notes consider that in this case and soone can verify that / in zadeh presents examples of networks that require / and / augmentationsbut these examples are dependent on choice of the shortest path we made worst-case tests with and vertexes and compared the running times of the improved version of the algorithm against the unimproved one as you can see on figure the improved algorithm is much faster on the network with nodes it works times faster practice analysis shows us thatin generalthe improved algorithm works / times faster figure -axis is the number of nodes -axis is working time in milliseconds blue colour indicates the shortest augmenting path algorithm and red does it improved version howeverour comparison in not definitivebecause we used only one kind of networks we just wanted to justify that the ( malgorithm works (ntimes faster than the (nm on dense network "network flow is an advanced branch of graph theory discuss network flow problem let (vebe directed network defined by set of vertexes (nodesand set of edges (arcsfor each edge (ije we associate capacity uij that denotes the maximum amount that can flow on the edge each edge (ije also has an associated cost cij that denotes the cost per unit flow on that edge we associate with each vertex number bi this value represents supply/demand of the vertex if bi node is supply nodeif bi node is demand node (its demand is equal to -biwe call vertex transshipment if bi is zero for simplificationlet' call transportation network and write (veucbin case we want to show all the network parameters explicitly lovely professional university