CS252 Algorithms
Friday, 19 April 2024

+ What's up?

+ Shortest paths with Old Man Edsger
- Walk through an example
- What the heck is going on with that notation?
- Why does it work?
- Is it greedy?
- How are we gonna implement this thing?
    P = priority queue of edges
        (the edges leaving the orange blob)
    priority for edge (u,v) will be d[u] + w(u,v)
    "remove from P" -- grab the minimum-endpoint-distance edge e

- Negative edges?

+ Applications?

[Jeff's version for today]
Given:
    connected* graph** G = (V, E)
    weight function W: E -> R+
        (i.e., positive real numbers on each edge)
    starting node s ∈ V

Produce:
    shortest-path distances*** from s to all nodes

Algorithm:
    S = {s} (the "settled" nodes)                          O(1)
    d[s] = 0; d[u] = infinity for all u != s               O(|V|)
    while S != V:                                          |V|-1 iterations
        let e = (u,v) be such that u ∈ S, v !∈ S,          ???
            and d[u] + w(u,v) is as small as possible
        d[v] = d[u] + w(u, v)                              O(1) (assuming w is O(1)???)
        add v to S                                         O(1)

* connected makes the loop condition easier; we can modify to handle arbitrary G
** same algorithm works for digraphs
*** we can modify to produce the paths themselves in addition to the distances