algorithm design techniques and am (nulland if am am egg if egg if am and if if am egg (null figure computation of table entry ( andfor am if performs correctly if there are negative edge costs but no negative-cost cyclesdijkstra' algorithm fails in this case let us recall the important details of dijkstra' algorithm (the reader may wish to review section dijkstra' algorithm starts at vertexsand works in stages each vertex in the graph is eventually selected as an intermediate vertex if the current selected vertex is vthen for each vwe set dw min(dw dv cv, this formula says that the best distance to (from sis either the previously known distance to from sor the result of going from to (optimallyand then directly from to dijkstra' algorithm provides the idea for the dynamic programming algorithmwe select the vertices in sequential order we will define dk, , to be the weight of the shortest path from vi to vj that uses only vk as intermediates by this definitiond , , ci, where ci, is if (vi vj is not an edge in the graph alsoby definitiond| |, , is the shortest path from vi to vj in the graph as figure showswhen we can write simple formula for dk, , the shortest path from vi to vj that uses only vk as intermediates is the shortest path that either does not use vk as an intermediate at allor consists of the merging of the two paths vi vk and vk vj each of which uses only the first - vertices as intermediates this leads to the formula dk, , min{dk- , , dk- , , dk- , , the time requirement is once again (| | unlike the two previous dynamic programming examplesthis time bound has not been substantially lowered by another approach