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---
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id: sp
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title: "Shortest Paths"
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author: Benjamin Qi
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prerequisites:
-
- Gold - Breadth First Search
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---
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< module-excerpt >
- Shortest Path Without Negative Edge Weights
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- Shortest Path With Negative Edge Weights
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- All Pairs Shortest Path
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< / module-excerpt >
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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 )).
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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
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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)
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- Timeline (Camp)
- equivalent to [Timeline (Gold) ](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017 ) except $N,C\le 5000$ and negative values of $x$ are possible.