111 lines
No EOL
3.9 KiB
Text
111 lines
No EOL
3.9 KiB
Text
---
|
||
id: sp
|
||
title: "Shortest Paths with Non-Negative Edge Weights"
|
||
author: Benjamin Qi
|
||
prerequisites:
|
||
- Gold - Breadth First Search
|
||
description: Introduces Dijkstra's Algorithm for a single source as well as Floyd-Warshall for All-Pairs Shortest Path.
|
||
frequency: 3
|
||
---
|
||
|
||
import { Problem } from "../models";
|
||
|
||
export const metadata = {
|
||
problems: {
|
||
sample: [
|
||
new Problem("CSES", "Shortest Routes I", "1671", "Easy", false, ["SP"], "equivalent to [Kattis SSSP Non-Negative](https://open.kattis.com/problems/shortestpath1)"),
|
||
],
|
||
dijk: [
|
||
new Problem("CSES", "Flight Discount", "1195", "Easy", false, ["SP"], "one edge modified"),
|
||
new Problem("CSES", "Flight Routes", "1196", "Easy", false, ["SP"], "$k$ smallest paths"),
|
||
new Problem("CSES", "Investigation", "1202", "Easy", false, ["SP"], ""),
|
||
new Problem("Gold", "Milk Pumping", "969", "Easy", false, ["SP"], ""),
|
||
new Problem("Gold", "Visit FJ", "717", "Easy", false, ["SP"], ""),
|
||
new Problem("Gold", "Shortcut", "899", "Easy", false, ["SP"], ""),
|
||
new Problem("Gold", "Fine Dining", "861", "Easy", false, ["SP"], ""),
|
||
new Problem("Kattis", "Robot Turtles", "robotturtles", "Easy", false, ["SP"], ""),
|
||
new Problem("Kat", "Lane Switching", "https://open.kattis.com/contests/acpc17open/problems/laneswitching", "Normal", false, ["SP"], ""),
|
||
],
|
||
apspSam: [
|
||
new Problem("CSES", "Shortest Routes II", "1672", "Easy", false, ["APSP"], ""),
|
||
],
|
||
apsp: [
|
||
new Problem("Gold", "Moortal Cowmbat", "971", "Hard", false, ["APSP", "DP"], ""),
|
||
],
|
||
}
|
||
};
|
||
|
||
<problems-list problems={metadata.problems.sample} />
|
||
|
||
|
||
## Tutorial
|
||
|
||
Use *Dijkstra's Algorithm*.
|
||
|
||
<resources>
|
||
<resource source="CPH" title="13.2 - Dijkstra" starred>code</resource>
|
||
<resource source="cp-algo" title="Dijkstra (Dense Graphs)" url="graph/dijkstra_dense.html"></resource>
|
||
<resource source="cp-algo" title="Dijkstra (Sparse Graphs)" url="graph/dijkstra_sparse.html"></resource>
|
||
<resource source="CPC" title="8 - Graphs 2" url="08_graphs_2"></resource>
|
||
</resources>
|
||
|
||
|
||
## Implementation
|
||
|
||
### Part 1: $O(N^2)$
|
||
|
||
The USACO training pages present a $O(N^2)$ version, although this is rarely used nowadays. (but describe anyways?)
|
||
|
||
### Part 2: $O(M\log N)$
|
||
|
||
<resources>
|
||
<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>
|
||
</resources>
|
||
|
||
(comments in code?)
|
||
|
||
<optional-content title="Faster Dijkstra">
|
||
|
||
Can be done in $O(M+N\log N)$ with [Fibonacci heap](https://en.wikipedia.org/wiki/Fibonacci_heap).
|
||
|
||
</optional-content>
|
||
|
||
## Problems
|
||
|
||
<problems-list problems={metadata.problems.dijk} />
|
||
|
||
## All Pairs Shortest Path (APSP)
|
||
|
||
<problems-list problems={metadata.problems.apspSam} />
|
||
|
||
Use the *Floyd-Warshall* algorithm.
|
||
|
||
### Tutorial
|
||
|
||
<resources>
|
||
<resource source="CPH" title="13.3 - Floyd-Warshall" starred>example calculation, code</resource>
|
||
<resource source="PAPS" title="12.3.3"></resource>
|
||
<resource source="cp-algo" title="Floyd-Warshall" url="graph/all-pair-shortest-path-floyd-warshall.html"></resource>
|
||
</resources>
|
||
|
||
<optional-content title="Incorrect Floyd-Warshall">
|
||
|
||
[Paper](https://arxiv.org/pdf/1904.01210.pdf)
|
||
|
||
> A common mistake in implementing the Floyd–Warshall algorithm is to
|
||
misorder the triply nested loops (The correct order is `KIJ`). The incorrect
|
||
`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.
|
||
>
|
||
> It would be emphasized that these fixes (repeating incorrect algorithms three
|
||
times) have the same time complexity as the correct Floyd–Warshall algorithm
|
||
up to constant factors. Therefore, our results suggest that, if one is confused by
|
||
the order of the triply nested loops, one can repeat the procedure three times
|
||
just to be safe.
|
||
|
||
</optional-content>
|
||
|
||
### Problems
|
||
|
||
<problems-list problems={metadata.problems.apsp} />
|
||
|
||
(more?) |