$S$-Hypersimplices, Pulling Triangulations, and Monotone paths

An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.

A Monotone Path Proof of an Extremal Result for Long Markov Chains

We prove an extremal result for long Markov chains based on the monotone path argument, generalizing an earlier work by Courtade and Jiao.

Straight-line monotone grid drawings of series–parallel graphs

A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discovered as a new standard for visualizing graphs. In this paper we study monotone drawings of series–parallel graphs in a variable embedding setting. We show that a series–parallel graph of n vertices has a straight-line planar monotone drawing on a grid of size O(n) × O(n2) and such a drawing can be found in linear time.

On Computing the Degree of Convexity of Polyominoes

In this paper we present an algorithm which has as input a convex polyomino $P$ and computes its degree of convexity, defined as the smallest integer $k$ such that any two cells of $P$ can be joined by a monotone path inside $P$ with at most $k$ changes of direction. The algorithm uses space $O(m + n)$ to represent a polyomino $P$ with $n$ rows and $m$ columns, and has a running time $O(min(m; r k))$, where $r$ is the number of corners of $P$. Moreover, the algorithm leads naturally to a decomposition of $P$ into simpler polyominoes.

Monotone Descent Path Queries on Dynamic Terrains

Monotone paths are useful in many engineering design applications. In this paper, we address the problem of answering monotone descent path queries on terrains that are continually changing. A terrain can be represented by a unique contour tree. Such a contour tree belongs to a class of graphs called arbitrarily directed trees (ADTs). Let T be an ADT with n nodes. In this paper, we present a new linear time preprocessing algorithm for decomposing a static ADT T into a forest F, with which we can answer lowest common descendent (LCA) queries in O(1) time. This is useful in answering monotone path queries on the corresponding terrain. We show how to maintain this data structure, and thereby answer LCA queries efficiently, for dynamic ADTs. We also show how to maintain the data structure of dynamic terrains, while simultaneously maintaining the corresponding contour tree. This allows us to efficiently answer monotone path queries between any two points on dynamic terrains.