Bidirectional Collision Detection and Faster Deterministic Isomorphism Testing
Abstract
In this work, we introduce bidirectional collision detection  a new algorithmic tool that applies to the collision problems that arise in many isomorphism problems. For the group isomorphism problem, we show that bidirectional collision detection yields a deterministic n^((1 / 2) log n + O(1)) time algorithm whereas previously the n^(log n + O(1)) generatorenumeration algorithm was the best result for several decades. For the hard special case of solvable groups, we combine bidirectional collision detection with methods from the author's previous work to obtain a deterministic squareroot speedup over the best previous algorithm. We also show a deterministic squareroot speedup over the best previous algorithm for testing isomorphism of rings. We can even apply bidirectional collision detection to the graph isomorphism problem to obtain a deterministic T^(1 / sqrt(2)) speedup over the best previous deterministic algorithm. Although the space requirements for our algorithms are greater than those for previous deterministic isomorphism tests, we show timespace tradeoffs that interpolate between the resource requirements of our algorithms and previous work.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.3935
 Bibcode:
 2013arXiv1304.3935R
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity
 EPrint:
 18 pages. v1 shows the results. v2 makes minor corrections