[Paper Review] A New Heuristic Synchronizing Algorithm

Jakub Kowalski, Marek Szykuła|arXiv (Cornell University)|Aug 8, 2013

semigroups and automata theory14 references3 citations

TL;DR

This paper introduces CutOff-IBFS, a novel heuristic algorithm for finding short reset words in synchronizing automata using inverse breadth-first search in the power automaton. By maintaining a bounded list of the largest state sets and leveraging a trie for deduplication, it efficiently finds shorter reset words than existing methods, especially on random and extremal automata, with competitive runtime performance.

ABSTRACT

We present a new heuristic algorithm finding reset words. The algorithm called CutOff-IBFS is based on a simple idea of inverse breadth-first-search in the power automaton. We perform an experimental investigation of effectiveness compared to other algorithms existing in literature, which yields that our method generally finds a shorter word in an average case and works well in practice.

Motivation & Objective

• To develop a more effective heuristic algorithm for finding short reset words in synchronizing automata, which is a computationally hard problem.
• To improve upon existing heuristic and exact algorithms by reducing the length of found reset words while maintaining practical runtime efficiency.
• To design a method that performs well on both random automata and known extremal cases like Černý's automaton and slowly synchronizing series.
• To provide a practical, polynomial-time algorithm that can be integrated into exact solvers as a fast preprocessor or heuristic component.

Proposed method

• The algorithm performs inverse breadth-first search (IBFS) on the power automaton, starting from all singletons of states.
• At each step, it computes preimages of current state sets under all input letters using the transition function.
• A trie data structure is used to efficiently detect and eliminate duplicate sets during preimage computation.
• Only the top-`maxsize` largest sets (by cardinality) are retained at each level to limit state space growth.
• The process iterates up to a given `maxlen` (upper bound on reset word length), returning the length of the shortest found word.
• To recover the actual word, each set stores a pointer to its predecessor and the applied letter, enabling path reconstruction without storing full words.

Experimental results

Research questions

• RQ1 Can a heuristic algorithm based on inverse BFS with bounded set size outperform existing heuristics in terms of reset word length and execution time?
• RQ2 Does the algorithm reliably find the shortest reset word for known extremal automata such as Černý's automaton and other slowly synchronizing series?
• RQ3 How does the choice of the `maxsize` parameter affect the trade-off between solution quality and computational cost?
• RQ4 Can technical optimizations like trie-based deduplication, in-degree-based state ordering, and selective state initialization improve performance in practice?

Key findings

• CutOff-IBFS consistently finds shorter reset words than other heuristic algorithms on uniformly random automata with 100 to 1000 states.
• The version with `maxsize = n` guarantees finding the shortest reset word for Černý automaton $\mathrsfs{C}_n$ and other slowly synchronizing series like $\mathrsfs{D}'_n$, $\mathrsfs{D}''_n$, $\mathrsfs{B}_n$, $\mathrsfs{G}_n$, and $\mathrsfs{H}_n$.
• The `maxsize = log n` variant of CutOff-IBFS is only slightly slower than the two fastest algorithms (Eppstein and Cycle) but finds significantly shorter words.
• The algorithm's time complexity is $O(lckn + kn^2)$ and space complexity $O(ckn)$, where $l$ is the length of the found word, $c = \mathtt{maxsize}$, $k$ is the alphabet size, and $n$ is the number of states.
• With proper technical improvements—such as using separate tries per set size and sorting states by in-degree—the algorithm achieves faster execution due to reduced trie height and better cache behavior.
• The algorithm's performance is particularly strong on random automata, where it quickly finds short reset words due to the fast convergence of large sets in the IBFS process.