When reading about shortest paths in Cormen . I came across this paragraph which says that 'shortest path cannot contain' positive edge path even. But I don't understand the logic behind their explanation
Can a shortest path contain a cycle? As we have just seen, it cannot contain a negative-weight cycle. Nor can it contain a positive-weight cycle, since removing the cycle from the path produces a path with the same source and destination vertices and a lower path weight. That is, if p {vo,v1,v2,.... vk} is a path and c = {vi; v(i+1)....vj) is a positive-weight cycle on this path (so that vi = vj and w(c) > 0), then the path pnew = {v0,v1,.... vi, v(j+1) , ... vk} has weight w(pnew) = w(p)-w(c) < w(p), and so p cannot be a shortest path from v0 to vk.
The original snip from the book is here.
Could someone explain why it is so ? I don't understand because any way removing an edge will cause the path length to decrease right ? why are they seeing removing positive edge cycle as different ?
For eg in the following diagram
How are they removing cycle and connecting vi and v(j+1) ?