A Fast Incremental Cycle Ratio Algorithm

In this paper, we propose an algorithm to quickly find the maximum cycle ratio (MCR) on an incrementally changing directed cyclic graph. Compared with traditional MCR algorithms which have to recalculate everything from scratch at each incremental change, our algorithm efficiently finds the MCR by just leveraging the previous MCR and the corresponding largest cycle before the change. In particular, the previous MCR allows us to safely break the graph at the changed node. Then, we can detect the changing direction of the MCR by solving a single source longest path problem on a graph without positive cycle. A distance bucket approach is proposed to speed up the process of finding the longest paths. Our algorithm continues to search upward or downward based on whether the MCR is detected as increased or decreased. The downward search is quickly performed by a modified Karp-Orlin algorithm reusing the longest paths found during the cycle detection. In addition, a cost shifting idea is proposed to avoid calculating MCR on certain type of incremental changes. We evaluated our algorithm on both random graphs and circuit benchmarks. A timing-driven detailed placement approach which applies our algorithm is also proposed. Compared with Howard's and Karp-Orlin MCR algorithm, our algorithm shows much more efficiency on finding the MCR in both random graphs and circuit benchmarks.