1 research outputs found
Dynamic Bridge-Finding in Amortized Time
We present a deterministic fully-dynamic data structure for maintaining
information about the bridges in a graph. We support updates in
amortized time, and can find a bridge in the component
of any given vertex, or a bridge separating any two given vertices, in worst case time. Our bounds match the current best for bounds
for deterministic fully-dynamic connectivity up to factors. The
previous best dynamic bridge finding was an amortized
time algorithm by Thorup [STOC2000], which was a bittrick-based improvement on
the amortized time algorithm by Holm et al.[STOC98, JACM2001].
Our approach is based on a different and purely combinatorial improvement of
the algorithm of Holm et al., which by itself gives a new combinatorial
amortized time algorithm. Combining it with Thorup's
bittrick, we get down to the claimed amortized time.
Essentially the same new trick can be applied to the biconnectivity data
structure from [STOC98, JACM2001], improving the amortized update time to
.
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 . Finally,
we show how to obtain query time, matching the optimal
trade-off between update and query time.
Our result yields an improved running time for deciding whether a unique
perfect matching exists in a static graph.Comment: v1 Submitted to SODA'18 v2 Added note about improvement to
biconnectivity v3 Small changes. Drop unproven claim about finding first k
bridges in O(k) additional tim