The Language (and Series) of Hammersley-Type Processes

Abstract

We study languages and formal power series associated to (variants of) the Hammersley process. We show that the ordinary Hammersley process yields a regular language and the Hammersley tree process yields deterministic context-free (but non-regular) languages. For the Hammersley interval process we show that there are two relevant variants of formal languages. One of them leads to the same language as the ordinary Hammersley tree process. The other one yields non-context-free languages. The results are motivated by the problem of studying the analog of the famous Ulam-Hammersley problem for heapable sequences. Towards this goal we also give an algorithm for computing formal power series associated to the Hammersley process. We employ this algorithm to settle the nature of the scaling constant, conjectured in previous work to be the golden ratio. Our results provide experimental support to this conjecture.

Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree

“…Heapability of integer sequences was introduced in [BHMZ11] and has been investigated further in [IB15,Por15,IB16,BGGS16,BS18,BIR18,BBD + 20]. Heapability of integer sequences can be decided by a simple greedy algorithm [BHMZ11] (see also [IB16] for an alternate approach based on integer programming, and [BBD + 20] for connections with Dilworth's theorem and an algorithm based on network flows).…”