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.
Rooted initial configuration
verifiedAll \(k\) agents start on the same node, so a single DFS traversal that drops one agent per new node solves the problem; the question is how fast and with how little memory.
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: 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: rooted start
- All \(k\) agents start at the same node (\(\ell=1\)). Also called co-located (black hole search: all agents at the home base) or a rooted graph Molla & Moses Jr. 2019.
What is known
- worst case over inputs: a path with all \(k\) agents on one endpoint
- one edge per round
- holds for randomized algorithms, unbounded memory and global communication
- deterministic
- all agents have the same memory bound
- an initial configuration with co-located agents (rooted suffices)
- memory counts the identifier
Results, in order of publication
DFS that leaves the smallest-identifier agent at every newly discovered node; time \(O(\min(m, k\Delta))\) for \(k \le n\) in later restatements.
Also \(O(k)\) time with \(O(\Delta + \log k)\) bits, the first linear-time algorithm. Both terminate: the last settled agent detects completion and relays it through the DFS tree in \(O(k)\) extra rounds. Work under any fair scheduler with atomic moves.
Linear time with logarithmic memory.
Open questions
Is \(O(k)\) time achievable with \(O(\log k)\) bits? Memory \(O(\log(k+\Delta))\) is needed by every known algorithm to remember a port number.
Notes
The rooted case is the one solved first and the one where every measure is settled up to the \(\log\Delta\) term. In the asynchronous model the rooted case also has \(O(k)\) epochs Pattanayak et al. 2025 (and \(O(k\log k)\) in Kshemkalyani et al. 2025).
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
Lower bounds from stronger settings hold here
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.