# Topological Sort

Authors: Benjamin Qi, Michael Cao, Nathan Chen, Andi Qu, Andrew Wang

An ordering of vertices in a directed acyclic graph that ensures that a node is visited before every node it has a directed edge to.

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

Focus Problem – read through this problem before continuing!

A topological sort 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.

There are two common ways to topologically sort, one involving DFS and the other involving BFS.

Resources | |||
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CSA | interactive, both versions |

## DFS

Resources | |||
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CPH | example walkthrough | ||

CP2 | code | ||

cp-algo | code |

C++

1#include <bits/stdc++.h>2using namespace std;34#define pb push_back56int N; // Number of nodes7vector<int> graph[100000], top_sort; // Assume that this graph is a DAG8bool visited[100000];910void dfs(int node) {

Java

1import java.util.*;2import java.io.*;34public class CourseSchedule {5 public static ArrayList <Integer> g[];6 public static ArrayList <Integer> topo = new ArrayList < Integer > ();7 public static int N;8 public static boolean visited[];9 public static void main(String[] args) throws Exception {10 BufferedReader br = new BufferedReader(new InputStreamReader(System.in));

### Finding a Cycle

Focus Problem – read through this problem before continuing!

We can modify the algorithm above to return a directed cycle in case a topological sort does not exist. To find the cycle, we add each node we visit onto the stack until we detect a node already on the stack. Once we find the cycle exists, we mark the start of the cycle as not part of the stack and revursively backtrack until we reach the start node again.

C++

1#include <vector>2#include <iostream>3#include <algorithm>4using namespace std;56bool visited[(int)10e5+5], on_stack[(int)10e5+5];7vector<int> adj[(int)10e5+5];8vector<int> cycle;9int N, M;10bool dfs(int n) {

Java

1import java.io.*;2import java.util.*;34public class cycle {5 public static ArrayList < Integer > [] adj;6 public static boolean visited[], on_stack[];7 public static ArrayList < Integer > cycle;8 public static int N, M;9 public static void main(String[] args) throws IOException {10 BufferedReader sc = new BufferedReader(new InputStreamReader(System.in));

## BFS

The BFS version, known as Kahn's Algorithm, makes it obvious how to extract the lexicographically minimum topological sort.

C++

1int in_degree[100000];2vector<int> edge[100000];34int N; //number of nodes56void compute() {7 queue<int> q;8 for (int i = 0; i < N; i++) {9 if (in_degree[i] == 0) {10 q.push(i);

Java

1static int in_degree[];2static ArrayList<Integer> edge[]; //adjacency list34static int N; //number of nodes56static void topological_sort() {7 Queue<Integer> q = new ArrayDeque<Integer>();8 for (int i = 0; i < N; i++) {9 if (in_degree[i] == 0) {10 q.add(i);

## Dynamic Programming

Resources | |||
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CPH |

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!

Focus Problem – read through this problem before continuing!

In this task, we must find the longest path in a DAG.

Solution

## Problems

Status | Source | Problem Name | Difficulty | Tags | Solution |
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CSES | Easy | ## Show TagsTopoSort | |||

Kattis | Easy | ## Show TagsTopoSort | |||

Gold | Easy | ## Show TagsTopoSort | External Sol | ||

Gold | Normal | ## Show TagsTopoSort, Binary Search | External Sol | ||

CSES | Hard | ## Show TagsTopoSort |