usaco-guide/content/5_Gold/4_Gold_SP.md

id	title	author	prerequisites
sp	Shortest Paths	Benjamin Qi	Gold - Breadth First Search

• Shortest Path Without Negative Edge Weights
• Shortest Path With Negative Edge Weights
• All Pairs Shortest Path

Non-Negative Edge Weights

Use Dijkstra's Algorithm.

Standard

• Kattis SSSP Non-Negative
• CSES Shortest Routes I

Tutorial

• CSES 13.2
• cp-algo Dijkstra (Dense Graphs)
• cp-algo Dijkstra (Sparse Graphs)
  • Usually, it's this one that's applicable.
• CPC.8

Problems

• CSES
  • CSES Flight Discount
  • CSES Flight Routes
  • CSES Investigation
• USACO
  • Milk Pumping
    • fairly standard application
  • Shortcut
  • Fine Dining
• Other
  • Lane Switching
  • Robot Turtles

All Pairs Shortest Path (APSP)

Use the Floyd-Warshall algorithm.

Standard

• CSES Shortest Routes II
• Kattis APSP (with negative weights)

Tutorial

• CPH 13.3
• cp-algo Floyd-Warshall

Problems

• USACO Moortal Cowmbat
  • Use APSP before running DP.
• SPOJ Arbitrage

Negative Edge Weights

• Hasn't appeared in recent USACO Gold as far as I know.
• Usually Bellman-Ford is used.
• If no negative cycles, can use Shortest Path Faster Algorithm or modify Dijkstra slightly (though the same running time bound no longer applies).

Tutorial

• cp-algo Bellman Ford
• Topcoder Graphs Pt 3

You can also use shortest path algorithms to solve the following problem (a very simple linear program).

Given variables x_1,x_2,\ldots,x_N with constraints in the form x_i-x_j\ge c, compute a feasible solution.

• Linear Programming Trick

Problems

• General
  • CSES High Score
  • Kattis SSSP Negative
  • CSES (Negative) Cycle Finding
• Simple Linear Programming
  • Restore Array
  • Art (basically same as above)
  • Timeline (Camp)
    • equivalent to Timeline (Gold) except N,C\le 5000 and negative values of x are possible.