2020-06-04 01:42:57 +00:00
---
2020-06-15 23:19:07 +00:00
id: sp
2020-06-04 21:42:30 +00:00
title: "Shortest Paths"
2020-06-04 01:42:57 +00:00
author: Benjamin Qi
2020-06-05 00:21:03 +00:00
prerequisites:
-
- Gold - Breadth First Search
2020-06-04 01:42:57 +00:00
---
2020-06-17 22:18:07 +00:00
< module-excerpt >
- Shortest Path Without Negative Edge Weights
2020-06-05 00:21:03 +00:00
- Shortest Path With Negative Edge Weights
2020-06-04 21:42:30 +00:00
- All Pairs Shortest Path
2020-06-03 20:56:04 +00:00
2020-06-17 22:18:07 +00:00
< / module-excerpt >
2020-06-05 00:28:32 +00:00
2020-06-03 21:08:42 +00:00
## Non-Negative Edge Weights
2020-06-03 20:56:04 +00:00
2020-06-03 21:08:42 +00:00
Use *Dijkstra's Algorithm* .
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
### Standard
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
- [Kattis SSSP Non-Negative ](https://open.kattis.com/problems/shortestpath1 )
- [CSES Shortest Routes I ](https://cses.fi/problemset/task/1671 )
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
### Tutorial
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
- 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)
2020-06-03 20:56:04 +00:00
2020-06-03 21:08:42 +00:00
## All Pairs Shortest Path (APSP)
2020-06-03 20:56:04 +00:00
Use the *Floyd-Warshall* algorithm.
2020-06-04 21:42:30 +00:00
### Standard
- [CSES Shortest Routes II ](https://cses.fi/problemset/task/1672 )
- [Kattis APSP (with negative weights) ](https://open.kattis.com/problems/allpairspath )
2020-06-03 21:08:42 +00:00
### Tutorial
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
- CPH 13.3
- [cp-algo Floyd-Warshall ](https://cp-algorithms.com/graph/all-pair-shortest-path-floyd-warshall.html )
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
### Problems
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
- [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/ )
2020-06-03 20:56:04 +00:00
2020-06-03 21:08:42 +00:00
## Negative Edge Weights
2020-06-03 20:56:04 +00:00
2020-06-08 19:51:58 +00:00
- 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).
2020-06-04 21:42:30 +00:00
### Tutorial
2020-06-08 19:51:58 +00:00
- [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/ )
2020-06-04 21:42:30 +00:00
You can also use shortest path algorithms to solve the following problem (a very simple [linear program ](https://en.wikipedia.org/wiki/Linear_programming )).
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
> Given variables $x_1,x_2,\ldots,x_N$ with constraints in the form $x_i-x_j\ge c$, compute a feasible solution.
2020-06-08 19:51:58 +00:00
- [Linear Programming Trick ](https://www.cs.rit.edu/~spr/COURSES/ALG/MIT/lec18.pdf )
2020-06-04 21:42:30 +00:00
### Problems
2020-06-03 20:56:04 +00:00
2020-06-04 21:42:30 +00:00
- 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)
2020-06-10 21:54:52 +00:00
- 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.
