1. Shortest paths & Weighted graphs
Based on Data Structures, Algorithms & Software Principles in C T. A. Standish, A-W (Pearson), 1998

2. Terms Weighted graph Weighted adjacency matrix
Numerical values attached to edges are called weights
Represent: costs, distances, times, etc.
Weighted adjacency matrix
A row/col intersection (edge exists from A-> B) contains the weight of the edge (vs a 1/0 of a non-weighted matrix)
A row/col intersection for a non-existent edge contains a value ≡ ∞

3. Usage Distance on a map (GPS implications) Cost to fly a plane
Computer network shortest # hops or delay time
Social media: degree of separation
Arbitrage (currency exchange)
convert (USD->EUR->CNY->USD)
1000 US$ to ₠ 810 (*.81)
₠ 810 to CNY (*7.73, 4/22/'18)
CNY back to US$ (rate=.16)
Profit of 1.80 USD (known as) in 1 msec
(find a negative cycle)

4. Weighted Di-Graph & it's Adj1 Shortest path A→D, is: ACEFDTo 3 B A B C D E F ∞ 9 3 5 4 7 8 1 6 9 8 4 D 5 C 6
From 7 F 1 E

5. Finding the shortest path
Easy if all weights are equal (simple BFS)
Dijkstra's algorithm - shortest path for ONE pair
See pg 426 for example and proof of correctness
O(V2), but O(V+ E* log V) if a min-priority queue is used
Floyd-Warshall algorithm - shortest path for ALL pairs O(V3)
See