On the Capacity of Sequences of Coloring Channels

TL;DR

This paper determines the exact capacities of various sequences of coloring channels, which are models for molecular information storage and protein identification, by relating them to a graph called the pairs graph.

Contribution

It introduces a graph-based approach to compute capacities of coloring channel sequences, providing exact values and bounds for multiple configurations.

Findings

Exact capacities for uniform sunflowers, intersecting sets, and paths.

Capacity depends solely on the pairs graph associated with the sequence.

Bounds for cycle sequences, with exact capacity for all but one case.

Abstract

A single coloring channel is defined by a subset of letters it allows to pass through, while deleting all others. A sequence of coloring channels provides multiple views of the same transmitted letter sequence, forming a type of sequence-reconstruction problem useful for protein identification and information storage at the molecular level. We provide exact capacities of several sequences of coloring channels: uniform sunflowers, two arbitrary intersecting sets, and paths. We also show how this capacity depends solely on a related graph we define, called the pairs graph. Using this equivalence, we prove lower and upper bounds on the capacity, and a tailored bound for a coloring-channel sequence forming a cycle. In particular, for an alphabet of size $4$, these results give the exact capacity of all coloring-channel sequences except for a cycle of length $4$, for which we only provide bounds.