# Greedy-Merge Degrading has Optimal Power-Law

## TL;DR

This paper proves that the greedy-merge algorithm for degrading channels is asymptotically optimal in reducing mutual information, with bounds close to the best possible, especially as the output alphabet size varies.

## Contribution

The paper establishes that the greedy-merge algorithm achieves an optimal power-law bound on mutual information reduction, matching a fundamental lower bound.

## Key findings

- Greedy-merge is within a constant factor of the optimal lower bound.
- The bounds on mutual information reduction are tight in the power-law sense.
- The results hold for fixed input alphabet size with varying output size.

## Abstract

Consider a channel with a given input distribution. Our aim is to degrade it to a channel with at most L output letters. One such degradation method is the so called "greedy-merge" algorithm. We derive an upper bound on the reduction in mutual information between input and output. For fixed input alphabet size and variable L, the upper bound is within a constant factor of an algorithm-independent lower bound. Thus, we establish that greedy-merge is optimal in the power-law sense.