Lecture #4

Two things I wanted to re-emphasize about today’s lecture:

1. Yu asked, “where does the algorithm use the undirectedness”? The undirectedness was used in the Claim 2 towards the end. In particular, for the first part of the claim, we used that for any neighbor z of y (in G), d(x,y) <= d(x,z) +1. (Because z can reach y in one hop.) In the second part, we used that for a neighbor z of y (in G), d(x,z) <= d(x,y) +1. (Because y can reach z in one hop). These are not satisfied if the graph is directed.
But maybe there’s a clever way to extend this idea for directed graphs. Feel free to throw out other ideas, we can discuss them in the comments…
2. Once we know the claim, one naive way to go from D to d would be this: for each x and y, iterate over all neighbors z of y and check the condition. That would take too much time: O(n^3). The matrix multiply (DA) allows us to use that either the “average” D(x,z)-value of the neighbors is at least D(x,y), or strictly less. And this we can look up in constant time.

Edit: answers to two more questions:

1. Euiwoong’s question about the fastest SSSP (negative weights) algorithm: if you don’t have a bound on the weights, then Bellman-Ford $O(mn)$ still seems the best. If you know that the weights are integers and the minimum edge weight in the graph is $-U$, then Goldberg has an $O(m\sqrt{n} \log U)$-time algorithm. (Uri Zwick’s notes.)
2. Yan’s question about parallelizability of Seidel’s algorithm: since it’s a bunch of matrix multiplies, and a small amount of extra work, it should be easily parallelizable.

See you on Friday!