Dynamic Shortest Paths and Transitive Closure: an Annotated Bibliography (Draft)

This is an annotated bibliography on fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. 1 Dynamic Path Problems A dynamic graph algorithm maintains a given property P on a graph subject to dynamic changes, such as edge insertions, edge deletions and edge weight updates. A dynamic graph algorithm should process queries on property P quickly, and perform update operations faster than recomputing from scratch, as carried out by the fastest static algorithm. We say that an algorithm is fully dynamic if it can handle both edge insertions and edge deletions. A partially dynamic algorithm can handle either edge insertions or edge deletions, but not both: we say that it is incremental if it supports insertions only, and decremental if it supports deletions only. In this annotated bibliography, we focus on fully dynamic algorithms for maintaining path problems on general directed graphs. In particular, we consider two fundamental problems. In the fully dynamic transitive closure problem we wish to maintain a directed graph G = (V,E) under an intermixed sequence of the following operations: Insert(x, y): insert an edge from x to y; Delete(x, y): delete the edge from x to y; Query(x, y): return yes if y is reachable from x, and return no otherwise. ∗ Email: [email protected]. URL: http://www.dis.uniroma1.it/~demetres. † Email: [email protected]. URL: http://www.disp.uniroma2.it/users/italiano.