Flexible Word Design and Graph Labeling

Motivated by emerging applications for DNA code word design, we consider a generalization of the code word design problem in which an input graph is given which must be labeled with equal length binary strings of minimal length such that the Hamming distance is small between words of adjacent nodes and large between words of non-adjacent nodes. For general graphs we provide algorithms that bound the word length with respect to either the maximum degree of any vertex or the number of edges in either the input graph or its complement. We further provide multiple types of recursive, deterministic algorithms for trees and forests, and provide an improvement for forests that makes use of randomization. We also consider generalizations of this problem to weighted graphs and graphs with optional edges. Finally, we explore the extension from simple adjacency queries to more general distance queries and show how to obtain distance labelings for rings and paths by applying properties of hypercube traversal. ∗Supported in part by NSF Grant EIA-0112934. †Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL 60208, USA. Email: [email protected]. ‡Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL 60208, USA. Email: [email protected]. §Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL 60208, USA. Email: [email protected].