In the Bellman-Ford algorithm's relaxation part, I can't seem to understand why we need to use (d[v] = min{ d[v], d[u] + c(u, v) }), the (c(u,v)) part. This uses the original weight. Why can't it use the least cost(shortest) between (u,v) instead of c(u,v)?
In the relaxation step we're trying to figure out if the shortest path to the source node from v needs to go through the edge (u, v).
If the current shortest path we know - d(v) is longer than the shortest path from u d(u) plust the cost of the adge c(u,v) then we need to update d(v)'s value. Otherwise we keep d(v) as it was before.
We can't just do the minimum between d[v] and d[u] because for v to reach the source through u it needs to go through the edge (u,v) which's cost needs to be incorporated into its final distance d[v]. So if you just update d[v] to be d[u] you're ignoring the cost of the edge between them.