Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0809.2319

A Log-space Algorithm for Canonization of Planar Graphs

Samir Datta·Nutan Limaye·Prajakta Nimbhorkar·Thomas Thierauf·Fabian Wagner
View original source ↗Read source document ↗Export metadata ↓

Abstract

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.

Explore related subjects

cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, Fabian Wagner. 2009-01-30. A Log-space Algorithm for Canonization of Planar Graphs. https://arxiv.org/abs/0809.2319

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bit-counting complexity classes

We define bit-counting complexity classes whose membership depends on the binary profile of the number of accepting paths of non-deterministic polynomial time Turing machines. We study the relationship between this new family of complexity classes and the classical complexity classes. We prove that the complexity class ${\bf PP}$ is contained in our comparison based bit-counting complexity classes ${\bf B_{|0|=|1|}P}$, ${\bf B_{|0|<|1|}P}$ and ${\bf B_{|0|>|1|}P}$. We then show that the comparison based bit-counting complexity classes and the complexity class ${\bf PP}$ are Turing equivalent, that is ${\bf P}^{\bf PP} = {\bf P}^{{\bf B_{|0|=|1|}P}}={\bf P}^{{\bf B_{|0|>|1|}P}}={\bf P}^{{\bf B_{|0|<|1|}P}}$. We then prove that the complexity classes ${\bf NP}$ and ${\bf CoNP}$ are contained in both of our parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We also show that the Turing closures of the parity based bit-counting complexity classes coincide, that is ${\bf P}^{{\bf B_{|0|\oplus}P}}={\bf P}^{{\bf B_{|1|\oplus}P}}$. We do this by proving that when either parity based bit-counting complexity class is provided as an oracle for a polynomial time Turing machine, then it can simulate the other one, that is ${\bf B_{|1| \oplus}P}\subseteq {\bf P}^{{\bf B_{|0| \oplus}P}}$ and ${\bf B_{|0| \oplus}P}\subseteq {\bf P}^{{\bf B_{|1| \oplus}P}}$.

cs.CC↗
arXiv2026

Formalizing PARITY Circuit Lower Bounds in Lean

We formalize Hastad's PARITY lower bound in Lean using the switching lemma. For every fixed d >= 2, formulas and DAG circuits of computation depth at most d computing PARITY on n inputs require size exp(Omega_d(n^(1/(d-1)))) for all sufficiently large n. This matches the classical upper bound up to constants in the exponent and implies that PARITY is not in nonuniform AC0. We also construct a polynomial-size, logarithmic-depth bounded-fan-in formula family for PARITY, providing a witness to NC1 is not a subset of AC0 for the formalized models. The Lean source code is available at https://github.com/formalcs/circuit-complexity and is checked with Lean 4.33.1 and mathlib 4.33.1.

cs.CC↗
arXiv2026

Constant-Coin Complete-Information Debates for $\mathsf{P}$ with Arbitrarily Small Strong Error

We study complete-information debate systems in which a probabilistic finite-state verifier reads the alternating messages of a prover and a refuter. Demirci, Say, and Yakaryılmaz showed that every language in $\mathsf{P}$ has such debates checkable with a constant number of random bits and arbitrarily small weak error. Their strong-error construction, which also counts nontermination as failure, did not permit arbitrary error reduction. We close this gap: for every $L\in\mathsf{P}$ and every $\varepsilon>0$, there is a constant-space verifier using a constant number of private coin tosses that has perfect completeness and strong error at most $\varepsilon$. The verifier simulates a polynomial-time alternating multihead finite automaton, privately spot-checking one of its input heads. The key observation is that, on a nonmember, the refuter may concede any round in which the prover first misreports a head reading. This ensures termination against every prover when the refuter follows the specified strategy, and permits strong-error reduction by repetition.

cs.CC↗