Dynamic Approximate All-Pairs Shortest Paths in Undirected Graphs

We obtain three new dynamic algorithms for the approximate all-pairs shortest paths problem in unweighted undirected graphs: (i) For any fixed $\varepsilon>0$, a decremental algorithm with an expected total running time of $\tilde{O}(mn)$, where $m$ is the number of edges and $n$ is the number of vertices in the initial graph. Each distance query is answered in $O(1)$ worst-case time, and the stretch of the returned distances is at most $1+\varepsilon$. The algorithm uses $\tilde{O}(n^2)$ space. (ii) For any fixed integer $k\geq1$, a decremental algorithm with an expected total running time of $\tilde{O}(mn)$. Each query is answered in $O(1)$ worst-case time, and the stretch of the returned distances is at most $2k-1$. This algorithm, however, uses only $O(m+n^{1+1/k})$ space. It is obtained by dynamizing techniques of Thorup and Zwick. In addition to being more space efficient, this algorithm is also one of the building blocks used to obtain the first algorithm. (iii) For any fixed $\varepsilon,\delta>0$ and every $t\leq m^{1/2-\delta}$, a fully dynamic algorithm with an expected amortized update time of $\tilde{O}(mn/t)$ and worst-case query time of $O(t)$. The stretch of the returned distances is at most $1+\varepsilon$. All algorithms can also be made to work on undirected graphs with small integer edge weights. If the largest edge weight is $b$, then all bounds on the running times are multiplied by $b$.