---
id: toposort
title: "Topological Sort"
author: Benjamin Qi, Michael Cao
prerequisites: 
 - Gold - Breadth First Search
 - Gold - Introduction to Dynamic Programming
description: "?"
---

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. 

## Topological Sort

 - [CSES Course Schedule](https://cses.fi/problemset/task/1679)

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. 

### Tutorial

 - CPH 16.1, 16.2
   - DFS
 - [cp-algorithms](https://cp-algorithms.com/graph/topological-sort.html)
   - DFS
 - [CSAcademy](https://csacademy.com/lesson/topological_sorting)
   - both BFS, DFS
   
Consider [Khan's Algorithm](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm) for topological sorting.

### Problems

  - [Milking Order](http://www.usaco.org/index.php?page=viewproblem2&cpid=838)
    - Binary search and check if a valid topological sort exists. 
  - [CSES Course Schedule II](https://cses.fi/problemset/task/1757)
    - Tricky!

## Dynamic Programming

 - [PAPS 9.1](https://www.csc.kth.se/~jsannemo/slask/main.pdf)

One useful property of directed acyclic graphs is, as the name suggests, that there exists no cycles. 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!

Let's consider the classical problem [CSES Longest Flight Route](https://cses.fi/problemset/task/1680), where we must find the longest path in a DAG.

<spoiler title="Solution">
Let `dp[curr] = longest path ending at the node curr`. Then, if we process states in topological order, the transition is relatively straightforward: `dp[curr] = max of all dp[prev] where prev represents a node with an edge going into the current node` (word better?). To reiterate, since the states a processed in topological order, we can guarantee that all possible `dp[prev]` are computed before we compute `dp[curr]`.
</spoiler>

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.

(Ben - this last paragraph doesn't seem very helpful.)

## Problems

  - [CSES Game Routes](https://cses.fi/problemset/task/1681)
    - counting paths on DAG
  - [Quantum](https://open.kattis.com/contests/acpc17open/problems/quantumsuperposition)
    - enumerating paths on DAG
  - [USACO Gold - Timeline](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017)
    - not explicitly given, but graph is a DAG

## Problems

 - Other
   - [Minimal Labels](http://codeforces.com/contest/825/problem/E) [](53)