We give a linear-time algorithm for single-source shortest paths in planar graphs with nonnegative edge-lengths. Our algorithm also yields a linear-time algorithm for maximum flow in a planar graph with the source and sink on the same face. For the case where negative edge-lengths are allowed, we give an algorithm requiring O(n 4Â3 log(nL)) time, where L is the absolute value of the most negative length. This algorithm can be used to obtain similar bounds for computing a feasible flow in a planar network, for finding a perfect matching in a planar bipartite graph, and for finding a maximum flow in a planar graph when the source and sink are not on the same face. We also give parallel and dynamic versions of these algorithms. ] 1997 Academic Press

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In this paper we show that, given a graph and parameters 6 and r, we can find either a K,,. minor or an edge-cut of size O(mT/6) whose removal yields components of weak diameter O(T-26); i.e., every pair of nodes in such a component are at distance 0(r26) in the original graph.Using this lemma, we improve the best known bounds for the rein-cut max-flow ratio for mukicommodity flows in graphs with forbidden small minors. In general graphs, it was known that the ratio is O(log k) for the uniform-demand case (the case where there is a unit-demand commodity between every pair of nodes), and that the ratio is 0(log2 k) for arbitrary demands, where k is the number of commodities.In this paper we show that for graphs excluding any fixed graph as a minor (e.g. planar graphs or boundedgenus graphs), the ratio is O(1) for the uniform-demand case and O(log k) for the arbitrary demand case.For such graphs, our method yields rein-ratio cut approximation algorithms with performance bounds that match the above ratios. Computation of such cuts is a basic step for a variety of approximation algorithms for NP-complete problems.

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This paper focuses on space efficient representations of rooted trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (or k-ary tries), where each node has k slots, labelled {1, . . . , k}, each of which may have a reference to a child, and ordinal trees, where the children of each node are simply ordered. Our representations use a number of bits close to the information theoretic lower bound and support operations in constant time. For ordinal trees we support the operations of finding the degree, parent, ith child, and subtree size. For cardinal trees the structure also supports finding the child labelled i of a given node apart from the ordinal tree operations. These representations also provide a mapping from the n nodes of the tree onto the integers {1, . . . , n}, giving unique labels to the nodes of the tree. This labelling can be used to store satellite information with the nodes efficiently.