68 lines
2.4 KiB
Text
68 lines
2.4 KiB
Text
---
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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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- Gold - Shortest Paths with Non-Negative Edge Weights
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description: Applications of Bellman-Ford.
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frequency: 0
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---
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import { Problem } from "../models";
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export const metadata = {
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problems: {
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sam: [
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new Problem("Kattis", "SSSP Negative", "shortestpath3", "Easy", false, [], ""),
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new Problem("Kattis", "APSP (with negative weights)", "allpairspath", "Easy", false, [], ""),
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new Problem("CSES", "(Negative) Cycle Finding", "1197", "Easy", false, [], ""),
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],
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probs: [
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new Problem("SPOJ", "Arbitrage", "ARBITRAG", "Easy", false, [], ""),
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new Problem("CSES", "High Score", "1673", "Easy", false, [], ""),
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],
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linear: [
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new Problem("ojuz", "Restore Array", "RMI19_restore", "Normal", false, [], "similar to [Art](https://codeforces.com/gym/102394/problem/A)"),
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],
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}
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};
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<problems-list problems={metadata.problems.sam} />
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<br/>
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### Tutorial
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<resources title="SSSP Negative">
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<resource source="CPH" title="13.1"></resource>
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<resource source="cp-algo" title="Bellman-Ford" url="graph/bellman_ford.html"></resource>
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<resource source="cp-algo" title="Finding Negative Cycle" url="finding-negative-cycle-in-graph.html"></resource>
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<resource source="TC" title="Intro to Graphs Section 3" url="introduction-to-graphs-and-their-data-structures-section-3/"></resource>
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</resources>
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Can also 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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(code)?
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### Problems
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<problems-list problems={metadata.problems.probs} />
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## Simple Linear Programming
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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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<resources>
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<resource source="MIT" title="Slides from Intro to Algorithms" url="https://www.cs.rit.edu/~spr/COURSES/ALG/MIT/lec18.pdf">Linear Programming Trick</resource>
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</resources>
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**Example:** Timeline (USACO 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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### Problems
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<problems-list problems={metadata.problems.linear} />
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