Dispersion
Spread \(k \le n\) agents over the nodes of an \(n\)-node anonymous graph so that no node hosts more than one agent, as fast as possible and with as little memory per agent as possible.
Problem statement and measuresexpand
Given a connected graph \(G\) with \(n\) nodes and \(m\) edges, and \(k \le n\) agents placed on the nodes of \(G\) in any initial configuration (the node of each agent; several agents may share a node), the agents must reach a configuration in which every node hosts at most one agent, and stay there.
Time is the number of rounds (synchronous) or epochs (asynchronous; an epoch is an interval in which every agent completes at least one full Communicate–Compute–Move step, not necessarily the shortest one) until the dispersed configuration is reached; where a result also makes every agent terminate explicitly, the notes say so. Memory is the number of bits of persistent memory each agent needs, counting its \(O(\log k)\)-bit identifier. Unless a variant or a result says otherwise, agents know nothing about \(n\), \(m\), \(k\), \(\Delta\) or \(D\).
Dispersion is the natural "one agent per node" problem: it is equivalent to leaving one agent at every node of a depth-first search (DFS) tree of \(k\) nodes, which is why depth-first search bounds such as \(\min(m, k\Delta)\) appear throughout, and it is a building block for exploration, leader election and self-deployment of agents.
Measures: Time, rounds in the synchronous model, epochs in the asynchronous model, until dispersion is reached; Memory, bits of persistent memory per agent, including its identifier.
Silent agents (no communication)
verifiedAgents cannot exchange any information; at its node an agent only learns whether it is alone and whether the number of agents there went up or down since the last round.
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.
- Number of agents: \(k \le n\)
- The number of agents satisfies \(1\le k\le n\).
- Identifiers: distinct IDs (eponymous)
- Distinct identifiers (eponymous agents): the label function \(\mathrm{id}:\{1,\dots,k\}\to\{1,\dots,L\}\) is injective; usually \(L=k^{O(1)}\), so a label takes \(O(\log k)\) bits.
- Randomness: deterministic
- The next state and move of an agent are a function of its current state and what it observes.
- Faults: no faults
- All agents follow the algorithm forever.
- Communication: no communication
- Agents exchange no messages; an agent at \(v\) observes only the number of agents at \(v\) (and, where stated, whether that number changed).
- Timing: synchronous
- Time proceeds in rounds \(t=1,2,\dots\); in every round every agent communicates, computes and moves (at most one edge) simultaneously.
- Node memory: no whiteboards
- Nodes have no storage: no information can be left at a node except by an agent staying there.
- Node identifiers: anonymous nodes
- Nodes have no identifiers; an agent at \(v\) sees only \(\deg(v)\) and the port numbers \(1,\dots,\deg(v)\), which are assigned locally and independently at each endpoint of an edge.
- Topology: any topology
- \(G\) is any connected graph in the class allowed by the other axes.
- Initial configuration: any start
- Agents start at arbitrary nodes; \(\ell\le k\) nodes are initially occupied, possibly with several agents each.
Statement for this variant
No communication at all. At the start of a round an agent at \(v\) learns (i) whether it is alone at \(v\) and (ii) whether the number of agents at \(v\) increased or decreased since the previous round. Agents have identifiers in \([0, L]\) and know only their own; memory is counted as additional bits beyond the identifier.
What is known
- no communication: only alone / count-change sensing
- identifiers in \([0,L]\), unknown to the other agents
- \(\Omega(k)\) from distance; \(\Omega(\log L)\) from distinguishing identifiers by waiting
- proven on rings, hence for arbitrary graphs
- additional bits beyond the identifier
- rooted configuration
- deterministic
Results, in order of publication
Introductory and current best for general graphs; the agents terminate. Information is conveyed only by the timing of moves and by counts of co-located agents.
Open questions
Close the time gap on general graphs: is \(O(k + \log L)\) achievable, as on rings?
Arbitrary (multi-source) initial configurations on general graphs; only rings are solved Das et al. 2025.
Weaker sensing: agents that only learn whether they are alone and whether the count decreased, and agents that wake up at different times (a 2026 ICDCIT paper by Gorain, Mandal and Sharma treats this; its bounds are not recorded here).
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
Branches from here
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.