95 lines
No EOL
3.5 KiB
Markdown
95 lines
No EOL
3.5 KiB
Markdown
---
|
|
id: sp
|
|
title: "Shortest Paths"
|
|
author: Benjamin Qi
|
|
prerequisites:
|
|
-
|
|
- Gold - Breadth First Search
|
|
---
|
|
|
|
- Shortest Path Without Negative Edge Weights
|
|
- Shortest Path With Negative Edge Weights
|
|
- All Pairs Shortest Path
|
|
|
|
<!-- END DESCRIPTION -->
|
|
|
|
## Non-Negative Edge Weights
|
|
|
|
Use *Dijkstra's Algorithm*.
|
|
|
|
### Standard
|
|
|
|
- [Kattis SSSP Non-Negative](https://open.kattis.com/problems/shortestpath1)
|
|
- [CSES Shortest Routes I](https://cses.fi/problemset/task/1671)
|
|
|
|
### Tutorial
|
|
|
|
- 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)
|
|
|
|
## All Pairs Shortest Path (APSP)
|
|
|
|
Use the *Floyd-Warshall* algorithm.
|
|
|
|
### Standard
|
|
|
|
- [CSES Shortest Routes II](https://cses.fi/problemset/task/1672)
|
|
- [Kattis APSP (with negative weights)](https://open.kattis.com/problems/allpairspath)
|
|
|
|
### Tutorial
|
|
|
|
- CPH 13.3
|
|
- [cp-algo Floyd-Warshall](https://cp-algorithms.com/graph/all-pair-shortest-path-floyd-warshall.html)
|
|
|
|
### Problems
|
|
|
|
- [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/)
|
|
|
|
## 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](https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm) or modify Dijkstra slightly (though the same running time bound no longer applies).
|
|
|
|
### Tutorial
|
|
|
|
- [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/)
|
|
|
|
You can also use shortest path algorithms to solve the following problem (a very simple [linear program](https://en.wikipedia.org/wiki/Linear_programming)).
|
|
|
|
> 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](https://www.cs.rit.edu/~spr/COURSES/ALG/MIT/lec18.pdf)
|
|
|
|
### Problems
|
|
|
|
- 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)
|
|
- 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. |