Graduate Thesis Or Dissertation

Maximum flow in planar digraphs

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  • Worst-case analysis is often meaningless in practice. Some problems never reach the anticipated worst-case complexity. Other solutions get bogged down with impractical constants during implementation, despite having favorable asymptotic running times. In this thesis, we investigate these contrasts in the context of finding maximum flows in planar digraphs. We suggest analytic techniques that adapt to the problem instance, and present a structural property that concludes equivalence between shortest paths and maximum st-flow in planar graphs. The best known algorithm for maximum st-flow in directed planar graphs is an augmenting- paths algorithm with O(n) iterations. Using dynamic trees, each iteration can be implemented in O(log n) time. Long before, Itai and Shiloach showed that when s and t are on the boundary of a common face, the O(n)-iteration augmenting-paths algorithm is equivalent to Dijkstra's algorithm in the graph’s dual: the max st-planar st-flow problem can be solved with one single-source shortest-path computation. In this thesis we show that (a) when s and t are separated by p faces, the max st-flow can be found with at most 2p single-source shortest-path computations, which, using the linear-time shortest-paths algorithm for planar graphs, results in an O(np)-time algorithm, and (b) that the equivalence between augmenting-paths and Dijkstra's extends to the most general non-st-planar digraphs, using their half-infinite universal cover graph.
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