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Topological Sort

Authors: Benjamin Qi, Nathan Chen

Contributors: Michael Cao, Andi Qu, Andrew Wang, Qi Wang, Maggie Liu, Martin Lin

Ordering the vertices of a directed acyclic graph such that each vertex is visited before its children.

To review, a directed graph consists of edges that can only be traversed in one direction. Additionally, an 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 – try your best to solve 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 uvu\to v from vertex uu to vertex vv, uu comes before vv in the ordering.

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

Resources
CSA

interactive, both versions

DFS

Resources
CPH

example walkthrough

CP2

code

cp-algo

code

#include <bits/stdc++.h>
using namespace std;
#define pb push_back
int N; // Number of nodes
vector<int> graph[100000], top_sort; // Assume that this graph is a DAG
bool visited[100000];
void dfs(int node) {

Finding a Cycle

Focus Problem – try your best to solve this problem before continuing!

We can modify the algorithm above to return a directed cycle in the case where 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.

For example, suppose that our stack currently consists of s1,s2,,sis_1,s_2,\ldots,s_i and we then visit u=sju=s_j for some jij\le i. Then sjsj+1sisjs_j\to s_{j+1}\to \cdots\to s_i\to s_j is a cycle. We can reconstruct the cycle without explicitly storing the stack by marking uu as not part of the stack and recursively backtracking until uu is reached again.

#include <algorithm>
#include <iostream>
#include <vector>
using namespace std;
bool visited[(int)1e5 + 5], on_stack[(int)1e5 + 5];
vector<int> adj[(int)1e5 + 5];
vector<int> cycle;
int N, M;
bool dfs(int n) {

Warning!

This code assumes that there are no self-loops.

BFS

The BFS version is known as Kahn's Algorithm.

Course Schedule Solution

#include <iostream>
#include <queue>
#include <vector>
using namespace std;
const int MAX_N = 100000;
int main() {
int n, m;
cin >> n >> m;

Pro Tip

If the length of the array containing the end order does not equal the number of elements that need to be sorted, then there is a cycle in the graph.

Optional

We can also use Kahn's algorithm to extract the lexicographically minimum topological sort by breaking ties lexographically.

Although the above code does not do this, one can simply replace the queue with a priority_queue to implement this extension.

Dynamic Programming

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 uvu\to v that uu is processed before vv. Fortunately, this is the exact definition of a topological sort!

Focus Problem – try your best to solve this problem before continuing!

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

Solution - Longest Flight Route

Let dp[v]dp[v] denote the length of the longest path ending at the node vv. Clearly

dp[v]=maxedge uv existsdp[u]+1,dp[v]=\max_{\text{edge } u\to v \text{ exists}}dp[u]+1,

or 11 if vv is node 11. If we process the states in topological order, it is guaranteed that dp[u]dp[u] will already have been computed before computing dp[v]dp[v].

Note that the implementation of this idea below uses Kahn's algorithm for topological sorting:

#include <bits/stdc++.h>
using namespace std;
int prev_flight[100000];
int dist[100000];
int in_degree[100000];
vector<int> edge[100000];
vector<int> backEdge[100000];

Problems

StatusSourceProblem NameDifficultyTags
CSESEasy
Show TagsTopoSort
KattisEasy
Show TagsTopoSort
GoldEasy
Show TagsTopoSort
CFEasy
Show TagsTopoSort
CFEasy
Show TagsTopoSort
CFEasy
Show TagsTopoSort
GoldNormal
Show TagsBinary Search, TopoSort
CFHard
Show TagsBitwise, TopoSort
CSESHard
Show TagsTopoSort

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