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.
Dynamic graphs (1-interval connected)
verifiedAn adversary changes the edge set every round, keeping the graph connected in every round. With only local communication the problem is impossible.
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.
- 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: local communication
- Agents at the same node at the same time can read each other's entire memory (face-to-face communication). Agents crossing on an edge do not meet.
- 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
The node set is fixed; in every round the adversary chooses the edge set, subject to the graph being connected in that round (1-interval connectivity). Agents see the ports of the current round. 1-neighbourhood knowledge means an agent knows which neighbouring nodes are occupied and by whom.
What is known
- \(k \ge 5\)
- deterministic
- local communication
- holds even with 1-neighbourhood knowledge and unbounded memory
- \(k \ge 5\)
- deterministic
- local communication
Results, in order of publication
No result recorded yet.
Open questions
Which combination of communication range and visibility is necessary for dispersion on \(T\)-interval connected graphs with \(T > 1\), where the impossibility proof for \(T = 1\) does not apply? Saxena & Mondal 2025
Dynamic graphs with a bounded number of missing edges per round: Saxena et al. 2025 gives impossibilities on 1-bounded 1-interval connected graphs for several weaker communication and visibility combinations and an \(O(k)\) algorithm with global communication and 1-hop visibility; the local-communication case with few missing edges is not settled.
Notes
The impossibility is the reason the dynamic-graph literature works with global communication. Rings are the exception: with full visibility of the ring, dispersion on a dynamic ring is possible with no communication model at all (see the dynamic ring branch).
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.