Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

TL;DR

The numbers of non-crossing structures to two easily-computed parameters of the point set are related: the minimum number of points whose removal results in a collinear set, and the number of Points interior to the convex hull.

Abstract

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.

Figures (9)

• Figure 1: Left: A point $p$ (blue), a point set $S$ (red and yellow), and the visible vertices of $S$ from $p$ (the three red vertices). Note that points of $S$ that lie within convex hull edges are not counted as vertices. Right: A maximal visible-vertex path (red edges) for $S\cup\{p\}$ starting from $p$, showing for one of its steps the hull (light blue) and visible vertices (red) of the remaining points.
• Figure 2: A tree of the visible-vertex paths in a point set, where the parent of each path is obtained by removing its last point. This point set has $\textsc{offline}(S)=2$. The root and the next two levels of nodes each have multiple children, but some nodes on the last level shown in the figure have only one child, because the last point on the path is collinear with all remaining points.
• Figure 3: Partition of the halfspace above $L$ into convex subsets, a non-crossing Hamiltonian path respecting the partition, and its signature, a $010$-avoiding binary sequence.
• Figure 4: Illustration for the proof of \ref{['lem:distinct-paths']}: The green vertex-visible path is labeled by the sequence $\sigma=10011100101001101100$ for the points shown, split by the vertical line through $p$ (the bottom green point) into $A$ (the red points left of the line) and $B$ (the blue points right of the line). One of the convex hulls of the remaining points partway through the sequence is shown; the path reaches this hull at one endpoint of its unique edge with endpoints in both $A$ and $B$.
• Figure 5: The case of \ref{['lem:nongen']} when there are many points on edge $e$.

Theorems & Definitions (44)

• Definition 1
• Theorem 5
• proof
• Definition 6
• Theorem 9
• proof
• Definition 10
• Lemma 11
• proof
• Lemma 12