Fast Distributed Computation of Distances in Networks

This paper presents a distributed algorithm to simultaneously compute the diameter, radius and node eccentricity in all nodes of a synchronous network. Such topological information may be useful as input to configure other algorithms. Previous approaches have been modular, progressing in sequential phases using building blocks such as BFS tree construction, thus incurring longer executions than strictly required. We present an algorithm that, by timely propagation of available estimations, achieves a faster convergence to the correct values. We show local criteria for detecting convergence in each node. The algorithm avoids the creation of BFS trees and simply manipulates sets of node ids and hop counts. For the worst scenario of variable start times, each node i with eccentricity ecc(i) can compute: the node eccentricity in diam(G)+ecc(i)+2 rounds; the diameter in 2diam(G)+ecc(i)+2 rounds; and the radius in diam(G)+ecc(i)+2radius(G) rounds.

💡 Research Summary

The paper introduces a novel distributed algorithm that enables every node in a synchronous, undirected network to compute three fundamental topological metrics—its own eccentricity, the network’s diameter, and the network’s radius—simultaneously. Traditional approaches typically rely on a multi‑phase scheme: each node first builds a breadth‑first‑search (BFS) tree rooted at itself, then uses the collection of these trees to derive eccentricities, and finally aggregates the eccentricities to obtain the diameter and radius. Although such methods achieve a message complexity of Θ(|V|·|E|·log|V|) bits, they require up to 4·D rounds (where D = diam(G)) to converge, which can be prohibitive for large‑scale systems.

The proposed algorithm eliminates the explicit construction of BFS trees and instead propagates three simple message types—bfs, diam, and rad—throughout the network. When a node becomes active (either by receiving a special environment message or by hearing from an already active neighbor), it immediately broadcasts a bfs message containing its identifier and a hop counter initialized to zero. Each node maintains a set I_i of identifiers whose bfs messages it has seen, a variable e_i that records the maximum hop count observed (an upper bound on its eccentricity), and two additional variables: d_i (a lower bound on the diameter) and r_i (an upper bound on the radius). Upon receiving a bfs message from a previously unknown identifier, a node increments the hop count and rebroadcasts the message. If its current e_i exceeds d_i, it updates d_i and sends a diam message with the new estimate. Similarly, once a node’s eccentricity has stabilized, it may lower r_i and broadcast a rad message. All messages are sent to every neighbor in each round, and nodes do not need to distinguish the source of each incoming message.

A key contribution is the introduction of a local convergence detector based on the counter c_i, which records the number of consecutive rounds in which no new bfs messages have been received. The authors prove three local convergence criteria:

1. If c_i ≥ 2, then the node’s view of the network is complete (A_i(r) = V), and consequently e_i = ecc(i).
2. If c_i ≥ 2 and c_i > d_i, then d_i = diam(G).
3. If c_i ≥ 2·r_i, then r_i = radius(G).

These results are derived through a series of lemmas that relate the set of received bfs identifiers to the “area of visibility” A_i(r), and by exploiting monotonicity properties of e_i, d_i, and r_i. The proofs show that once a node has not seen any new bfs for two rounds, all bfs messages have already traversed the entire graph, guaranteeing that the node’s eccentricity estimate is exact. The diameter and radius criteria follow from the fact that d_i can only increase and r_i can only decrease, respectively, and that any discrepancy would eventually be corrected by a newly arriving diam or rad message, contradicting the observed stability of c_i.

Regarding time bounds, the paper analyzes the worst‑case scenario where nodes may start at arbitrary rounds. It demonstrates that for any node i:

• Eccentricity is known after at most diam(G) + ecc(i) + 2 rounds.
• Diameter is known after at most 2·diam(G) + ecc(i) + 2 rounds.
• Radius is known after at most diam(G) + ecc(i) + 2·radius(G) rounds.

These bounds improve upon the classic 4·D + 2 round guarantee, achieving roughly a 30‑40 % reduction in convergence time while preserving the same Θ(|V|·|E|·log|V|) bit message complexity. The additional diam and rad messages contribute only O(D·|E|·log D) bits, which is negligible compared to the bfs traffic.

The algorithm also accommodates variable start times: nodes may become active at different rounds without any prior coordination, and the convergence detection remains correct. Although the model assumes synchrony, the authors note that a standard α‑synchronizer can be employed to emulate the algorithm in asynchronous environments, preserving its correctness and performance guarantees.

In summary, the paper delivers a clean, single‑phase distributed protocol that simultaneously computes eccentricities, diameter, and radius with provably fast convergence, low communication overhead, and simple local termination conditions. Its design is well‑suited for large‑scale distributed systems such as sensor networks, peer‑to‑peer overlays, and cloud infrastructures where rapid acquisition of global topological information is essential for higher‑level algorithmic decisions.