A fully dynamic algorithm for distributed shortest paths

Serafino Cicerone, Gabriele Di Stefano, Daniele Frigioni, Umberto Nanni

Abstract: We propose a fully dynamic distributed algorithm for the all-pairs shortest paths problem on general networks with positive real edge weights. If Δσ is the number of pairs of nodes changing the distance after a single edge modification σ (insert, delete, weight decrease, or weight increase) then the message complexity of the proposed algorithm is O(nΔσ) in the worst case, where n is the number of nodes of the network. If Δσ=o(n2)<math><mtext>Δ</mtext><msub><mi></mi><mn>σ</mn></msub><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>=</mtext><mspace xmlns="true" sp="0.16" width="2px" linebreak="nobreak" is="true"></mspace><mtext>o</mtext><mtext>(n</mtext><msup><mi></mi><mn>2</mn></msup><mtext>)</mtext></math>, this is better than recomputing everything from scratch after each edge modification. Up to now only a result of Ramarao and Venkatesan was known, stating that the problem of updating shortest paths in a dynamic distributed environment is as hard as that of computing shortest paths.