Fully dynamic cycle-equivalence in graphs

Two edges e_1 and e_2 of an undirected graph are cycle-equivalent iff all cycles that contain e_1 also contain e_2, i.e., iff e_1 and e_2 are a cut-edge pair. The cycle-equivalence classes of the control-flow graph are used in optimizing compilers to speed up existing control-flow and data-flow algorithms. While the cycle-equivalence classes can be computed in linear time, we present the first fully dynamic algorithm for maintaining the cycle-equivalence relation. In an n-node graph our data structure executes an edge insertion or deletion in O(sqrt(n.log n)) time and answers the query whether two given edges are cycle-equivalent in O(pow2(log(n))) time. We also present an algorithm for plane graphs with O(log n) update and query time and for planar graphs with O(log n) insertion time and O(log2 n) query and deletion time. Additionally, we show a lower bound of Ω(log n/log log n) for the amortized time per operation for the dynamic cycle-equivalence problem in the cell probe model