How can we exploit the knowledge we have on the solution of instance I to compute a (approximate) solution of instance I ′ in an efficient way?

This computation model is called reoptimization and is of practical interest in various circumstances.

The previous best algorithms for these problems required\O ..." We give a linear-time algorithm for single-source shortest paths in planar graphs with nonnegative edge-lengths.

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.

Consider a set S of n data points in real d-dimensional space, R d , where distances are measured using any Minkowski metric.

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 expected time for p updates is O(p log 3 n) and expected time for q queries is O(qk log 3 n). In the last decade there has been a growing interest in such dynamicall ..." Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices.In the last decade there has been a growing interest in such dynamically changing graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered. We show that a preprocessing of Θ(n λ(k,n)) time and space is both necessary and sufficient to answer each such query in at most k steps, for any fixed k.Gabow has given an O(m log m km#(m,n))-time algorithm [G], Katoh, Ibaraki, and Mine have given... This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion.The algorithms are designed using a new d ..." This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion.