Poly - Logarithmic Deterministic Fully - Dynamic GraphAlgorithms I : Connectivity and Minimum Spanning Tree

Deterministic fully dynamic graph algorithms are presented for con-nectivity and minimum spanning forest. For connectivity, starting with no edges, the amortized cost for maintaining a spanning forest is O(log 2 n) per update, i.e. per edge insertion or deletion. Deciding connectivity between any two given vertices is done in O(log n= log log n) time. This matches the previous best randomized bounds. The previous best deterministic bound was O(3 p n log n) amortized time per update but constant time for connectivity queries. For minimum spanning trees, rst a deletions-only algorithm is presented supporting deletes in amortized time O(log 2 n). Applying a general reduction from Henzinger and King, we then get a fully dynamic algorithm such that starting with no edges, the amortized cost for maintaining a minimum spanning forest is O(log 4 n) per update. The previous best deterministic bound was O(3 p n log n) amortized time per update, and no better randomized bounds were known. Corresponding O(log 2 n) algorithms for 2-edge connectivity and bicon-nectivity will be presented in a subsequent report.