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.

💡 Research Summary

The paper introduces the notion of a “heapable” sequence: a list of numbers that can be inserted one by one into a binary tree while preserving the heap property, with each new element placed at a leaf of the current tree. This definition restricts the usual heap insertion operation to a fixed order of arrivals, turning the classic heap construction problem into a combinatorial question about the order of the input.

The authors first present a deterministic algorithm that decides whether a given sequence is heapable. The algorithm maintains the set of current leaves in a priority queue (or balanced search tree). When a new element arrives, it is compared with the smallest (for a min‑heap) or largest (for a max‑heap) leaf. If the heap condition is satisfied, that leaf becomes an internal node and two new leaves are created; otherwise the sequence is declared non‑heapable. Because each step requires only a logarithmic‑time update of the leaf set, the whole procedure runs in O(n log n) time and is provably correct by an inductive argument that the leaf set always represents the minimal feasible positions for future insertions.

Next, the paper tackles a more constrained variant: can the elements be permuted so that they form a complete binary heap (all levels full, last level filled left‑to‑right)? By constructing a polynomial‑time reduction from 3‑SAT (or Partition), the authors show this decision problem is NP‑hard. The reduction encodes variable assignments and clause satisfaction into placement constraints of the heap, demonstrating that requiring a perfectly balanced shape dramatically increases combinatorial complexity.

The probabilistic part of the work studies random permutations of n distinct numbers. The authors prove that the length of the longest heapable subsequence (LHS) is asymptotically n; more precisely, with high probability the LHS has length (1 − o(1))·n. Their proof combines a martingale analysis with a coupling to a Poisson process that models the evolution of the leaf set. Consequently, almost all elements of a random permutation can be kept while still respecting heapability.

Moreover, the authors design an online algorithm that observes the permutation element by element and decides immediately whether to keep it. The rule is simple: keep the element if it exceeds the smallest current leaf; otherwise discard it. They show that this online strategy also yields a heapable subsequence of length (1 − o(1))·n with high probability. Thus, even without hindsight, one can achieve near‑optimal performance.

Finally, the paper extends the random‑permutation analysis to the complete‑heap setting. For a constant α ∈ (0,1), they prove that a subsequence of size α n that can be arranged into a complete heap exists with high probability. The construction proceeds level by level, ensuring that each level receives enough elements to satisfy the left‑filled property. The result indicates that, despite the stricter shape requirement, a linear‑size complete heap can still be extracted from a random permutation.

Overall, the work establishes a new combinatorial framework that parallels the classic longest increasing subsequence problem but replaces the linear order constraint with a hierarchical heap constraint. It provides (i) an O(n log n) decision algorithm for heapability, (ii) an NP‑hardness proof for complete‑heap arrangement, (iii) high‑probability guarantees for both offline longest heapable subsequences and online near‑optimal constructions, and (iv) analogous guarantees for complete heaps. The authors conclude by outlining several open directions, such as extensions to multi‑ary heaps, weighted elements, and dynamic insertion/deletion models, suggesting a rich landscape for future research.