Black hole search
A black hole destroys every agent that enters it, without trace; a team must locate it, losing as few agents as possible.
Problem statement and measuresexpand
A connected graph contains one black hole: a node that destroys any agent arriving there, leaving no trace. All other nodes are safe. Agents start at a safe node \(s\) (the home base) and must determine the location of the black hole; at least one agent must survive and know it. The primary measure is the team size (number of agents needed); the secondary measure is the cost (total number of moves, a move being one traversal of one edge) or, for synchronous agents, the time. Some papers also start the agents scattered, at distinct safe nodes; those results are recorded in the notes, because every catalogued variant starts at a home base.
Measures: Agents, number of agents needed (team size); Cost, total moves (asynchronous) or rounds (synchronous).
Tokens, no map
verifiedWith a constant number of tokens and no map, \(\Delta+2\) agents and 3 tokens are necessary and sufficient.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Graph dynamics: static graph
- \(G_t=G\) for all \(t\): the edge set never changes.
- Edge direction: undirected
- \(G=(V,E)\) with \(E\subseteq\binom V2\); every edge can be traversed in both directions.
- Faults: no faults
- All agents follow the algorithm forever.
- Timing: asynchronous
- Each agent's Communicate–Compute–Move steps take finite but unbounded, adversarially chosen time. Time is measured in epochs. An epoch is an interval of time in which every agent completes at least one full Communicate–Compute–Move step; it need not be the shortest such interval. Time is counted by dividing the execution into consecutive epochs, and a bound of \(X\) epochs holds for every such division.
- Knowledge of the graph: topology unknown
- Agents know \(n\) but not the topology of \(G\).
- Marking nodes: tokens
- Agents carry tokens (pebbles) they can place at nodes or on ports; tokens are identical and carry no other information.
- Topology: any topology
- \(G\) is any connected graph in the class allowed by the other axes.
What is known
- \(O(1)\) tokens
- \(\Delta+1\) agents are not enough; 2 tokens never suffice
- teams of \((2-\varepsilon)\Delta\) agents
Results, in order of publication
First token-based exploration of an unknown dangerous graph; polynomial cost.
Open questions
None recorded for this variant.
What carries over from neighbouring settings
Read off the order of assumptions, not checked case by case: results may carry extra assumptions (shown on their pages), and asynchronous time is measured in epochs.
Algorithms from weaker settings work here
No catalogued weaker setting.
Lower bounds from stronger settings hold here
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