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

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---
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slug: /gold/sp
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title: "Shortest Paths"
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author: Benjamin Qi
prereqs
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order: 4
prerequisites:
-
- Gold - Breadth First Search
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---
prereqs
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- Shortest Path Without Negative Edge Weights
- Shortest Path With Negative Edge Weights
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- All Pairs Shortest Path
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<!-- END DESCRIPTION -->
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## Non-Negative Edge Weights
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Use *Dijkstra's Algorithm*.
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### Standard
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- [Kattis SSSP Non-Negative](https://open.kattis.com/problems/shortestpath1)
- [CSES Shortest Routes I](https://cses.fi/problemset/task/1671)
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### Tutorial
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- 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.
- [CPC.8](https://github.com/SuprDewd/T-414-AFLV/tree/master/08_graphs_2)
### Problems
- CSES
- [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)
- USACO
- [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)
- Other
- [Lane Switching](https://open.kattis.com/contests/acpc17open/problems/laneswitching)
- [Robot Turtles](https://open.kattis.com/problems/robotturtles) [](100)
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## All Pairs Shortest Path (APSP)
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Use the *Floyd-Warshall* algorithm.
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### Standard
- [CSES Shortest Routes II](https://cses.fi/problemset/task/1672)
- [Kattis APSP (with negative weights)](https://open.kattis.com/problems/allpairspath)
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### Tutorial
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- CPH 13.3
- [cp-algo Floyd-Warshall](https://cp-algorithms.com/graph/all-pair-shortest-path-floyd-warshall.html)
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### Problems
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- [USACO Moortal Cowmbat](http://usaco.org/index.php?page=viewproblem2&cpid=971)
- Use APSP before running DP.
- [SPOJ Arbitrage](https://www.spoj.com/problems/ARBITRAG/)
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## Negative Edge Weights
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- 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](https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm) or modify Dijkstra slightly (though the same running time bound no longer applies).
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### Tutorial
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- [cp-algo Bellman Ford](https://cp-algorithms.com/graph/bellman_ford.html)
- [Topcoder Graphs Pt 3](https://www.topcoder.com/community/data-science/data-science-tutorials/introduction-to-graphs-and-their-data-structures-section-3/)
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You can also use shortest path algorithms to solve the following problem (a very simple [linear program](https://en.wikipedia.org/wiki/Linear_programming)).
Gold_Graphs
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> Given variables $x_1,x_2,\ldots,x_N$ with constraints in the form $x_i-x_j\ge c$, compute a feasible solution.
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- [Linear Programming Trick](https://www.cs.rit.edu/~spr/COURSES/ALG/MIT/lec18.pdf)
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### Problems
Gold_Graphs
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- General
- [CSES High Score](https://cses.fi/problemset/task/1673)
- [Kattis SSSP Negative](https://open.kattis.com/problems/shortestpath3)
- [CSES (Negative) Cycle Finding](https://cses.fi/problemset/task/1197)
- Simple Linear Programming
- [Restore Array](https://oj.uz/problem/view/RMI19_restore)
- [Art](https://codeforces.com/gym/102394/problem/A) (basically same as above)
- [Timeline (Camp)](https://probgate.org/viewproblem.php?pid=524&cid=80)
minor
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- equivalent to [Timeline (Gold)](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017) except negative values of $x$ are possible.
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