Sorting and Ranking of Self-Delimiting Numbers with Applications to Outerplanar Graph Isomorphism

Abstract

Assume that an $N$-bit sequence $S$ of $k$ numbers encoded as Elias gamma codes is given as input. We present space-efficient algorithms for sorting, dense ranking and competitive ranking on $S$ in the word RAM model with word size $\Omega(\log N)$ bits. Our algorithms run in $O(k + \frac{N}{\log N})$ time and use $O(N)$ bits. The sorting algorithm returns the given numbers in sorted order, stored within a bit-vector of $N$ bits, whereas our ranking algorithms construct data structures that allow us subsequently to return the dense/competitive rank of each number $x$ in $S$ in constant time. For numbers $x \in \mathbb{N}$ with $x > N$ we require the position $p_x$ of $x$ as the input for our dense-/competitive-rank data structure. As an application of our algorithms above we give an algorithm for tree isomorphism, which runs in $O(n)$ time and uses $O(n)$ bits on $n$-node trees. Finally, we generalize our result for tree isomorphism to forests and outerplanar graphs, while maintaining a space-usage of $O(n)$ bits. The previous best linear-time algorithms for trees, forests and outerplanar graph isomorphism all use $\Theta(n \log n)$ bits.