117 lines
4 KiB
Text
117 lines
4 KiB
Text
---
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id: sp
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title: "Shortest Paths with Non-Negative Edge Weights"
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author: Benjamin Qi
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prerequisites:
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- bfs
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- stacks-queues
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description: "Introduces Dijkstra's Algorithm for a single source shortest path as well as Floyd-Warshall for All-Pairs Shortest Path."
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frequency: 3
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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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sample: [
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new Problem("CSES", "Shortest Routes I", "1671", "Easy", false, ["SP"], "equivalent to [Kattis SSSP Non-Negative](https://open.kattis.com/problems/shortestpath1)"),
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],
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dijk: [
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new Problem("CSES", "Flight Discount", "1195", "Easy", false, ["SP"], "one edge modified"),
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new Problem("CSES", "Flight Routes", "1196", "Easy", false, ["SP"], "$k$ smallest paths"),
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new Problem("CSES", "Investigation", "1202", "Easy", false, ["SP"], ""),
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new Problem("Gold", "Milk Pumping", "969", "Easy", false, ["SP"], ""),
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new Problem("Gold", "Why ... (visitfj)", "717", "Easy", false, ["SP"], ""),
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new Problem("Gold", "Shortcut", "899", "Easy", false, ["SP"], ""),
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new Problem("Gold", "Fine Dining", "861", "Easy", false, ["SP"], ""),
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new Problem("Kattis", "Robot Turtles", "robotturtles", "Easy", false, ["SP"], ""),
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new Problem("Kattis", "Lane Switching", "https://open.kattis.com/contests/acpc17open/problems/laneswitching", "Normal", false, ["SP"], ""),
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],
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apspSam: [
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new Problem("CSES", "Shortest Routes II", "1672", "Easy", false, ["APSP"], ""),
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],
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apsp: [
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new Problem("Gold", "Moortal Cowmbat", "971", "Hard", false, ["APSP", "DP"], ""),
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],
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}
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};
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## Single-Source Shortest Path
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<Problems problems={metadata.problems.sample} />
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### Tutorial
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Use *Dijkstra's Algorithm*.
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<Resources>
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<Resource source="CPH" title="13.2 - Dijkstra" starred>code</Resource>
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<Resource source="cp-algo" title="Dijkstra (Dense Graphs)" url="graph/dijkstra_dense.html"></Resource>
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<Resource source="cp-algo" title="Dijkstra (Sparse Graphs)" url="graph/dijkstra_sparse.html"></Resource>
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<Resource source="CPC" title="8 - Graphs 2" url="08_graphs_2"></Resource>
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</Resources>
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### Implementation
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#### Part 1: $O(N^2)$
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The USACO training pages present a $O(N^2)$ version, although this is rarely used nowadays. (but describe anyways?)
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<IncompleteSection />
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#### Part 2: $O(M\log N)$
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<Resources>
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<Resource source="Benq" title="Dijkstra" url="https://github.com/bqi343/USACO/blob/master/Implementations/content/graphs%20(12)/Basics/Dijkstra%20(7.3).h"> </Resource>
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</Resources>
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<IncompleteSection />
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(comments in code?)
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<Optional title="Faster Dijkstra">
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Can be done in $O(M+N\log N)$ with [Fibonacci heap](https://en.wikipedia.org/wiki/Fibonacci_heap).
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</Optional>
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### Problems
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<Problems problems={metadata.problems.dijk} />
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## All Pairs Shortest Path (APSP)
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<Problems problems={metadata.problems.apspSam} />
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Use the *Floyd-Warshall* algorithm.
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### Tutorial
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<Resources>
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<Resource source="CPH" title="13.3 - Floyd-Warshall" starred>example calculation, code</Resource>
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<Resource source="PAPS" title="12.3.3"></Resource>
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<Resource source="cp-algo" title="Floyd-Warshall" url="graph/all-pair-shortest-path-floyd-warshall.html"></Resource>
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</Resources>
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<Optional title="Incorrect Floyd-Warshall">
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[Paper](https://arxiv.org/pdf/1904.01210.pdf)
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> A common mistake in implementing the Floyd–Warshall algorithm is to
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misorder the triply nested loops (The correct order is `KIJ`). The incorrect
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`IJK` and `IKJ` algorithms do not give correct solutions for some instance. However, we can prove that if these are repeated three times, we obtain the correct solutions.
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>
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> It would be emphasized that these fixes (repeating incorrect algorithms three
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times) have the same time complexity as the correct Floyd–Warshall algorithm
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up to constant factors. Therefore, our results suggest that, if one is confused by
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the order of the triply nested loops, one can repeat the procedure three times
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just to be safe.
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</Optional>
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### Problems
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<Problems problems={metadata.problems.apsp} />
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(more?) |