Use Knowledge to Learn Faster: Topology Recognition with Advice

Abstract

Abstract. Topology recognition is one of the fundamental distributed tasks in networks. Each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes. We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D ≤ αn, for any constant α < 1. In most cases, our bounds are asymptotically tight. More precisely, if the allotted time is D − k, where 0 < k ≤ D, then the optimal size of advice is Θ((n 2 log n)/(D − k + 1)). If the allotted time is D, then this optimal size is Θ(n log n). If the allotted time is D + k, where 0 < k ≤ D/2, then the optimal size of advice is Θ(1 + (log n)/k). The only remaining gap between our bounds is for time D + k, where D/2 < k ≤ D. In this time interval our upper bound remains O(1 + (log n)/k), while the lower bound (that holds for any time) is 1. This leaves a gap if D ∈ o(log n). Finally, we show that for time 2D + 1, one bit of advice is both necessary and sufficient. Our results show how sensitive is the minimum size of advice to the time allowed for topology recognition: allowing just one round more, from D to D + 1, decreases exponentially the advice needed to accomplish this task.

Deterministic Graph Exploration with Advice

“…Providing nodes or agents with arbitrary kinds of information that can be used to perform network tasks more efficiently has been previously proposed in [1,13,18,19,20,21,22,24,25,26,28,29,30] in contexts ranging from graph coloring to broadcasting and leader election. This approach was referred to as algorithms with advice.…”

“…They proved that constant time is achievable with O(n) bits of advice in such networks, while o(n) bits are not enough. In [25], the authors studied the problem of topology recognition with advice given to nodes. The topic of [27] and [15] was the size of advice needed for fast leader election, resp.…”

Tradeoffs between cost and information for rendezvous and treasure hunt

“…Following the paradigm of algorithms with advice [1,14,16,21,26,28,29,30,31,32,33,34,36,37,38,44,49], this information is provided to the agents at the start of their navigation by an oracle that knows the network, the starting positions of the agents and, in the case of treasure hunt, the node where the treasure is hidden. The oracle assists the agents by providing them with a binary string called advice, which can be used by the agent during the algorithm execution.…”

“…Providing nodes or agents with information of arbitrary type that can be used to perform network tasks more efficiently has been proposed in [1,14,16,21,26,28,29,30,31,32,33,34,36,37,38,42,44,49]. This approach was referred to as algorithms with advice.…”

“…They proved that O(n) bits of advice allow to obtain constant time in such networks, while o(n) bits are not enough. In [33], the authors studied the problem of topology recognition with advice given to nodes. In [21], the authors considered the task of drawing an isomorphic map by an agent in a graph, and their goal was to determine the minimum amount of advice that has to be given to the agent for the task to be feasible.…”