We have introduced Graph implementation using array of vectors in Graph implementation using STL for competitive programming | Set 1. In this post, a different implementation is used which can be used to implement graphs using sets. The implementation is for adjacency list representation of graph.
A set is different from a vector in two ways: it stores elements in a sorted way, and duplicate elements are not allowed. Therefore, this approach cannot be used for graphs containing parallel edges. Since sets are internally implemented as binary search trees, an edge between two vertices can be searched in O(logV) time, where V is the number of vertices in the graph. Sets in python are unordered and not indexed. Hence, for python we will be using dictionary which will have source vertex as key and its adjacency list will be stored in a set format as value for that key.
Following is an example of an undirected and unweighted graph with 5 vertices.
Below is adjacency list representation of this graph using array of sets.
Below is the code for adjacency list representation of an undirected graph using sets:
C++
// A C++ program to demonstrate adjacency list// representation of graphs using sets#include <bits/stdc++.h>using namespace std;struct Graph { int V; set<int>* adjList;};// A utility function that creates a graph of V verticesGraph* createGraph(int V){ Graph* graph = new Graph; graph->V = V; // Create an array of sets representing // adjacency lists. Size of the array will be V graph->adjList = new set<int >[V]; return graph;}// Adds an edge to an undirected graphvoid addEdge(Graph* graph, int src, int dest){ // Add an edge from src to dest. A new // element is inserted to the adjacent // list of src. graph->adjList[src].insert(dest); // Since graph is undirected, add an edge // from dest to src also graph->adjList[dest].insert(src);}// A utility function to print the adjacency// list representation of graphvoid printGraph(Graph* graph){ for (int i = 0; i < graph->V; ++i) { set<int> lst = graph->adjList[i]; cout << endl << "Adjacency list of vertex " << i << endl; for (auto itr = lst.begin(); itr != lst.end(); ++itr) cout << *itr << " "; cout << endl; }}// Searches for a given edge in the graphvoid searchEdge(Graph* graph, int src, int dest){ auto itr = graph->adjList[src].find(dest); if (itr == graph->adjList[src].end()) cout << endl << "Edge from " << src << " to " << dest << " not found." << endl; else cout << endl << "Edge from " << src << " to " << dest << " found." << endl;}// Driver codeint main(){ // Create the graph given in the above figure int V = 5; struct Graph* graph = createGraph(V); addEdge(graph, 0, 1); addEdge(graph, 0, 4); addEdge(graph, 1, 2); addEdge(graph, 1, 3); addEdge(graph, 1, 4); addEdge(graph, 2, 3); addEdge(graph, 3, 4); // Print the adjacency list representation of // the above graph printGraph(graph); // Search the given edge in the graph searchEdge(graph, 2, 1); searchEdge(graph, 0, 3); return 0;} |
Java
// A Java program to demonstrate adjacency// list using HashMap and TreeSet// representation of graphs using setsimport java.util.*;class Graph { // TreeSet is used to get clear // understand of graph. HashMap<Integer, TreeSet<Integer> > graph; static int v; // Graph Constructor public Graph() { graph = new HashMap<>(); for (int i = 0; i < v; i++) { graph.put(i, new TreeSet<>()); } } // Adds an edge to an undirected graph public void addEdge(int src, int dest) { // Add an edge from src to dest into the set graph.get(src).add(dest); // Since graph is undirected, add an edge // from dest to src into the set graph.get(dest).add(src); } // A utility function to print the graph public void printGraph() { for (int i = 0; i < v; i++) { System.out.println("Adjacency list of vertex " + i); Iterator set = graph.get(i).iterator(); while (set.hasNext()) System.out.print(set.next() + " "); System.out.println(); System.out.println(); } } // Searches for a given edge in the graph public void searchEdge(int src, int dest) { Iterator set = graph.get(src).iterator(); if (graph.get(src).contains(dest)) System.out.println("Edge from " + src + " to " + dest + " found"); else System.out.println("Edge from " + src + " to " + dest + " not found"); System.out.println(); } // Driver code public static void main(String[] args) { // Create the graph given in the above figure v = 5; Graph graph = new Graph(); graph.addEdge(0, 1); graph.addEdge(0, 4); graph.addEdge(1, 2); graph.addEdge(1, 3); graph.addEdge(1, 4); graph.addEdge(2, 3); graph.addEdge(3, 4); // Print the adjacency list representation of // the above graph graph.printGraph(); // Search the given edge in the graph graph.searchEdge(2, 1); graph.searchEdge(0, 3); }}// This code is contributed by rexj8 |
Python3
# Python3 program to represent adjacency# list using dictionaryfrom collections import defaultdictclass graph(object): def __init__(self, v): self.v = v self.graph = defaultdict(set) # Adds an edge to undirected graph def addEdge(self, source, destination): # Add an edge from source to destination. # If source is not present in dict add source to dict self.graph.add(destination) # Add an dge from destination to source. # If destination is not present in dict add destination to dict self.graph[destination].add(source) # A utility function to print the adjacency # list representation of graph def print(self): for i, adjlist in sorted(self.graph.items()): print("Adjacency list of vertex ", i) for j in sorted(adjlist, reverse = True): print(j, end = " ") print('\n') # Search for a given edge in graph def searchEdge(self,source,destination): if source in self.graph: if destination in self.graph: if destination in self.graph: if source in self.graph[destination]: print("Edge from {0} to {1} found.\n".format(source, destination)) return else: print("Edge from {0} to {1} not found.\n".format(source, destination)) return else: print("Edge from {0} to {1} not found.\n".format(source, destination)) return else: print("Destination vertex {} is not present in graph.\n".format(destination)) return else: print("Source vertex {} is not present in graph.\n".format(source)) # Driver codeif __name__=="__main__": g = graph(5) g.addEdge(0, 1) g.addEdge(0, 4) g.addEdge(1, 2) g.addEdge(1, 3) g.addEdge(1, 4) g.addEdge(2, 3) g.addEdge(3, 4) # Print adjacenecy list # representation of graph g.print() # Search the given edge in a graph g.searchEdge(2, 1) g.searchEdge(0, 3)#This code is contributed by Yalavarthi Supriya |
C#
// A C# program to demonstrate adjacency// list using HashMap and TreeSet// representation of graphs using setsusing System;using System.Collections.Generic;class Graph { // TreeSet is used to get clear // understand of graph. Dictionary<int, HashSet<int>> graph; static int v; // Graph Constructor public Graph() { graph = new Dictionary<int, HashSet<int> >(); for (int i = 0; i < v; i++) { graph.Add(i, new HashSet<int>()); } } // Adds an edge to an undirected graph public void addEdge(int src, int dest) { // Add an edge from src to dest into the set graph[src].Add(dest); // Since graph is undirected, add an edge // from dest to src into the set graph[dest].Add(src); } // A utility function to print the graph public void printGraph() { for (int i = 0; i < v; i++) { Console.WriteLine("Adjacency list of vertex " + i); foreach(int set_ in graph[i]) Console.Write(set_ + " "); Console.WriteLine(); Console.WriteLine(); } } // Searches for a given edge in the graph public void searchEdge(int src, int dest) { // Iterator set = graph.get(src).iterator(); if (graph[src].Contains(dest)) Console.WriteLine("Edge from " + src + " to " + dest + " found"); else Console.WriteLine("Edge from " + src + " to " + dest + " not found"); Console.WriteLine(); } // Driver code public static void Main(String[] args) { // Create the graph given in the above figure v = 5; Graph graph = new Graph(); graph.addEdge(0, 1); graph.addEdge(0, 4); graph.addEdge(1, 2); graph.addEdge(1, 3); graph.addEdge(1, 4); graph.addEdge(2, 3); graph.addEdge(3, 4); // Print the adjacency list representation of // the above graph graph.printGraph(); // Search the given edge in the graph graph.searchEdge(2, 1); graph.searchEdge(0, 3); }}// This code is contributed by Abhijeet Kumar(abhijeet19403) |
Javascript
<script>// A Javascript program to demonstrate adjacency list// representation of graphs using setsclass Graph { constructor() { this.V = 0; this.adjList = new Set(); }};// A utility function that creates a graph of V verticesfunction createGraph(V){ var graph = new Graph(); graph.V = V; // Create an array of sets representing // adjacency lists. Size of the array will be V graph.adjList = Array.from(Array(V), ()=>new Set()); return graph;}// Adds an edge to an undirected graphfunction addEdge(graph, src, dest){ // Add an edge from src to dest. A new // element is inserted to the adjacent // list of src. graph.adjList[src].add(dest); // Since graph is undirected, add an edge // from dest to src also graph.adjList[dest].add(src);}// A utility function to print the adjacency// list representation of graphfunction printGraph(graph){ for (var i = 0; i < graph.V; ++i) { var lst = graph.adjList[i]; document.write( "<br>" + "Adjacency list of vertex " + i + "<br>"); for(var item of [...lst].reverse()) document.write( item + " "); document.write("<br>"); }}// Searches for a given edge in the graphfunction searchEdge(graph, src, dest){ if (!graph.adjList[src].has(dest)) document.write( "Edge from " + src + " to " + dest + " not found.<br>"); else document.write( "<br> Edge from " + src + " to " + dest + " found." + "<br><br>");}// Driver code// Create the graph given in the above figurevar V = 5;var graph = createGraph(V);addEdge(graph, 0, 1);addEdge(graph, 0, 4);addEdge(graph, 1, 2);addEdge(graph, 1, 3);addEdge(graph, 1, 4);addEdge(graph, 2, 3);addEdge(graph, 3, 4);// Print the adjacency list representation of// the above graphprintGraph(graph);// Search the given edge in the graphsearchEdge(graph, 2, 1);searchEdge(graph, 0, 3);// This code is contributed by rutvik_56.</script> |
Output
Adjacency list of vertex 0 1 4 Adjacency list of vertex 1 0 2 3 4 Adjacency list of vertex 2 1 3 Adjacency list of vertex 3 1 2 4 Adjacency list of vertex 4 0 1 3 Edge from 2 to 1 found. Edge from 0 to 3 not found.
Pros: Queries like whether there is an edge from vertex u to vertex v can be done in O(log V).
Cons:
- Adding an edge takes O(log V), as opposed to O(1) in vector implementation.
- Graphs containing parallel edge(s) cannot be implemented through this method.
Space Complexity: O(V+E), where V is the number of vertices and E is the number of edges in the graph. This is because the code uses an adjacency list to store the graph, which takes linear space.
Further Optimization of Edge Search Operation using unordered_set (or hashing): The edge search operation can be further optimized to O(1) using unordered_set which uses hashing internally.
Implementation:
C++
// A C++ program to demonstrate adjacency list// representation of graphs using sets#include <bits/stdc++.h>using namespace std;struct Graph { int V; unordered_set<int>* adjList;};// A utility function that creates a graph of // V verticesGraph* createGraph(int V){ Graph* graph = new Graph; graph->V = V; // Create an array of sets representing // adjacency lists. Size of the array will be V graph->adjList = new unordered_set<int>[V]; return graph;}// Adds an edge to an undirected graphvoid addEdge(Graph* graph, int src, int dest){ // Add an edge from src to dest. A new // element is inserted to the adjacent // list of src. graph->adjList[src].insert(dest); // Since graph is undirected, add an edge // from dest to src also graph->adjList[dest].insert(src);}// A utility function to print the adjacency// list representation of graphvoid printGraph(Graph* graph){ for (int i = 0; i < graph->V; ++i) { unordered_set<int> lst = graph->adjList[i]; cout << endl << "Adjacency list of vertex " << i << endl; for (auto itr = lst.begin(); itr != lst.end(); ++itr) cout << *itr << " "; cout << endl; }}// Searches for a given edge in the graphvoid searchEdge(Graph* graph, int src, int dest){ auto itr = graph->adjList[src].find(dest); if (itr == graph->adjList[src].end()) cout << endl << "Edge from " << src << " to " << dest << " not found." << endl; else cout << endl << "Edge from " << src << " to " << dest << " found." << endl;}// Driver codeint main(){ // Create the graph given in the above figure int V = 5; struct Graph* graph = createGraph(V); addEdge(graph, 0, 1); addEdge(graph, 0, 4); addEdge(graph, 1, 2); addEdge(graph, 1, 3); addEdge(graph, 1, 4); addEdge(graph, 2, 3); addEdge(graph, 3, 4); // Print the adjacency list representation of // the above graph printGraph(graph); // Search the given edge in the graph searchEdge(graph, 2, 1); searchEdge(graph, 0, 3); return 0;} |
Java
import java.util.HashSet;import java.util.Set;class Graph { int V; Set<Integer>[] adjList; public Graph(int V) { this.V = V; adjList = new HashSet[V]; for (int i = 0; i < V; i++) { adjList[i] = new HashSet<Integer>(); } } // Adds an edge to an undirected graph void addEdge(int src, int dest) { // Add an edge from src to dest. A new // element is inserted to the adjacent // list of src. adjList[src].add(dest); // Since graph is undirected, add an edge // from dest to src also adjList[dest].add(src); } // A utility function to print the adjacency // list representation of graph void printGraph() { for (int i = 0; i < V; i++) { Set<Integer> lst = adjList[i]; System.out.println("Adjacency list of vertex " + i); for (Integer itr : lst) { System.out.print(itr + " "); } System.out.println(); } } // Searches for a given edge in the graph void searchEdge(int src, int dest) { if (!adjList[src].contains(dest)) { System.out.println("Edge from " + src + " to " + dest + " not found."); } else { System.out.println("Edge from " + src + " to " + dest + " found."); } }}public class Main { public static void main(String[] args) { // Create the graph given in the above figure int V = 5; Graph graph = new Graph(V); graph.addEdge(0, 1); graph.addEdge(0, 4); graph.addEdge(1, 2); graph.addEdge(1, 3); graph.addEdge(1, 4); graph.addEdge(2, 3); graph.addEdge(3, 4); // Print the adjacency list representation of // the above graph graph.printGraph(); // Search the given edge in the graph graph.searchEdge(2, 1); graph.searchEdge(0, 3); }}// This code is contributed by divya_p123. |
Python3
import collectionsclass Graph: def __init__(self, V): self.V = V self.adjList = [set() for _ in range(V)] def add_edge(self, src, dest): self.adjList[src].add(dest) self.adjList[dest].add(src) def print_graph(self): for i in range(self.V): print("Adjacency list of vertex {}".format(i)) for vertex in self.adjList[i]: print(vertex, end=' ') print() def search_edge(self, src, dest): if dest in self.adjList[src]: print("Edge from {} to {} found.".format(src, dest)) else: print("Edge from {} to {} not found.".format(src, dest))# Driver codeif __name__ == "__main__": V = 5 graph = Graph(V) graph.add_edge(0, 1) graph.add_edge(0, 4) graph.add_edge(1, 2) graph.add_edge(1, 3) graph.add_edge(1, 4) graph.add_edge(2, 3) graph.add_edge(3, 4) # Print the adjacency list representation of the above graph graph.print_graph() # Search the given edge in the graph graph.search_edge(2, 1) graph.search_edge(0, 3) |
C#
using System;using System.Collections.Generic;class Graph { int V; HashSet<int>[] adjList; public Graph(int V) { this.V = V; adjList = new HashSet<int>[ V ]; for (int i = 0; i < V; i++) { adjList[i] = new HashSet<int>(); } } // Adds an edge to an undirected graph void addEdge(int src, int dest) { // Add an edge from src to dest. A new // element is inserted to the adjacent // list of src. adjList[src].Add(dest); // Since graph is undirected, add an edge // from dest to src also adjList[dest].Add(src); } // A utility function to print the adjacency // list representation of graph void printGraph() { for (int i = 0; i < V; i++) { HashSet<int> lst = adjList[i]; Console.WriteLine("Adjacency list of vertex " + i); foreach(int itr in lst) { Console.Write(itr + " "); } Console.WriteLine(); } } // Searches for a given edge in the graph void searchEdge(int src, int dest) { if (!adjList[src].Contains(dest)) { Console.WriteLine("Edge from " + src + " to " + dest + " not found."); } else { Console.WriteLine("Edge from " + src + " to " + dest + " found."); } } public static void Main(string[] args) { // Create the graph given in the above figure int V = 5; Graph graph = new Graph(V); graph.addEdge(0, 1); graph.addEdge(0, 4); graph.addEdge(1, 2); graph.addEdge(1, 3); graph.addEdge(1, 4); graph.addEdge(2, 3); graph.addEdge(3, 4); // Print the adjacency list representation of // the above graph graph.printGraph(); // Search the given edge in the graph graph.searchEdge(2, 1); graph.searchEdge(0, 3); }}// THIS CODE IS CONTRIBUTED BY YASH// AGARWAL(YASHAGARWAL2852002) |
Javascript
<script>// A JavaScript program to demonstrate adjacency list// representation of graphs using sets// Struct to represent a graphclass Graph { constructor(V) { this.V = V; this.adjList = new Array(V).fill().map(() => new Set()); }}// Adds an edge to an undirected graphfunction addEdge(graph, src, dest) { // Add an edge from src to dest. A new element is inserted // to the adjacent list of src. graph.adjList[src].add(dest); // Since graph is undirected, add an edge from dest to src also graph.adjList[dest].add(src);}// A utility function to print the adjacency list representation of graphfunction printGraph(graph) { for (let i = 0; i < graph.V; ++i) { const lst = graph.adjList[i]; console.log(`\nAdjacency list of vertex ${i}\n`); for (const element of lst) { console.log(element); } }}// Searches for a given edge in the graphfunction searchEdge(graph, src, dest) { if (graph.adjList[src].has(dest)) { console.log(`\nEdge from ${src} to ${dest} found.\n`); } else { console.log(`\nEdge from ${src} to ${dest} not found.\n`); }}// Test code// Create the graph given in the above figureconst V = 5;const graph = new Graph(V);addEdge(graph, 0, 1);addEdge(graph, 0, 4);addEdge(graph, 1, 2);addEdge(graph, 1, 3);addEdge(graph, 1, 4);addEdge(graph, 2, 3);addEdge(graph, 3, 4);// Print the adjacency list representation of the above graphprintGraph(graph);// Search the given edge in the graphsearchEdge(graph, 2, 1);searchEdge(graph, 0, 3);</script> |
Output
Adjacency list of vertex 0 4 1 Adjacency list of vertex 1 4 3 2 0 Adjacency list of vertex 2 3 1 Adjacency list of vertex 3 4 2 1 Adjacency list of vertex 4 3 1 0 Edge from 2 to 1 found. Edge from 0 to 3 not found.
Time Complexity: The time complexity of creating a graph using adjacency list is O(V + E), where V is the number of vertices and E is the number of edges in the graph.
Space Complexity: The space complexity of creating a graph using adjacency list is O(V + E), where V is the number of vertices and E is the number of edges in the graph.
Pros:
- Queries like whether there is an edge from vertex u to vertex v can be done in O(1).
- Adding an edge takes O(1).
Cons:
- Graphs containing parallel edge(s) cannot be implemented through this method.
- Edges are stored in any order.
Note : adjacency matrix representation is the most optimized for edge search, but space requirements of adjacency matrix are comparatively high for big sparse graphs. Moreover adjacency matrix has other disadvantages as well like BFS and DFS become costly as we can’t quickly get all adjacent of a node.
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