An optimal algorithm for solving the searchlight guarding problem on weighted trees

doi:10.1016/0020-0255(95)00122-0

Information Sciences

Volume 87, Issues 1–3, November 1995, Pages 79-105

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Abstract

In this paper, we propose a new graph problem on a connected weighted undirected graph, called the searchlight guarding problem. Consider that there is a fugitive who moves along the edges of a graph at any speed. We want to place a set of searchlights at vertices to search the edges of the graph and capture the fugitive. Suppose that it costs some building cost to place a searchlight at some vertex. The searchlight guarding problem is to allocate a set S of searchlights such that the summation of the building costs is minimized. If there is more than one set of searchlights with the minimum building cost, then find the one with the minimum searching time, that is, where the time slot needed to capture the fugitive is minimum.

We first prove that the problem is NP-hard on weighted bipartite graphs. Then an O(n) time optimal algorithm is designed to solve the problem on weighted trees, where n is the number of the vertices of the given weighted tree. The algorithm is divided into two phases: In the first phase, we find the set of searchlights with the minimum guarding cost and assign the searching directions of all edges by the dynamic programming strategy. In the second phase, the searched time slots of each edge are determined. Both phases take linear time.

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Cited by (4)

The searchlight problem for road networks
2015, Theoretical Computer Science
Citation Excerpt :
In particular, our searchlight problem is a variant of the art gallery problem for lines and line segments, which was first formulated by Ntafos [17], and then extensively studied in its several variants in [1–3,6,7,9,14,19,29,30,36]. Furthermore, our problem is also related to the searchlight problem for graphs introduced and studied in [38–40]. The difference between the two lies in the visibility model: in the searchlight problem for graphs visibility is restricted to within the neighborhood of a vertex where a searchlight is located, while visibility in our problem is based on a straight line of sight within a line or a line segment.
We consider the problem of searching for a mobile intruder hiding in a road network given as the union of two or more lines, or two or more line segments, in the plane. Some of the intersections of the road network are occupied by stationary guards equipped with a number of searchlights, each of which can emit a single ray of light in any direction along the lines (or line segments) it is on. The goal is to detect the intruder, that is, to illuminate its location. Guards may alter the direction in which they aim a searchlight, but need to switch it off for some finite time interval to effect the change. In contrast, the intruder may move with arbitrary speed along the network (but cannot pass guards) and exploit this time interval to recontaminate previously illuminated sections of the network. For various classes of road networks characterized by the number n of lines (or line segments) comprising it and the number g ( $\leq n - 1$ ) of possible locations of guards (fixed in advance and guaranteed to give complete coverage), we present several upper and lower bounds on the worst-case number of searchlights, each placed at one of the guard positions, required to successfully search a given road network. In particular, we prove the following results:
- 1.
  $\min {2 g - 1, n - 2}$ searchlights are sometimes necessary and $\min {\frac{7}{3} g, n} - 1$ are always sufficient for searching a road network given as the union of n lines;
- 2.
  $Ω (g \cdot \log \frac{n}{g})$ searchlights are sometimes necessary and $O (g^{2} \cdot \log n)$ searchlights are always sufficient for searching a road network given as the union of n line segments, and
- 3.
  at most one searchlight per guard position, and hence a total of at most g searchlights, is always sufficient for searching a road network given as the union of axis-aligned lines or line segments.
The proofs of the upper bounds induce algorithms for generating a search schedule for detecting the intruder using the claimed number of searchlights.
An optimal algorithm for solving the searchlight guarding problem on weighted interval graphs
1997, Information Sciences
In this paper, a graph problem on connected, weighted, undirected graphs, called the searchlight guarding problem, is considered. Assume that there is a fugitive who moves along the edges of the graph at a random speed. The task involves placing a set of searchlights at vertices to search the edges of the graph and to spot the fugitive. Suppose that placing a searchlight at some vertex incurs some building cost. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total cost of the vertices in S is minimized. If there is more than one set of searchlights, each with a minimum building cost, then identify the set with the minimum search time, that is, where the time slots needed to spot the fugitive is the minimum. As is well established, the problem is NP-hard on weighted bipartite graphs but is linear-time solvable on weighted trees. In this paper, the design of a linear-time optimal algorithm for the searchlight guarding problem on weighted interval graphs is presented. It entails two phases. In the first phase, a set of searchlights with minimum guarding cost is identified and the search directions of all edges are assigned. To achieve this task, a new problem, called the edge-direction assignment problem, is first defined and the problem on weighted complete-split graphs is solved by the greedy strategy. Based on this computational result, the problem of finding the set of searchlights with minimum guarding cost and assigning the search directions of all edges is solved by the dynamic programming strategy. Then, in the second phase, the search time slots of each edge are determined on the basis of the results of the first phase and on some properties of interval graphs.
The searchlight guarding problem on weighted split graphs and weighted cographs
2000, Networks
An optimal algorithm for solving the searchlight guarding problem on weighted two-terminal series-parallel graphs
1999, Acta Informatica

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Information Sciences

An optimal algorithm for solving the searchlight guarding problem on weighted trees

Abstract

An O(|V|3) algorithm for finding maximum flows in networks

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An optimal algorithm for solving the searchlight guarding problem on weighted interval graphs

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An O(|V|³) algorithm for finding maximum flows in networks