# Distributed Voting/Ranking with Optimal Number of States per Node

## TL;DR

This paper introduces a scalable distributed algorithm for multi-choice voting and ranking in networks, using a minimal and optimal number of node states, with proven efficiency and complexity analysis.

## Contribution

It presents a novel distributed voting/ranking algorithm that is simple, scalable, and uses an optimal number of states, with proven optimality in ranking and conjectured optimality in voting.

## Key findings

- Algorithm uses only K×2^{K-1} states for voting.
- Time complexity is O(log(n)) in complete graphs.
- Number of states is proven optimal for ranking.

## Abstract

Considering a network with $n$ nodes, where each node initially votes for one (or more) choices out of $K$ possible choices, we present a Distributed Multi-choice Voting/Ranking (DMVR) algorithm to determine either the choice with maximum vote (the voting problem) or to rank all the choices in terms of their acquired votes (the ranking problem). The algorithm consolidates node votes across the network by updating the states of interacting nodes using two key operations, the union and the intersection. The proposed algorithm is simple, independent from network size, and easily scalable in terms of the number of choices $K$, using only $K\times 2^{K-1}$ nodal states for voting, and $K\times K!$ nodal states for ranking. We prove the number of states to be optimal in the ranking case, this optimality is conjectured to also apply to the voting case. The time complexity of the algorithm is analyzed in complete graphs. We show that the time complexity for both ranking and voting is $O(\log(n))$ for given vote percentages, and is inversely proportional to the minimum of the vote percentage differences among various choices.