99 lines
4 KiB
Text
99 lines
4 KiB
Text
---
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id: toposort
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title: "Topological Sort"
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author: Benjamin Qi, Michael Cao
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prerequisites:
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- Gold - Breadth First Search
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- Gold - Introduction to Dynamic Programming
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description: "An ordering of vertices in a directed acyclic graph that ensures that a node is visited before a node it has a directed edge to."
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frequency: 1
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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", "Course Schedule", "1679", "Easy", false, []),
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],
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dp: [
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new Problem("CSES", "Longest Flight Route", "1680", "Easy", false, []),
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],
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general: [
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new Problem("CSES", "Game Routes", "1681", "Easy", false, [], "counting paths on DAG"),
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new Problem("Kattis", "Quantum", "https://open.kattis.com/contests/acpc17open/problems/quantumsuperposition", "Easy", false, [], "enumerating paths on DAG"),
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new Problem("Gold", "Timeline", "1017", "Easy", false, [], "not explicitly given, but graph is a DAG"),
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new Problem("Gold", "Milking Order", "838", "Normal", false, ["TopoSort", "Binary Search"]),
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new Problem("CSES", "Course Schedule II", "1681", "Hard", false, [], "equivalent to [Minimal Labels](https://codeforces.com/contest/825/problem/E)"),
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],
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}
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};
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To review, a **directed** graph consists of edges that can only be traversed in one direction. Additionally, a **acyclic** graph defines a graph which does not contain cycles, meaning you are unable to traverse across one or more edges and return to the node you started on. Putting these definitions together, a **directed acyclic** graph, sometimes abbreviated as DAG, is a graph which has edges which can only be traversed in one direction and does not contain cycles.
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## Topological Sort
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<Problems problems={metadata.problems.sample} />
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A [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) of a directed acyclic graph is a linear ordering of its vertices such that for every directed edge $u\to v$ from vertex $u$ to vertex $v$, $u$ comes before $v$ in the ordering.
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## Tutorial
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<Resources>
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<Resource source="CSA" title="Topological Sorting" url="topological_sorting" starred>both BFS, DFS</Resource>
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</Resources>
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### DFS
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<Resources>
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<Resource source="CPH" title="16.1, 16.2 - Topological Sorting">DFS</Resource>
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<Resource source="cp-algo" title="Topological Sort" url="graph/topological-sort.html">DFS</Resource>
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</Resources>
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(implementation)
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<IncompleteSection />
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### BFS
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The BFS version, known as [Kahn's Algorithm](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm), makes it obvious how to extract the lexicographically minimum topological sort.
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(implementation)
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<IncompleteSection />
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## Dynamic Programming
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<Resources>
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<Resource source="PAPS" title="9.1">Best Path in a DAG</Resource>
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</Resources>
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One useful property of directed acyclic graphs is, as the name suggests, that no cycles exist. If we consider each node in the graph as a state, we can perform dynamic programming on the graph if we process the states in an order that guarantees for every edge $u\to v$ that $u$ is processed before $v$. Fortunately, this is the exact definition of a topological sort!
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<Problems problems={metadata.problems.dp} />
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In this task, we must find the longest path in a DAG.
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<Spoiler title="Solution">
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Let $dp[v]$ denote the length of the longest path ending at the node $v$. Clearly
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$$
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dp[v]=\max_{\text{edge } u\to v \text{ exists}}dp[u]+1,
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$$
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or zero if $v$ has in-degree $0$. If we process the states in topological order, it is guarangeed that $dp[u]$ will already have been computed before computing $dp[v]$.
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(implementation?)
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<IncompleteSection />
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</Spoiler>
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<!-- However, not all problems clearly give you directed acyclic graphs (ex. [Plat - Cave Paintings](http://usaco.org/index.php?page=viewproblem2&cpid=996)). An important step in many problems is to reduce the statement into a directed acyclic graph. See the editorial of the linked problem for more information.
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(Ben - this last paragraph doesn't seem very helpful.) -->
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## Problems
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<Problems problems={metadata.problems.general} /> |