Decremental associative algorithm for updating the shortest paths tree

The paper proposes an efficient associative algorithm for dynamic update of the shortest paths tree of a directed weighted graph after deletion of an edge. To this end, we use the STAR–machine that simulates the run of associative (content addressable) parallel systems of the SIMD type with bit–serial (vertical) processing. We provide a data structure that allows us to perform access to data by contents. On the STAR–machine, the associative algorithm is represented as the main procedure DeleteArcSPT that uses a group of auxiliary procedures. By means of the auxiliary procedures, we execute some parts of the associative parallel algorithm for dynamic update of the shortest paths tree after deleting an arc from the graph. We prove correctness of the procedure DeleteArcSPT and all its parts. On the STAR–machine, this procedure takes O(hk) time, where h is the number of bits required for coding the maximum of the shortest paths weights and k is the number of vertices, whose shortest paths change after deleting an edge from the given graph.