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 trees
verifiedAll agents start at one node of a tree; the time is governed by the diameter, as in collaborative tree exploration.
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: tree
- \(G\) is connected and acyclic: \(|E|=n-1\).
- 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
- \(k = n\): every node must be reached
- deterministic
- existential in \(D\): the hard tree has height \(D=\lceil\sqrt{\log n}\rceil\), so the bound is about \(\log n\) rounds; it is not shown for larger heights
- proved for collaborative exploration with labelled nodes and global communication, so it holds for dispersion Augustine & Moses Jr. 2018 Kshemkalyani et al. 2022
- 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; the full (arXiv) version improves the memory from \(O(\log n + \Delta)\) in the conference version to \(O(\log n)\).
Inherited; better than \(O(D^2)\) only when \(k = o(D^2)\).
Open questions
For \(k < n\) the lower bound \(\Omega(D^2)\) no longer applies (only \(k\) nodes must be reached). Whether \(O(\min(k, D^2))\) time is achievable, and whether it is necessary, has not been addressed.
For \(k=n\) agents rooted on a tree of height \(D\) much larger than \(\sqrt{\log n}\), is \(\Omega(D^2)\) time necessary? The known bound is existential in \(D\) Disser 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.