# Gold - Shortest Path

Author: Benjamin Qi

[CPC.7](https://github.com/SuprDewd/T-414-AFLV/tree/master/07_graphs_1)

## Breadth First Search

Find the shortest path where all edge weights are 1.

 * [CSES Message Route](https://cses.fi/problemset/task/1667)

### Tutorial

 - [CSAcademy BFS](https://csacademy.com/lesson/breadth_first_search)
 - [cp-algo BFS](https://cp-algorithms.com/graph/breadth-first-search.html)
 - [cp-algo 0/1 BFS](https://cp-algorithms.com/graph/01_bfs.html)

### Problems

 - [Cow Navigation](http://www.usaco.org/index.php?page=viewproblem2&cpid=695)
 - [Dream](http://www.usaco.org/index.php?page=viewproblem2&cpid=575)
 - [Lasers](http://www.usaco.org/index.php?page=viewproblem2&cpid=671)
 * [Monsters](https://cses.fi/problemset/task/1194)

## Non-Negative Edge Weights

 * [Kattis SSSP Non-Negative](https://open.kattis.com/problems/shortestpath1)
 * [CSES Shortest Routes I](https://cses.fi/problemset/task/1671)
 * [CSES Flight Discount](https://cses.fi/problemset/task/1195)
 * [CSES Flight Routes](https://cses.fi/problemset/task/1196)
 * [CSES Investigation](https://cses.fi/problemset/task/1202)

Use *Dijkstra's Algorithm*.

### Tutorial

 * CSES 13.2
 * [cp-algo Dijkstra (Dense Graphs)](https://cp-algorithms.com/graph/dijkstra_sparse.html)
 * [cp-algo Dijkstra (Sparse Graphs)](https://cp-algorithms.com/graph/dijkstra_sparse.html)
   * Usually, it's this one that's applicable.

### USACO Gold Problems 

 * [Milk Pumping](http://www.usaco.org/index.php?page=viewproblem2&cpid=969)
   * fairly standard application
 * [Shortcut](http://usaco.org/index.php?page=viewproblem2&cpid=899)
 * [Fine Dining](http://usaco.org/index.php?page=viewproblem2&cpid=861)

## All Pairs Shortest Path (APSP)

 * [CSES Shortest Routes II](https://cses.fi/problemset/task/1672)
 * [Kattis APSP (with negative weights)](https://open.kattis.com/problems/allpairspath)

Use the *Floyd-Warshall* algorithm.

### Tutorial

 * CSES 13.3
 * [cp-algo Floyd-Warshall](https://cp-algorithms.com/graph/all-pair-shortest-path-floyd-warshall.html)

### USACO Gold Problems 

 * [Moortal Cowmbat](http://usaco.org/index.php?page=viewproblem2&cpid=971)
   * Use APSP before running DP.

## Negative Edge Weights

Hasn't appeared in recent USACO Gold as far as I know.

 * [CSES High Score](https://cses.fi/problemset/task/1673)
 * [Kattis SSSP Negative](https://open.kattis.com/problems/shortestpath3)

Can also modify Dijkstra's so it works with negative edge weights (but not negative cycles). The same running time bound no longer applies.