Topological Sort
| Key points of topological sorting algorithm |
|---|
| - Topological sort runs on a Directed Acyclic Graph (DAG) and returns a sequence of vertices. Each vertex in the topological sorting order comes prior to the vertices that it points to. - For a given Directed Acyclic Graph (DAG) there could be atleast 1 topological sorting order - Rule for topological sorting : Vertex with no incoming edges is accessed first followed by the vertices on the outgoing paths. Example 1 : In the below graph vertex 1 has no incoming edges and can be reached before 3, 6, 0, 5, 2 and 4. Example 2 : In the below graph vertex 0 has no incoming edges and can be reached before 5, 2, 1, 3, 6 and 4.  Topological sorting using Depth First Search (DFS) - A topologically sorted order could be found by doing a DFS on the graph. - While doing a DFS, a stack is maintained to store the nodes in a reverse topologically sorted order. - A node is pushed into the stack only after all the adjacent nodes on its outgoing edges are visited. - When the DFS ends, the nodes are popped off the stack to get a topologically sorted order. Applications of Topological Sorting: - Many job scheduling applications use Topological Sorting for scheduling jobs that have dependencies on other jobs. - Many complex build systems use Topological Sorting for resolving the build dependencies. |
Algorithm : Topological Sort ( Graph G, Source_Vertex S )
- Mark the source vertex S as visited.
- For every vertex V adjacent to the source vertex S.
- If the vertex V is not visited, then
- Topological Sort ( Graph G, Source_Vertex V )
- Push the source vertex S into the stack STK.
Printing the topological sorting order
- While the stack STK is not empty.
- Print the vertex V at the stack ( STK ) top.
- Pop the vertex V at the stack ( STK ) top.
Topological traversal example
Data structure used for storing graph: Adjacency list
Data structure used for DFS: Stack
Time complexity of topological sort : O ( V + E ) for an adjacency list implementation of a graph. ‘V’ is the number of vertices and ‘E’ is the number of edges in a graph.
Topological Sort implementation
from collections import deque, defaultdict
class Graph :
def __init__(self, arg_nodes : int) :
self.nodes = arg_nodes
self.visited = [False] * self.nodes
# The default dictionary would create an empty list as a default (value)
# for the nonexistent keys.
self.adjlist = defaultdict(list)
self.stack = deque()
def AddEdge(self, src : int, dst : int) :
self.adjlist[src].append(dst)
def TopologicalSort(self, src : int) :
self.visited[src] = True
# Check if there is an outgoing edge for a node in the adjacency list
if src in self.adjlist :
for node in self.adjlist[src] :
if self.visited[node] == False :
self.TopologicalSort(node)
# Only after all the nodes on the outgoing edges are visited push the
# source node in the stack
self.stack.appendleft(src)
def Traverse(self) :
for node in range(self.nodes) :
if self.visited[node] == False :
self.TopologicalSort(node)
print("Topological Sorting Order : ", end = ' ')
while self.stack :
print(self.stack.popleft(),end=' ')
def main() :
node_cnt = 7
g = Graph(node_cnt)
g.AddEdge(0,2)
g.AddEdge(0,5)
g.AddEdge(1,3)
g.AddEdge(1,6)
g.AddEdge(2,4)
g.AddEdge(3,5)
g.AddEdge(5,2)
g.AddEdge(5,4)
g.AddEdge(6,2)
g.Traverse()
if __name__ == "__main__" :
main()
Output
Topological Sorting Order : 1 6 3 0 5 2 4
#include<iostream>
#include<list>
#include<vector>
#include<stack>
using namespace std;
class Graph{
private:
int nodes;
list<int> *adjlist;
vector<bool> visited;
stack<int> stack_topological_order;
public:
Graph () {
}
Graph (int nodes) {
adjlist = new list<int> [nodes];
visited.resize(nodes, false);
this->nodes = nodes;
}
~Graph(){
delete [] adjlist;
}
void AddEdge (int src, int dst) {
adjlist[src].push_back(dst);
}
void TopologicalSort (int src) {
visited[src] = true;
for (auto& itr : adjlist[src]) {
if (!visited[itr]) {
TopologicalSort (itr);
}
}
/* Only after all the outgoing edges are visited push the
source node in the stack */
stack_topological_order.push(src);
}
void Traverse () {
for(int i=0; i<nodes; i++) {
if (!visited[i]) {
TopologicalSort (i);
}
}
while (!stack_topological_order.empty()) {
cout << stack_topological_order.top() << " ";
stack_topological_order.pop();
}
}
};
int main()
{
Graph g(7);
// Store the edges of the directed graph in adjacency list.
g.AddEdge(0,2);
g.AddEdge(0,5);
g.AddEdge(1,3);
g.AddEdge(1,6);
g.AddEdge(2,4);
g.AddEdge(3,5);
g.AddEdge(5,2);
g.AddEdge(5,4);
g.AddEdge(6,2);
cout << "Topological Sorting Order: ";
g.Traverse();
return 0;
}
Output
Topological Sorting Order: 1 6 3 0 5 2 4
import java.util.List;
import java.util.ArrayList;
import java.util.Stack;
import java.util.Arrays;
class Graph {
Integer nodes;
List<List<Integer>> adjlist;
boolean[] visited;
Stack<Integer> stack_topological_order;
Graph (Integer arg_nodes) {
nodes = arg_nodes;
adjlist = new ArrayList<>(nodes);
for (int i=0; i<nodes; i++)
adjlist.add(new ArrayList<>());
visited = new boolean[nodes];
stack_topological_order = new Stack<Integer>();
}
void AddEdge (Integer src, Integer dst) {
adjlist.get(src).add(dst);
}
void TopologicalSort (Integer src) {
visited[src] = true;
for (Integer adj_node : adjlist.get(src)) {
if (!visited[adj_node]) {
TopologicalSort (adj_node);
}
}
/* Only after all the outgoing edges are visited push the
source node in the stack */
stack_topological_order.push(src);
}
void Traverse () {
for(int i=0; i<nodes; i++) {
if (!visited[i]) {
TopologicalSort (i);
}
}
while (!stack_topological_order.empty()) {
System.out.print(stack_topological_order.peek() + " ");
stack_topological_order.pop();
}
}
public static void main (String[] args) {
Graph g = new Graph(7);
// Store the edges of the directed graph in adjacency list.
g.AddEdge(0,2);
g.AddEdge(0,5);
g.AddEdge(1,3);
g.AddEdge(1,6);
g.AddEdge(2,4);
g.AddEdge(3,5);
g.AddEdge(5,2);
g.AddEdge(5,4);
g.AddEdge(6,2);
System.out.print("Topological Sorting Order: ");
g.Traverse();
}
}
Output
Topological Sorting Order: 1 6 3 0 5 2 4