A Distributed Message-Optimal Assignment on Rings

Consider a set of items and a set of $m$ colors, where each item is associated to one color. Consider also $n$ computational agents connected by a ring. Each agent holds a subset of the items and items of the same color can be held by different agents. We analyze the problem of distributively assigning colors to agents in such a way that (a) each color is assigned to one agent only and (b) the number of different colors assigned to each agent is minimum. Since any color assignment requires the items be distributed according to it (e.g. all items of the same color are to be held by only one agent), we define the cost of a color assignment as the amount of items that need to be moved, given an initial allocation. We first show that any distributed algorithm for this problem requires a message complexity of $\Omega(n\cdot m)$ and then we exhibit an optimal message complexity algorithm for synchronous rings that in polynomial time determines a color assignment with cost at most three times the optimal. We also discuss solutions for the asynchronous setting. Finally, we show how to get a better cost solution at the expenses of either the message or the time complexity.

💡 Research Summary

The paper addresses a distributed optimization problem that arises when a set of n computational agents, arranged in a logical ring, must reorganize a collection of items according to a predefined set of m colors (or classes). Initially each agent holds an arbitrary multiset of items; items of the same color may be scattered across many agents. The goal is two‑fold: (1) assign each color to exactly one agent (so that after reorganization all items of that color reside on that agent), and (2) keep the number of colors assigned to each agent as balanced as possible, i.e., each agent receives either ⌊m/n⌋ or ⌈m/n⌉ colors. The cost of an assignment is the total number of items that must be moved across agents to satisfy the color‑to‑agent mapping. Formally, for a mapping π: {0,…,m‑1}→{0,…,n‑1}, the cost is
 Cost(π)=∑{j=0}^{m‑1}∑{i≠π(j)} Q_{j,i},
where Q_{j,i} denotes the number of items of color j initially stored at agent i.

The authors first establish a lower bound on the communication required to solve the problem in any distributed setting (both synchronous and asynchronous). By constructing a family of instances in which colors are partitioned into n/2 disjoint pairs of agents and each pair holds all items of its own t colors, they prove that any optimal solution must exchange information about every color across the ring, leading to a message‑complexity lower bound of Ω(n·m). This bound holds even when agents know the total number of agents n and the set of colors C, but not the global distribution of items.

Having identified the fundamental communication cost, the paper presents a distributed algorithm that essentially matches this lower bound when m is polynomial in n. The algorithm works in two phases on a synchronous ring:

1. Color Partitioning – Each agent locally counts how many items it holds for each color. Using basic messages of size O(log n) bits, agents collectively partition the set of m colors into roughly log m / log n groups, each group containing Θ(n) colors. This partitioning can be performed in O(log m / log n) rounds with O(n·m·log m / log n) messages.

2. Local β‑Assignment – Within each group, agents solve a weighted β‑assignment problem on the complete bipartite graph between the group’s colors and the agents, where β = ⌈m/n⌉. When m = n, this reduces to a maximum‑weight perfect matching, solvable locally by the Hungarian algorithm. For the general case, the algorithm of Chang and Ho is used. The solution of each group is then communicated to the neighboring agents to resolve any cross‑group conflicts.

The total message complexity of the algorithm is O(n·m·log m / log n). When m =O(poly(n)), this matches the Ω(n·m) lower bound up to a logarithmic factor, which is optimal in the asymptotic sense. The algorithm runs in polynomial time (the dominant computation is the local β‑assignment, which is O(m²n) in the worst case but can be reduced with faster matching techniques).

A key theoretical contribution is the analysis of the approximation ratio. Because each group is solved optimally but colors may be reassigned across group boundaries, the overall cost can be at most three times the optimal cost (a 3‑approximation). The authors provide a constructive worst‑case instance showing that this factor is tight for their approach.

The paper also adapts the protocol to an asynchronous ring. The same two‑phase structure is retained, but agents react to incoming messages without a global clock. This incurs a modest increase in message complexity—an additional factor of log p (where p = max_j ∑i Q{j,i}, the maximum number of items of any color) appears—but the asymptotic bound remains O(n·m·log m) unless p is extremely large (super‑polynomial in m). The time complexity in the asynchronous model is measured in the number of message‑delivery events; it grows proportionally to the number of rounds needed in the synchronous version.

Finally, the authors explore trade‑offs between communication, time, and solution quality. By increasing the granularity of the color partition (i.e., using more, smaller groups) and performing additional coordination rounds, the approximation factor can be improved to 2 or even (1 + ε) at the cost of higher message volume (up to O(n·m·log m·log n)) and longer execution time (polylogarithmic factors). Conversely, a very lightweight protocol that assigns colors based solely on local information uses no communication but offers no guarantee on the cost.

In summary, the paper makes three major contributions: (1) a tight Ω(n·m) lower bound on the communication required for balanced color assignment on rings; (2) a near‑optimal distributed algorithm achieving O(n·m·log m / log n) messages and a provable 3‑approximation of the minimum movement cost; (3) extensions to asynchronous rings and a systematic analysis of how additional resources (messages or time) can be traded for better approximation ratios. The work bridges a gap between classic centralized matching theory (maximum‑weight perfect matching, weighted β‑assignment) and practical distributed systems where communication is the primary bottleneck, and it opens avenues for future research on other topologies, dynamic item arrivals, and energy‑aware protocols.