Partitioning a sequence into few monotone subsequences

Abstract.

In this paper we consider the problem of finding sets of long disjoint monotone subsequences of a sequence of $n$ numbers. We give an algorithm that, after $O(n \log n)$ preprocessing time, finds and deletes an increasing subsequence of size $k$ (if it exists) in time $O(n + k^2)$. Using this algorithm, it is possible to partition a sequence of $n$ numbers into $2 \lfloor \sqrt n \rfloor$ monotone subsequences in time $O(n^{1.5})$. Our algorithm yields improvements for two applications: The first is constructing good splitters for a set of lines in the plane. Good splitters are useful for two dimensional simplex range searching. The second application is in VLSI, where we seek a partitioning of a given graph into subsets, commonly refered to as the pages of a book, where all the vertices can be placed on the spine of the book, and each subgraph is planar.