On Covering Points with Minimum Turns

  title={On Covering Points with Minimum Turns},
  author={Minghui Jiang},
  booktitle={Int. J. Comput. Geom. Appl.},
  • Minghui Jiang
  • Published in Int. J. Comput. Geom. Appl. 14 May 2012
  • Computer Science, Mathematics
We study the problem of finding a polygonal chain of line segments to cover a set of points in ℝd, d≥2, with the goal of minimizing the number of links or turns in the chain. A chain of line segments that covers all points in the given set is called a covering tour if the chain is closed, and is called a covering path if the chain is open. A covering tour or a covering path is rectilinear if all segments in the chain are axis-parallel. We prove that the two problems Minimum-Link Rectilinear… 

Covering Paths for Planar Point Sets

It is shown that computing a noncrossing covering path for n points in the plane requires Ω(nlogn) time in the worst case, and it is proved that (1−ε)n straight line segments suffice for a small constant ε>0.

On the Minimum Link-Length Rectilinear Spanning Path Problem: Complexity and Algorithms

A new parameterized algorithm with running time O*((2d )k) for the constrained -RSP problem is presented, which significantly improves the previous best result and is the first parameterized algorithms of running timeO*(2O(k))) for the constraints of the constrained d-R SP problem for a fixed .

Minimum-Link Rectilinear Covering Tour is NP-hard in R4

The problems of defining a path and a tour with minimum number of links, also known as Minimum-link Covering Path and Minimum-Link Covering Tour respectively are proven to be NP-hard in $R^2 and the corresponding rectilinear versions are also NP- hard.

Fixed-Parameter Tractable Algorithms for Corridor Guarding Problems

This work proves the fixed parameter tractability of the parameterized version of MCC, namely k-MCC with respect to the parameter k, where k is the number of rooms, and proposes a variant of CTSP, in which the link-distance of the closed walk has to be minimized.

Size constrained k simple polygons

This paper proposes a novel approach for finding k simple polygons that maximize the total weights under the size constraint and demonstrates that the proposed algorithm outperforms baseline approaches and reduces the computational cost to create a SCSP.


  • Marco Ripà
  • Computer Science
    Journal of Fundamental Mathematics and Applications (JFMA)
  • 2021
An optimization problem related to the extension in k-dimensions of the well known 3x3 points problem by Sam Loyd is considered, thanks to a variation of the so called “clockwise-algorithm”, it is shown how it is possible to visit all the 3^k points of the k-dimensional grid given by the Cartesian product of (0, 1, 2).



Covering Paths for Planar Point Sets

It is shown that computing a noncrossing covering path for n points in the plane requires Ω(nlogn) time in the worst case, and it is proved that (1−ε)n straight line segments suffice for a small constant ε>0.

Traversing a Set of Points with a Minimum Number of Turns

For arbitrary point sets, the general problem of traversing an arbitrary set of points in ℝd with an axis-aligned spanning path having a minimum number of links with a constant ratio (depending on the dimension d) approximation algorithm is presented.

Optimal covering tours with turn costs

This work gives the first algorithmic study of a class of “covering tour” problems related to the geometric Traveling Salesman Problem and gives efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.

Angle-Restricted Tours in the Plane

Minimum-link watchman tours

Covering a Set of Points with a Minimum Number of Lines

The minimum line covering problem is considered and it is shown that this problem can be solved in O(n log l) time if l ∈ O(log1−en), and that this is optimal in the algebraic computation tree model.

Link length of rectilinear Hamiltonian tours in grids

An asymptotically optimal algorithm for constructing rectilinear walks traversing all the vertices of complete multidimensional grids is developed and the worst-case behavior of s(G), when G is a multiddimensional grid is analyzed.

Covering a set of points with a minimum number of turns

Some new upper and lower bounds are proved for a restricted version of the piecewise-linear problem in which all motion is orthogonal to the coordinate axes.

NP-completeness and FPT Results for Rectilinear Covering Problems

The proof that the Rectilinear Minimum Link Traveling Salesman Problem and the Recti- linear Hyperplane Cover are NP-complete by a reduction from the One-In-Three 3-SAT problem is provided and this suggests dealing with the intractability just discovered with fixed-parameter tractability.

Fpt-Algorithms for Minimum-Bends Tours

It is proved that this problem is fixed-parameter tractable (FPT), based on the kernelization approach, and two types of constraints derived from the distinction between line-segments and lines are introduced.