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Computing a Hamiltonian Path of Minimum Euclidean Length Inside a Simple Polygon

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Abstract

Given an n-vertex convex polygon, we show that a shortest Hamiltonian path visiting all vertices without imposing any restriction on the starting and ending vertices of the path can be found in O(nlogn) time and Θ(n) space. The time complexity increases to O(nlog2 n) for computing this path inside an n-vertex simple polygon. The previous best algorithms for these problems are quadratic in time and space. For our purposes, we reformulate the above shortest-path problems in terms of a dynamic programming scheme involving falling staircase anti-Monge weight-arrays, and, in addition, we provide an O(nlogn) time and Θ(n) space algorithm to solve the following one-dimensional dynamic programming recurrence

$$E[i] = \min _{1\le j\le k}\min _{k\le i} \{V[k-1] + b(i,j) + c(j,k)\},\quad i=1, \dots,n,$$

where V[0] is known, V[k], for k=1,…,n, can be computed from E[k] in constant time, and B={b(i,j)} and C={c(j,k)} are known falling staircase anti-Monge weight-arrays of size n×n.

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Acknowledgements

The authors express their sincere thanks to the anonymous referees for their insightful comments, which led to an improvement of the presentation of the paper.

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Authors and Affiliations

  1. IUMA, Dep. Métodos Estadísticos, Universidad de Zaragoza, Pedro Cerbuna 12, 50009, Zaragoza, Spain

    A. García & J. Tejel

  2. Centro Politécnico Superior, Dep. Métodos Estadísticos, Universidad de Zaragoza, María de Luna 3, 50018, Zaragoza, Spain

    P. Jodrá

Authors
  1. A. García
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  2. P. Jodrá
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  3. J. Tejel
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Corresponding author

Correspondence to P. Jodrá.

Additional information

Research of A. García and J. Tejel partially supported by the projects MTM2009-07242 and DGA E58 228/80.

Research of P. Jodrá partially supported by the projects MTM2009-14394-C02-01 and DGA E18 226/172.

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García, A., Jodrá, P. & Tejel, J. Computing a Hamiltonian Path of Minimum Euclidean Length Inside a Simple Polygon. Algorithmica 65, 481–497 (2013). https://doi.org/10.1007/s00453-011-9603-5

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