Time Optimal Algorithms for Black Hole Search in Rings

Abstract

Abstract. In a network environments supporting mobile entities (called robots or agents), a black hole is harmful site that destroys any incoming entity without leaving any visible trace. The black-hole search problem is the task of a team of k > 1 mobile entities, starting from the same safe location and executing the same algorithm, to determine within finite time the location of the black hole. In this paper we consider the black hole search problem in asynchronous ring networks of n nodes, and focus on the time complexity. It is known that any algorithm for black-hole search in a ring requires at least 2(n − 2) time in the worst case. The best algorithm achieves this bound with a team of n − 1 agents with an average time cost 2(n − 2), equal to the worst case. In this paper we first show how the same number of agents using 2 extra time units from optimal in the worst case, can solve the problem in only 7 4 n − O(1) time on the average. We then prove that the optimal average case complexity 3 2 n − O(1) can be achieved without increasing the worst case using 2(n − 1) agents Finally we design an algorithm that achieves asymptotically optimal both worst case and average case time complexity employing an optimal team of k = 2 agents, thus improving on the earlier results that required O(n) agents.

Explore and repair graphs with black holes using mobile entities

“…It can be known in advance [7,12] or just a bound on the number of nodes is provided [11]. It can refer to a specific topology like rings [9,13], trees [6], hypercubes [14], tori [15]. Entities may communicate only when they meet [16], or by means of white-boards associated to the nodes of the graph [13], or simply by opportunely disposing available pebbles [17] or tokens [18].…”

“…It can refer to a specific topology like rings [9,13], trees [6], hypercubes [14], tori [15]. Entities may communicate only when they meet [16], or by means of white-boards associated to the nodes of the graph [13], or simply by opportunely disposing available pebbles [17] or tokens [18]. The objective function may ask for the minimum number of entities, the minimum number of steps performed by all the entities, the minimum number of synchronous steps.…”

Asynchronous Gathering in a Dangerous Ring

“…This problem has been extensively investigated, both in synchronous and asynchronous networks (e.g., [13][14][15][16][17][18][19][20][21][22][23][24][25]; see [26] for a recent survey). The case when the network is an asynchronous ring (the setting considered here) has been investigated (e.g., [13,27,28]); however, with the exception of [28], these investigations assume that all agents start from the same node (i.e., they are initially gathered). The best known bounded time complexity for this problem is O(n log n) [13].…”