we show that, with high probability, the longest
heapable subsequence of a random permutation of n numbers has length (1 - o(1))
n, and a subsequence of length (1 - o(1)) n can in fact be found online with
high probability. We similarly show that for a random permutation a subsequence
that yields a complete heap of size \alpha n for a constant \alpha can be found
with high probability. Our work highlights the interesting structure underlying
this class of subsequence problems, and we leave many further interesting
variations open for future work.Comment: 15 pages, 6 figure