Average Case Analysis of Dynamic Graph Algorithms 25

[32] V. V. Vazirani. A theory of alternating paths and blossoms for proving correctness of the O(p V E) general graph maximum matching algorithm. [13] G. N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. Linear time algorithms for nding a sparse k-connected spanning subgraph of a k-connected graph. Re-randomization and average case analysis of fully dynamic graph algorithms. Average Case Analysis of Dynamic Graph Algorithms 23 In fact, backwards analysis rst was used in computational geometry for exactly this purpose by Chew [5]. Note that our technique can also be used to analyze the average case performance of randomized dynamic graph algorithms. (A randomized algorithm is an algorithm that makes use of random choices for computing the solution to a worst case input.) For the connectivity problems considered in this paper the running time of an update consists of two parts: an expected running time of O(n= p m +log n) (where m is the number of edges after the update) plus an amortized constant time for rebuilds. It is an interesting open question whether the data structure can be improved by distributing the costs of rebuilds over previous updates in a way that gives an expected time bound of O(n= p m + log n) per update. Eppstein [9] suggested that a good average case behavior for some of the above problems can also be shown for node insertions and deletions.