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.
Directed graphs
verifiedEdges can be traversed in one direction only, so an agent cannot backtrack; the problem is solvable only on strongly connected digraphs and the known time bounds are quadratic in \(k\).
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: directed
- \(G=(V,A)\) with \(A\subseteq V\times V\); an agent at \(u\) may move along \((u,v)\in A\) only, never back along it unless \((v,u)\in A\). \(G\) is strongly connected.
- 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: any start
- Agents start at arbitrary nodes; \(\ell\le k\) nodes are initially occupied, possibly with several agents each.
Statement for this variant
\(G\) is a directed graph with maximum out-degree \(\Delta_{out}\); at a node an agent sees only the out-ports \(1,\dots,\deg_{out}(v)\). Dispersion is not always solvable when \(G\) is not strongly connected, so results assume strong connectivity. \(\phi\) denotes the deficiency of \(G\), the number of arcs whose addition makes \(G\) Eulerian; \(f \le k\) agents may crash.
What is known
- Theorem 3: deterministic
- a directed cycle
- Theorem 4: deterministic, \(k = n\), \(D = \omega(1)\)
- a strongly connected digraph built from a tree, after the exploration bound of Disser et al. 2020
- Theorem 5: only for algorithms based on DFS traversal
- deficiency \(\phi > 0\)
- Theorem 2: on some digraphs that are not strongly connected
- a rooted start at a node from which some node is unreachable
- Theorem 6: deterministic
- a rooted start
- memory counts the identifier
Results, in order of publication
Introductory and current best (Theorem 1(b)): DFS with ranked backtracking paths of at most \(k-1\) arcs; \(O(f k^2\Delta_{out})\) with \(f\) crashes. Optimal memory.
Theorem 1(a), simulating Kwek's exploration; \(O(f \phi k^2)\) with \(f\) crashes. Better when \(\phi < \Delta_{out}\).
Open questions
Close the time gaps; in particular \(O((k+f)k\Delta_{out})\) with \(f\) crashes Italiano et al. 2022.
BFS-based dispersion in the local model, without 1-hop communication Italiano et al. 2022.
Dispersion on dynamic directed graphs, and in semi-synchronous and asynchronous settings Italiano et al. 2022.
Notes
The algorithms do not terminate on their own: agents cannot tell that dispersion is complete. With \(k\), \(\Delta_{out}\), \(f\) (and \(\phi\) or \(D\)) known in advance, a round counter makes them terminate (Corollary 1). Checked against the full text of the JPDC version.
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
No catalogued weaker setting.
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.