Computational geometry: an introduction
Computational geometry: an introduction
Fast heuristics for minimum length rectangular partitions of polygons
SCG '86 Proceedings of the second annual symposium on Computational geometry
The complexity of computing minimum separating polygons
Pattern Recognition Letters - Special issue on computational geometry
A pedestrian approach to ray shooting: shoot a ray, take a walk
SODA '93 Selected papers from the fourth annual ACM SIAM symposium on Discrete algorithms
Approximation algorithms for the geometric covering salesman problem
Discrete Applied Mathematics
Approximation algorithms for geometric tour and network design problems (extended abstract)
Proceedings of the eleventh annual symposium on Computational geometry
Bounds for partitioning rectilinear polygons
SCG '85 Proceedings of the first annual symposium on Computational geometry
Beyond Steiner's Problem: A VLSI Oriented Generalization
WG '89 Proceedings of the 15th International Workshop on Graph-Theoretic Concepts in Computer Science
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Shortest paths in simple polygons with polygon-meet constraints
Information Processing Letters
Query-point visibility constrained shortest paths in simple polygons
Theoretical Computer Science
Visiting a Polygon on the Optimal Way to a Query Point
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
Racetrack: an approximation algorithm for the mobile sink routing problem
ADHOC-NOW'10 Proceedings of the 9th international conference on Ad-hoc, mobile and wireless networks
Parameterized algorithms for generalized traveling salesman problems in road networks
Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
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In TSP with neighborhoods (TSPN) we are given a collection X of k polygonal regions, called neighborhoods, with totally n vertices, and we seek the shortest tour that visits each neighborhood. The Euclidean TSP is a special case of the TSPN problem, so TSPN is also NP-hard. In this paper we present a simple and fast algorithm that, given a start point, computes a TSPN tour of length O(log k) times the optimum in time O(n+k log k). When no start point is given we show how to compute a "good" start point in time O(n2 log n), hence we obtain a logarithmic approximation algorithm that runs in time O(n2 log n). We also present an algorithm which performs at least one of the following two tasks (which of these tasks is performed depends on the given input): (1) It outputs in time O(n log n) a TSPN tour of length O(log k) times the optimum. (2) It outputs a TSPN tour of length less than (1+Ɛ) times the optimum in cubic time, where Ɛ is an arbitrary real constant given as an optional parameter. The results above are significant improvements, since the best previously known logarithmic approximation algorithm runs in Ω(n5) time in the worst case.