Dynamic Bridge-Finding in Õ(log2 n) Amortized Time.

We present a deterministic fully-dynamic data structure for maintaining information about the bridges in a graph. We support updates in Õ((log n)2) amortized time, and can find a bridge in the component of any given vertex, or a bridge separating any two given vertices, in O(log n/ log log n) worst case time. Our bounds match the current best for bounds for deterministic fully-dynamic connectivity up to log log n factors. The previous best dynamic bridge finding was an Õ((log n)3) amortized time algorithm by Thorup [STOC2000], which was a bittrick-based improvement on the O((log n)4) amortized time algorithm by Holm et al.[STOC98, JACM2001]. Our approach is based on a different and purely combinatorial improvement of the algorithim of Holm et al., which by itself gives a new combinatorial Õ((log n)3) amortized time algorithm. Combining it with Thorup's bittrick, we get down to the claimed Õ((log n)2) amortized time. Essentially the same new trick can be applied to the biconnectivity data structure from [STOC98, JACM2001], improving the amortized update time to Õ((log n)3). We also offer improvements in space. We describe a general trick which applies to both of our new algorithms, and to the old ones, to get down to linear space, where the previous best use O(m + n log n log log n). Our result yields an improved running time for deciding whether a unique perfect matching exists in a static graph.