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. The previous best algorithms for these problems required\Omega\Gamma n p log n) time where n is the number of nodes in the input graph. For the case where negative edge-lengths are allowed, we give an algorithm requiring O(n 4=3 log nL) time, where L is the absolute value of the most negative length. Previous algorithms for shortest paths with negative edge-lengths required \Omega\Gamma n 3=2 ) time. Our shortest-path algorithm yields an O(n 4=3 log n)-time algorithm for finding a perfect matching in a planar bipartite graph. A similar improvement is obtained for maximum flow in a directed planar graph.

In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter ffl such that 0 ! ffl, our algorithm maintains approximate allpairs shortest-paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ffl factor. The time bounds for both query and update for our algorithm is O(ffl \Gamma1 n 2=3 log 2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the adds is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Research supported by NSF grant CCR-9012357 and NSF PYI award CCR-9157620, together with PYI matching funds from Thinking Machines Corporation and Xerox Corporation. Addit...

In this paper we propose a uniform approach to deal with incremental problems on digraphs and with decremental problems on dags generalizing a technique used by La Poutr'e and van Leeuwen in [17] for updating the transitive closure and the transitive reduction of a dag. We define a propagation property on a binary relationship over the vertices of a digraph as a simple sufficient condition to apply this approach. The proposed technique is suitable for a very simple implementation which does not depend on the particular problem; in other words, the same procedures can be used to deal with different problems by simply setting appropriate boundary conditions.

We give an algorithm for the Dynamic Transitive Closure Problem for planar directed acyclic graphs with one source and one sink. The graph can be updated in logarithmic time under arbitrary edge insertions and deletions that preserve the embedding. Queries of the form `is there a directed path from u to v?' for arbitrary vertices u and v can be answered in logarithmic time. The size of the data structure and the initialisation time are linear in the number of edges. We also give a lower bound of###26 n/ log log n) on the amortised complexity of the problem in the cell probe model with logarithmic word size.

In a weighted, directed graph an L-bounded leg path is one whose constituent edges have length at most L. For any fixed L, computing L-bounded leg shortest paths is just as easy as the standard shortest path algorithm. In this paper we study approximate distance oracles (and reachability oracles) for bounded leg path problems, where the leg bound L is not known in advance, but forms part of the query. Bounded-leg path problems are more complicated than standard shortest path problems because the number of distinct shortest paths between two vertices (over all leg bounds) could be as large as the number of edges in the graph. The bounded leg constraint models situations where there is some limited resource that must be spent when traversing an edge. For example, the size of a fuel tank or the life of a battery places a hard limit on how far a vehicle can travel in one leg before refueling or recharging. Someone making a long road trip may place a hard limit on how many hours they are willing to drive in any one day. Our main result is a nearly optimal algorithm for preprocessing a directed graph in order to answer approximate bounded leg distance and bounded leg shortest path queries. In particular, we can preprocess any graph in Õ(n3) time, producing a data structure with size Õ(n2) that answers (1 + ɛ)-approximate bounded leg distance queries in O(log log n) time. If the corresponding (1 + ɛ)-approximate shortest path has l edges it can be returned in O(l log log n) time. These bounds are all within polylog(n) factors of the best standard all-pairs shortest path algorithm and improve substantially the previous best bounded leg shortest path algorithm, whose preprocessing time and space are O(n 4) and Õ(n 2.5). We also consider bounded leg oracles in other situations. In the context of planar directed graphs we give a time-space tradeoff for answering bounded leg reachability queries. For any k ≥ 2 we can build a data structure with size O(kn 1+1/k) that answers reachability queries in time

. This progress report surveys some of my scientific work on part A of the Ph.D.-programme at the Computer Science Department of the University of Aarhus. I give a survey-like introduction to the field of Complexity Theory to which my results bear relevance. I give a general account of my contributions to the field, including the the Dynamic Transitive Closure Problem for spherical st-graphs, the Dynamic Circuit Value Problem, and some dynamic membership problems for formal languages. Contents I. Introduction 3 1. Dynamic Algorithms 2. Overview II. Reachability 6 1. The parallel world 2. Dynamic algorithms for Reachability 3. Planar dags with one source and one sink 4. The result: a detailed account III. Formal Languages 14 1. Other realms 2. Regular Languages 3. Dyck Languages 4. Deterministic context-free languages 5. Conclusion and open problems IV. Finite Index Properties on Graphs of Bounded Tree-width 21 1. Introduction 2. Preliminaries 3. Dynamic Algorithms...

Abstract. In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter ε such that 0 <ε, our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ε factor. The time bounds for both query and update for our algorithm is O(ε−1n2/3 log2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph. Key Words.