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.
Trees
verifiedNo cycles, so a DFS never needs to test for revisits; the difficulty is spreading agents that start far apart across long branches.
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: tree
- \(G\) is connected and acyclic: \(|E|=n-1\).
- Grid orientation: unoriented
- Port labels carry no global meaning: at every node \(v\) the \(\delta_v\) incident edges get the labels \(1,\dots,\delta_v\) in an arbitrary order, chosen independently at each node, so an agent that enters \(v\) cannot tell from the labels which port continues in the same row or column. This is the default on every graph; on a grid it is called an unoriented grid.
- 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.
- Initial configuration: any start
- Agents start at arbitrary nodes; \(\ell\le k\) nodes are initially occupied, possibly with several agents each.
What is known
- \(k = n\): 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
Memory-optimal; the authors ask whether \(o(n)\) rounds are possible.
Inherited from arbitrary graphs. For \(k = n\) this is \(O(n)\), no better than 2018.
Open questions
Augustine & Moses Jr. 2018 asked for \(o(n)\)-round dispersion on arbitrary trees. Precisely: for \(k = n\) and an arbitrary initial configuration, is there an algorithm whose time is bounded by a function of \(D\) alone, ideally \(O(D^2)\) as in the rooted case? No tree-specific algorithm has appeared since 2018; the linear-time general algorithms give only \(O(n)\).
With global communication, trees admit \(O(D\cdot\max(D,k))\) rounds with \(O(\max(D, \Delta\log k))\) bits (arXiv version of Kshemkalyani et al. 2020b); whether the \(\max(D,k)\) factor can be reduced to \(D\), and whether local communication suffices, is open.
Notes
Anonymous randomized agents from a rooted tree configuration: \(O(n)\) rounds with \(O(\log\Delta)\) bits Molla & Moses Jr. 2019.
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
Branches from here
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.