35 lines
1.6 KiB
Markdown
35 lines
1.6 KiB
Markdown
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---
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id: sp-neg
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title: "Shortest Paths with Negative Edge Weights"
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author: Benjamin Qi
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prerequisites:
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-
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- Gold - Shortest Paths with Non-Negative Edge Weights
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---
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- Hasn't appeared in recent USACO Gold as far as I know.
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- Usually Bellman-Ford is used.
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- 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)
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- [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
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- [CSES High Score](https://cses.fi/problemset/task/1673)
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- [Kattis SSSP Negative](https://open.kattis.com/problems/shortestpath3)
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- [CSES (Negative) Cycle Finding](https://cses.fi/problemset/task/1197)
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- Simple Linear Programming
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- [Restore Array](https://oj.uz/problem/view/RMI19_restore)
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- [Art](https://codeforces.com/gym/102394/problem/A) (basically same as above)
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- Timeline (Camp)
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- 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.
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