Generalizations and Variants of the Largest Non-crossing Matching Problem in Random Bipartite Graphs

TL;DR

This paper investigates the asymptotic behavior of the longest increasing subsequence in various random array models, extending classical problems like LIS and LCS to new settings with different distributions and array structures.

Contribution

It introduces new distribution families and analyzes their impact on the expected length of the LIS, including generalizations to multi-row and symmetry-restricted arrays.

Findings

Asymptotic characterizations for expected LIS length under various distributions

Extension of classical LIS and LCS problems to new array models

Analysis of multi-rowed and symmetry-restricted array variants

Abstract

We are interested in the statistics of the length of the longest increasing subsequence of 2-rowed lexicographically sorted arrays chosen according to distinct families of distributions D = (D_n)_n, and when n goes to infinity. This framework encompasses well studied problems such as the so called Longest Increasing Subsequence problem, the Longest Common Subsequence problem, problems concerning directed bond percolation models, among others. We define several natural families of distinct distributions and characterize the asymptotic behavior of the expected length of a longest increasing subsequence chosen according to them. In particular, we consider generalizations to d-rowed arrays as well as symmetry restricted two-rowed arrays.