Optimal Labeling Schemes for $\Gamma$-Free Matrices

Abstract

We study labeling schemes for $p \times q$ $\Gamma$-free binary matrices, which do not contain a submatrix of the form $\Gamma = \begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix}$. We give a labeling scheme that uses $O(\log^2 N)$-bit labels for rows and columns, where $N=p+q$, such that any matrix entry can be determined from the corresponding two labels in constant time in the word RAM model. The worst-case label size matches the counting lower bound up to constant factors. We also show that the entrywise least $\Gamma$-free supermatrix of an arbitrary $p\times q$ binary matrix, together with its labels, can be constructed in $O(pq)$ time. Our construction represents a $\Gamma$-free matrix by a sequence of rooted forests, where each column corresponds to a node and the $1$-entries of each row form a root path in its corresponding forest. For $n$-vertex strongly chordal and chordal bipartite graphs, our result gives asymptotically optimal $O(\log^2 n)$-bit adjacency labeling schemes with constant-time queries. These results answer both representation questions posed by Spinrad in 1995.