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.
Grids
verifiedA \(\sqrt n \times \sqrt n\) anonymous grid; the diameter gives an \(\Omega(\sqrt k)\) bound and the question is whether local communication can reach it for small \(k\).
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: grid
- \(G\) is a \(\sqrt n\times\sqrt n\) grid graph; whether ports are globally consistent (oriented) is stated per result.
- 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 graph is a \(\sqrt n\times\sqrt n\) grid with \(\Delta = 4\). Whether an agent can tell the four directions apart consistently (an oriented grid) or only sees locally labelled ports (an unoriented grid) is stated on each result.
What is known
- \(k\) agents on one node: a grid ball of radius \(r\) holds \(O(r^2)\) nodes
- any algorithm
- 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
Introductory and current best for square grids with local communication; optimal in both measures when \(k = \Omega(n)\). With global communication: \(O(\sqrt k)\) rounds and \(O(\log k)\) bits for every \(k\), optimal.
First result without a sense of direction; the authors note the algorithm can be adapted to unknown \(n\). Crash-tolerant versions are under the crash-fault branch.
No knowledge of \(n\); the agents compute \(\sqrt n\) themselves.
Open questions
Can \(k \le n\) agents disperse on a grid in \(O(\sqrt k)\) rounds with local communication? Global communication achieves it Kshemkalyani et al. 2020; asked again in Banerjee et al. 2025.
A lower bound in terms of \(k\) for \(k = o(\sqrt n)\) with \(O(\log n)\) memory on unoriented grids Banerjee et al. 2024.
Unoriented grids with no knowledge of \(n\) and with faults, matching the fault-free bounds Banerjee et al. 2024.
Notes
Grids are the one class where global communication is known to help: \(O(\sqrt k)\) versus \(O(\min(k,\sqrt n))\). Rectangular \(x\times y\) grids with \(x, y\) known: \(O(\min(k, \max(x,y)))\) rounds with \(\Theta(\log k)\) bits (arXiv version of Kshemkalyani et al. 2020).
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
No catalogued stronger setting.
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.