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In this thesis, we present new techniques to deal with fundamentalalgorithmic graph problems where graphs are directed and partially dynamic,i.e. undergo either a sequence of edge insertions or deletions:  - Single-Source Reachability (SSR),  - Strongly-Connected Components (SCCs), and  - Single-Source Shortest Paths (SSSP).  These problems have recently received an extraordinary amount of attentiondue to their role as subproblems in various more complex and notoriously hardgraph problems, especially to compute flows, bipartite matchings and cuts.  Our techniques lead to the first near-optimal data structures for theseproblems in various different settings. Letting $n$ denote the number ofvertices in the graph and by $m$ the maximum number of edges in any version ofthe graph, we obtain  - the first randomized data structure to maintain SSR and SCCs innear-optimal total update time $\tilde{O}(m)$ in a graph undergoing edgedeletions.  - the first randomized data structure to maintain SSSP in partially dynamicgraphs in total update time $\tilde{O}(n^2)$ which is near-optimal in densegraphs.  - the first deterministic data structures for SSR and SCC for graphsundergoing edge deletions, and for SSSP in partially dynamic graphs thatimprove upon the $O(mn)$ total update time by Even and Shiloach from 1981 thatis often considered to be a fundamental barrier.

Near-Optimal Algorithms for Reachability, Strongly-Connected Components and Shortest Paths in Partially Dynamic Digraphs