The Accuracy of One Polynomial Algorithm for the Convergecast Scheduling Problem on a Square Grid with Rectangular Obstacles
In the Convergecast Scheduling Problem, it is required to find in the communication graph an oriented spanning aggregation tree with a root in a base station and the arcs oriented to the root and to build a conflict-free min-length schedule for aggregating data along the arcs of the aggregation tree. This problem is NP-hard in general, however, if the communication graph is a unit square grid in each node of which there is a sensor and in which a data packet is transmitted along any edge during a one-time slot, the problem is polynomially solvable. In this paper, we consider a communication graph in the form of a square grid with rectangular obstacles impenetrable by the messages. In our previous paper, we proposed a polynomial algorithm for constructing a feasible schedule and intensive numerical experiment allowed us to make a hypothesis that the algorithm constructs an optimal solution. In this paper, we present a counterexample and prove that the proposed algorithm constructs a schedule of length at most one time round longer than the optimal schedule.
The research of A.I. Erzin is partly supported by the Russian Foundation for Basic Research, Projects 16-07-00552 and by the program of fundamental scientific researches of the SB RAS No. I.5.1. (project 0314-2016-0014). The research of R.V. Plotnikov is partly supported by the Russian Foundation for Basic Research, Project 16-37-60006.
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