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.
Crash faults on oriented grids
verifiedOn oriented grids crashes are free for \(k = \Omega(n)\): the fault-free \(\Theta(\sqrt n)\) bound survives any number of crashes.
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.
- Topology: grid
- \(G\) is a \(\sqrt n\times\sqrt n\) grid graph; whether ports are globally consistent (oriented) is stated per result.
- Grid orientation: oriented
- The graph is a grid with nodes \((x,y)\), and every node labels its ports North, East, South and West consistently: the port North of \((x,y)\) leads to \((x,y+1)\), East to \((x+1,y)\), South to \((x,y-1)\) and West to \((x-1,y)\), whenever that neighbour exists. The same direction carries the same label at every node, so an agent can keep moving along a row or column; a boundary node is recognised by its missing ports.
- Faults: at most crash faults
- Up to \(f\) agents may stop forever at an adversarially chosen time; a crashed agent stays at its node (or vanishes, as stated per result) and never acts again.
- 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.
- Initial configuration: any start
- Agents start at arbitrary nodes; \(\ell\le k\) nodes are initially occupied, possibly with several agents each.
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
Its Theorem 1 (Section 4): \(f \le k \le n\) crashed agents; the authors remark that the algorithm can be modified to work without knowing \(n\). The same paper treats unoriented grids with crashes (that setting).
Any \(f \le n\), no knowledge of \(n\); stated for \(k = n\) and, by its Remark 2, for every \(k \le n\); optimal in both measures for \(k = \Omega(n)\).
Open questions
Crash-tolerant dispersion in \(O(\sqrt k)\) rounds on oriented grids for \(k = o(n)\); the fault-free case is solved by Kumar et al. 2026a (preprint), which names fault tolerance as future work.
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
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