All previously known efftcient maximum-flow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortest-length augmenting paths at once (using the layered network approach of Dinic). An alternative method based on the preflow concept of Karzanov is introduced. A preflow is like a flow, except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out. The method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths. The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense. graphs, achieving an O(n)) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version of the algorithm running in O(nm log(n'/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efticient distributed and parallel implementations. A parallel implementation running in O(n'log n) time using n processors and O(m) space is obtained. This time bound matches that of the Shiloach-Vishkin algorithm, which also uses n processors but requires O(n') space.

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We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs. *

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All previously known efftcient maximum-flow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortest-length augmenting paths at once (using the layered network approach of Dinic). An alternative method based on the preflow concept of Karzanov is introduced. A preflow is like a flow, except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out. The method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths. The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense. graphs, achieving an O(n)) time bound on an n-vertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version of the algorithm running in O(nm log(n'/m)) time on an n-vertex, m-edge graph. This is as fast as any known method for any graph density and faster on graphs of moderate density. The algorithm also admits efticient distributed and parallel implementations. A parallel implementation running in O(n'log n) time using n processors and O(m) space is obtained. This time bound matches that of the Shiloach-Vishkin algorithm, which also uses n processors but requires O(n') space.