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Given a convex polygon with n vertices, we study the problem of identifying a triangle with its smallest side as large as possible among all the triangles that can be drawn inside the polygon. We show that at least one of the vertices of such a triangle must coincide with a vertex of the polygon. We also propose an O (n 2) time algorithm to compute such a triangle inside the given convex polygon. [ABSTRACT FROM AUTHOR]

In the paper "Linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares", Theor. Comput. Sci. 769 (2019) 63–74, the LHIT problem is proposed as follows: For a given set of non-intersecting line segments L = { ℓ 1 , ℓ 2 , ... , ℓ n } in Image 1 , compute two axis-parallel congruent squares S 1 and S 2 of minimum size whose union hits all the line segments in L , and a linear time algorithm was proposed. Later it was observed that the algorithm has a bug. In this corrigendum, we corrected the algorithm. The time complexity of the corrected algorithm is O (n 2). [ABSTRACT FROM AUTHOR]

Highlights • Algorithms for covering and hitting a given set of line segments by two congruent squares of minimum size is studied. • The algorithm produces optimum solution by sequentially reading the input array (read only) twice. • The extra space complexity of the algorithm is O (1). • A restricted version of covering is also studied where each object is to be completely covered by at least one square. • This algorithm gives 2 approximation result to cover/hit the segments with two congruent disks of minimum size. Abstract This paper discusses the problem of covering and hitting a set of line segments L in R 2 by a pair of axis-parallel congruent squares of minimum size. We also discuss the restricted version of covering, where each line segment in L is to be covered completely by at least one square. The proposed algorithms assume that the input segments are given in a read-only array. For each of these problems (i.e. covering , hitting and restricted covering problems), our proposed algorithm reports the optimum result by executing only two passes of reading the input array sequentially. All these algorithms need only O (1) extra space. The solution of these problems also give a 2 approximation for covering and hitting those line segments L by two congruent disks of minimum radius with same computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamic version of covering the convex hull of a point set P in R2 by two congruent disks of minimum size. Here, the points can be added or deleted in the set P, and the objective is to maintain a data structure that, at any instant of time, can efficiently report two disks of minimum size whose union completely covers the boundary of the convex hull of the point set P. We show that maintaining a linear size data structure, we can report a radius r satisfying r ≤ 2ropt at any query time, where ropt is the optimum solution at that instant of time. For each insertion or deletion of a point in P, the update time of our data structure is O (log n). Our algorithm can be tailored to work in the restricted streaming model where only insertions are allowed, using constant work-space. The problem studied in this paper has novelty in two ways: (i) it computes the covering of the convex hull of a point set P, which has lot of surveillance related applications, but not studied in the literature, and (ii) it also considers the dynamic version of the problem. In the dynamic setup, the extent measure problems are studied very little, and in particular, the k-center problem is not at all studied for any k ≥ 2. [ABSTRACT FROM AUTHOR]

We study broadcasting of information in a system of moving agents equipped with omnidirectional as well as directional antenna. The agent communication protocol is inspired by the classical SIRS epidemics dynamics. We assume that the antennas of all agents have a fixed transmitting power, while signal reception only occurs when the receivers sense signals with power exceeding a certain threshold. Thus, information exchange is a local phenomenon which depends on the relative distance and antenna orientation between the transmitting and the receiving agent. We derive an expression for the mean broadcasting time and study the information dissemination robustness of the system using elements of classical epidemiology and physics. In particular, we show that the mean broadcasting time depends upon ψ which quantifies the area the radiation pattern of the antenna sweeps as it moves. We report three important observations (a) directional antennas perform better than omnidirectional antennas, (b) directional antennas whose beam-width is narrower perform even better, and (c) the performance enhances a lot if directional antennas rotate. These behaviors can be understood in the light of the reported analytical findings. [ABSTRACT FROM AUTHOR]