Tight bounds for undirected graph exploration with pebbles and multiple agents
| Autor: | Yann Disser, Jan Hackfeld, Max Klimm |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
FOS: Computer and information sciences
0102 computer and information sciences 02 engineering and technology Computational Complexity (cs.CC) 01 natural sciences Upper and lower bounds 68Q17 Combinatorics Artificial Intelligence Log-log plot TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY Computer Science - Data Structures and Algorithms 0202 electrical engineering electronic engineering information engineering Data Structures and Algorithms (cs.DS) Undirected graph Time complexity Mathematics Multi-agent system Binary logarithm Vertex (geometry) Computer Science::Multiagent Systems Computer Science - Computational Complexity 010201 computation theory & mathematics Hardware and Architecture Control and Systems Engineering A priori and a posteriori 020201 artificial intelligence & image processing Software Information Systems MathematicsofComputing_DISCRETEMATHEMATICS |
| DOI: | 10.48550/arxiv.1805.03476 |
| Popis: | We study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles Θ(log n ) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory Θ(log log n ) pebbles are necessary and sufficient for exploration. We further prove that using collaborating agents instead of pebbles does not help as Θ(log log n ) agents with constant memory each are necessary and sufficient for exploration. For the upper bounds, we devise an algorithm for a single agent with constant memory that explores any n -vertex graph using O (log log n ) pebbles, even when n is not known a priori . The algorithm terminates after polynomial time and returns to the starting vertex. We further show that the algorithm can be realized with additional constant-memory agents rather than pebbles, implying that O (log log n ) agents with constant memory can explore any n -vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph with at most n vertices is already Ω(log log n ) when we allow each agent to have at most O ((log n ) 1 -ε) bits of memory for any ε > 0. Our argument also implies that a single agent with sublogarithmic memory needs Θ(log log n ) pebbles to explore any n -vertex graph. |
| Databáze: | OpenAIRE |
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