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. In most settings the black hole is present from the start; an eventually emerging black hole appears at an unknown time during the execution, and time is then counted from its appearance. 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).
Dynamic rings
verifiedIn a 1-interval-connected ring three agents are necessary and sufficient; with pebbles they need \(\Theta(n^{1.5})\) moves.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Graph dynamics: connected every round
- \(\mathcal G=(G_t)_{t\ge0}\) is 1-interval connected if every snapshot \(G_t=(V,E_t)\) is connected. More generally it is \(T\)-interval connected if for every \(t\) the graph \(\bigl(V,\bigcap_{i=t}^{t+T-1}E_i\bigr)\) is connected.
- 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.
- Black hole appearance: present from the start
- The black hole is present at time 0, and the home base \(s\) is safe throughout.
- Timing: synchronous
- Time proceeds in rounds \(t=1,2,\dots\); in every round every agent communicates, computes and moves (at most one edge) simultaneously.
- 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: ring
- \(G\) is a cycle \(C_n\) (every node of degree 2), or a path when stated.
What is known
- fewer agents fail even when co-located with whiteboards and identifiers
- \(n\) must be known
- co-located agents, whiteboards
Results, in order of publication
Introductory and current best; with face-to-face communication or vision \(\Theta(n^2)\).
Open questions
None recorded for this variant.
Notes
The ring is oriented and each agent carries one pebble; \(n\) must be known Di Luna et al. 2025b. With scattered agents (distinct arbitrary starting nodes, with visible identifiers) three agents with pebbles are still necessary and sufficient, and the cost rises to \(\Theta(n^2)\) moves and rounds; the lower bound holds even with whiteboards. With only face-to-face communication, three scattered agents fail even on a static oriented ring Di Luna et al. 2023 (Theorems 4 to 7 of its 2026 arXiv version; the OPODIS version, and Section 6 of the 2020 arXiv version of Di Luna et al. 2025b, state this impossibility for any constant number of agents). Arbitrary dynamic graphs with whiteboards Kaur et al. 2026b are a setting of their own.
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