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

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 l...

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Bibliographic Details
Published in:Algorithmica Vol. 86; no. 9; pp. 3027 - 3053
Main Author: Eppstein, David
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2024
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Convexity
Data Structures and Information Theory
Mathematics of Computing
Polynomials
Theory of Computation
Output-sensitive enumeration
Non-crossing Hamiltonian path
Polygonalization
Approximate counting
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary: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. We do not assume that the points are in general position. 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.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-024-01255-y