The Space Complexity of k-Tree Isomorphism

Abstract

Abstract. We show that isomorphism testing of k-trees is in the class StUSPACE(log n) (strongly unambiguous logspace). This bound follows from a deterministic logspace algorithm that accesses a strongly unambiguous logspace oracle for canonizing k-trees. Further we give a logspace canonization algorithm for k-paths.

The Isomorphism Problem for k-Trees Is Complete for Logspace

“…This improves the previously known upper bound of StUL [ADK07] and matches the lower bound. In fact, we prove the formally stronger result that a canonical labeling for a given k-tree is computable in logspace.…”

“…We note that k-trees can be recognized in logspace [ADK07], so we can safely assume that the input is indeed a k-tree.…”

“…The definition of T (G, π) is similar to the construction of the colored tree T (G, B, θ) in [ADK07]. The main advantage of our construction lies in the fact that T (G, π) can be directly constructed from G in logspace, whereas the tree representation used in [ADK07] (which in turn is related to the decomposition defined in [KCP82]) is defined as the reachable subgraph of a mangrove derived from G. This allows us to decide st-reachability in the tree T (G, π) in logspace, an essential step to achieve our upper bound.…”

“…This makes it superfluous to cycle through all k-cliques of G (as in [ADK07]), leaving only the permutations of the vertices within K to enumerate. For a k-tree G and a permutation π on the kernel K of G, T (G, π) can be computed in FL.…”

“…The main advantage of our construction lies in the fact that the tree T (G) can be directly constructed in logspace from G, whereas the tree representation of [ADK07] is obtained as a reachable subgraph of a mangrove 1 based on G and hence can only be derived from G with the help of an StUL oracle.…”