# Weak Coverage of a Rectangular Barrier

## TL;DR

This paper investigates optimization problems for deploying mobile sensors to achieve weak barrier coverage in a rectangular region, providing algorithms and complexity results for different scenarios.

## Contribution

It presents efficient algorithms for some sensor movement problems and proves NP-hardness for others, advancing understanding of barrier coverage optimization.

## Key findings

- O(n^{3/2}) algorithm for MinNum with integer-position sensors of diameter 1
- NP-hardness of MinNum for sensors of diameter 2
- O(n log n) solution for MinSum with homogeneous sensors

## Abstract

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak barrier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (MinMax). We give an O(n^{3/2}) time algorithm for the MinNum problem for sensors of diameter 1 that are initially placed at integer positions; in contrast we show that the problem is NP-hard even for sensors of diameter 2 that are initially placed at integer positions. We show that the MinSum problem is solvable in O(n log n) time for homogeneous range sensors in arbitrary initial positions, while it is NP-hard for heterogeneous sensor ranges. Finally, we prove that even very restricted homogeneous versions of the MinMax problem are NP-hard.