Maximum decimal equivalent possible among all connected components of a Binary Valued Graph
Given a binary-valued Undirected Graph with V vertices and E edges, the task is to find the maximum decimal equivalent among all the connected components of the graph. A binary-valued graph can be considered as having only binary numbers (0 or 1) as the vertex values.
Examples:
Input: E = 4, V = 7
Output: 3
Explanation:
Decimal equivalents of the connected components are as follows:
[0, 1] : Maximum possible decimal equivalent = 2 [(10)2]
[0, 0, 0] : Maximum possible decimal equivalent = 2
[1, 1] : Maximum possible decimal equivalent = 3
Hence, Maximum decimal equivalent of all components = 3Input: E = 6, V = 10
Output: 8
Explanation:
Connected Components and decimal equivalent are as follows:
[1] : Maximum possible decimal equivalent = 2
[0, 0, 1, 0] : Maximum possible decimal equivalent = 8 [(1000)2]
[1, 1, 0] : Maximum possible decimal equivalent = 6
[1, 0] : Maximum possible decimal equivalent = 2
Hence, Maximum decimal equivalent of all components = 8
Approach:
- The idea is to use Depth First Search Traversal to keep track of the connected components in the undirected graph as explained in this article.
- For each connected component, the binary string is stored and the equivalent decimal value is calculated.
- A global maximum is set that is compared to the maximum decimal equivalent obtained after every iteration to get the final result.
Below is the implementation of the above approach:
C++
// C++ Program to find // maximum decimal equivalent among // all connected components #include <bits/stdc++.h> using namespace std; // Function to traverse the undirected // graph using the Depth first traversal void depthFirst( int v, vector< int > graph[], vector< bool >& visited, vector< int >& storeChain) { // Marking the visited // vertex as true visited[v] = true ; // Store the connected chain storeChain.push_back(v); for ( auto i : graph[v]) { if (visited[i] == false ) { // Recursive call to // the DFS algorithm depthFirst(i, graph, visited, storeChain); } } } // Function to return decimal // equivalent of each connected // component int decimal( int arr[], int n) { int zeros = 0, ones = 0; // Storing the respective // counts of 1's and 0's for ( int i = 0; i < n; i++) { if (arr[i] == 0) zeros++; else ones++; } // If all are zero then // maximum decimal equivalent // is also zero if (zeros == n) return 0; int temp = n - ones; int dec = 0; // For all the 1's, calculate // the decimal equivalent by // appropriate multiplication // of power of 2's while (ones--) { dec += pow (2, temp); temp++; } return dec; } // Function to find the maximum // decimal equivalent among all // connected components void decimalValue( vector< int > graph[], int vertices, vector< int > values) { // Initializing boolean array // to mark visited vertices vector< bool > visited(10001, false ); // maxDeci stores the // maximum decimal value int maxDeci = INT_MIN; // Following loop invokes // DFS algorithm for ( int i = 1; i <= vertices; i++) { if (visited[i] == false ) { // Variable to hold // temporary length int sizeChain; // Variable to hold temporary // Decimal values int tempDeci; // Container to store // each chain vector< int > storeChain; // DFS algorithm depthFirst(i, graph, visited, storeChain); // Variable to hold each // chain size sizeChain = storeChain.size(); // Container to store values // of vertices of individual // chains int chainValues[sizeChain + 1]; // Storing the values of // each chain for ( int i = 0; i < sizeChain; i++) { int temp = values[storeChain[i] - 1]; chainValues[i] = temp; } // Function call to find // decimal equivalent tempDeci = decimal(chainValues, sizeChain); // Conditional to store maximum // value of decimal equivalent if (tempDeci > maxDeci) { maxDeci = tempDeci; } } } // Printing the decimal result // (global maxima) cout << maxDeci; } // Driver code int main() { // Initializing graph in the // form of adjacency list vector< int > graph[1001]; // Defining the number of // edges and vertices int E, V; E = 4; V = 7; // Assigning the values // for each vertex of the // undirected graph vector< int > values; values.push_back(0); values.push_back(1); values.push_back(0); values.push_back(0); values.push_back(0); values.push_back(1); values.push_back(1); // Constructing the // undirected graph graph[1].push_back(2); graph[2].push_back(1); graph[3].push_back(4); graph[4].push_back(3); graph[4].push_back(5); graph[5].push_back(4); graph[6].push_back(7); graph[7].push_back(6); decimalValue(graph, V, values); return 0; } |
Java
// Java program to find maximum // decimal equivalent among all // connected components import java.io.*; import java.util.*; class GFG{ // Function to traverse the undirected // graph using the Depth first traversal static void depthFirst( int v, List<List<Integer>> graph, boolean [] visited, List<Integer> storeChain) { // Marking the visited // vertex as true visited[v] = true ; // Store the connected chain storeChain.add(v); for ( int i : graph.get(v)) { if (visited[i] == false ) { // Recursive call to // the DFS algorithm depthFirst(i, graph, visited, storeChain); } } } // Function to return decimal // equivalent of each connected // component static int decimal( int arr[], int n) { int zeros = 0 , ones = 0 ; // Storing the respective // counts of 1's and 0's for ( int i = 0 ; i < n; i++) { if (arr[i] == 0 ) zeros++; else ones++; } // If all are zero then maximum // decimal equivalent is also zero if (zeros == n) return 0 ; int temp = n - ones; int dec = 0 ; // For all the 1's, calculate // the decimal equivalent by // appropriate multiplication // of power of 2's while (ones > 0 ) { dec += Math.pow( 2 , temp); temp++; ones--; } return dec; } // Function to find the maximum // decimal equivalent among all // connected components static void decimalValue(List<List<Integer>> graph, int vertices, List<Integer> values) { // Initializing boolean array // to mark visited vertices boolean [] visited = new boolean [ 10001 ]; // maxDeci stores the // maximum decimal value int maxDeci = Integer.MIN_VALUE; // Following loop invokes // DFS algorithm for ( int i = 1 ; i <= vertices; i++) { if (visited[i] == false ) { // Variable to hold // temporary length int sizeChain; // Variable to hold temporary // Decimal values int tempDeci; // Container to store // each chain List<Integer> storeChain = new ArrayList<Integer>(); // DFS algorithm depthFirst(i, graph, visited, storeChain); // Variable to hold each // chain size sizeChain = storeChain.size(); // Container to store values // of vertices of individual // chains int [] chainValues = new int [sizeChain + 1 ]; // Storing the values of // each chain for ( int j = 0 ; j < sizeChain; j++) { int temp = values.get( storeChain.get(j) - 1 ); chainValues[j] = temp; } // Function call to find // decimal equivalent tempDeci = decimal(chainValues, sizeChain); // Conditional to store maximum // value of decimal equivalent if (tempDeci > maxDeci) { maxDeci = tempDeci; } } } // Printing the decimal result // (global maxima) System.out.println(maxDeci); } // Driver code public static void main(String[] args) { // Initializing graph in the // form of adjacency list @SuppressWarnings ( "unchecked" ) List<List<Integer>> graph = new ArrayList(); for ( int i = 0 ; i < 1001 ; i++) graph.add( new ArrayList<Integer>()); // Defining the number // of edges and vertices int E = 4 , V = 7 ; // Assigning the values for each // vertex of the undirected graph List<Integer> values = new ArrayList<Integer>(); values.add( 0 ); values.add( 1 ); values.add( 0 ); values.add( 0 ); values.add( 0 ); values.add( 1 ); values.add( 1 ); // Constructing the // undirected graph graph.get( 1 ).add( 2 ); graph.get( 2 ).add( 1 ); graph.get( 3 ).add( 4 ); graph.get( 4 ).add( 3 ); graph.get( 4 ).add( 5 ); graph.get( 5 ).add( 4 ); graph.get( 6 ).add( 7 ); graph.get( 7 ).add( 6 ); decimalValue(graph, V, values); } } // This code is contributed by jithin |
Python3
# Python3 Program to find # maximum decimal equivalent among # all connected components import sys # Function to traverse the # undirected graph using # the Depth first traversal def depthFirst(v, graph, visited, storeChain): # Marking the visited # vertex as true visited[v] = True ; # Store the connected chain storeChain.append(v); for i in graph[v]: if (visited[i] = = False ): # Recursive call to # the DFS algorithm depthFirst(i, graph, visited, storeChain); # Function to return decimal # equivalent of each connected # component def decimal(arr, n): zeros = 0 ones = 0 # Storing the respective # counts of 1's and 0's for i in range (n): if (arr[i] = = 0 ): zeros + = 1 ; else : ones + = 1 ; # If all are zero then # maximum decimal equivalent # is also zero if (zeros = = n): return 0 ; temp = n - ones; dec = 0 ; # For all the 1's, calculate # the decimal equivalent by # appropriate multiplication # of power of 2's while (ones ! = 0 ): ones - = 1 dec + = pow ( 2 , temp); temp + = 1 ; return dec; # Function to find the maximum # decimal equivalent among all # connected components def decimalValue(graph, vertices, values): # Initializing boolean array # to mark visited vertices visited = [ False for i in range ( 10001 )] # maxDeci stores the # maximum decimal value maxDeci = - sys.maxsize; # Following loop invokes # DFS algorithm for i in range (vertices + 1 ): if (visited[i] = = False ): # Variable to hold # temporary length sizeChain = 0 ; # Variable to hold # temporary Decimal values tempDeci = 0 ; # Container to store # each chain storeChain = []; # DFS algorithm depthFirst(i, graph, visited, storeChain); # Variable to hold each # chain size sizeChain = len (storeChain) # Container to store values # of vertices of individual # chains chainValues = [ 0 for i in range (sizeChain + 1 )] # Storing the values of # each chain for i in range (sizeChain): temp = values[storeChain[i] - 1 ]; chainValues[i] = temp; # Function call to find # decimal equivalent tempDeci = decimal(chainValues, sizeChain); # Conditional to store maximum # value of decimal equivalent if (tempDeci > maxDeci): maxDeci = tempDeci; # Printing the decimal result # (global maxima) print (maxDeci) if __name__ = = "__main__" : # Initializing graph in the # form of adjacency list graph = [[] for i in range ( 1001 )] # Defining the number # of edges and vertices E = 4 ; V = 7 ; # Assigning the values # for each vertex of # the undirected graph values = []; values.append( 0 ); values.append( 1 ); values.append( 0 ); values.append( 0 ); values.append( 0 ); values.append( 1 ); values.append( 1 ); # Constructing the # undirected graph graph[ 1 ].append( 2 ); graph[ 2 ].append( 1 ); graph[ 3 ].append( 4 ); graph[ 4 ].append( 3 ); graph[ 4 ].append( 5 ); graph[ 5 ].append( 4 ); graph[ 6 ].append( 7 ); graph[ 7 ].append( 6 ); decimalValue(graph, V, values); # This code is contributed by rutvik_56 |
C#
// C# program to find maximum // decimal equivalent among all // connected components using System; using System.Collections; using System.Collections.Generic; class GFG{ // Function to traverse the undirected // graph using the Depth first traversal static void depthFirst( int v, ArrayList graph, bool [] visited, ArrayList storeChain) { // Marking the visited // vertex as true visited[v] = true ; // Store the connected chain storeChain.Add(v); foreach ( int i in (ArrayList)graph[v]) { if (visited[i] == false ) { // Recursive call to // the DFS algorithm depthFirst(i, graph, visited, storeChain); } } } // Function to return decimal_t // equivalent of each connected // component static int decimal_t( int []arr, int n) { int zeros = 0, ones = 0; // Storing the respective // counts of 1's and 0's for ( int i = 0; i < n; i++) { if (arr[i] == 0) zeros++; else ones++; } // If all are zero then maximum // decimal_t equivalent is also zero if (zeros == n) return 0; int temp = n - ones; int dec = 0; // For all the 1's, calculate // the decimal_t equivalent by // appropriate multiplication // of power of 2's while (ones > 0) { dec += ( int )Math.Pow(2, temp); temp++; ones--; } return dec; } // Function to find the maximum // decimal_t equivalent among all // connected components static void decimal_tValue(ArrayList graph, int vertices, ArrayList values) { // Initializing boolean array // to mark visited vertices bool [] visited = new bool [10001]; // maxDeci stores the // maximum decimal_t value int maxDeci = -100000000; // Following loop invokes // DFS algorithm for ( int i = 1; i <= vertices; i++) { if (visited[i] == false ) { // Variable to hold // temporary length int sizeChain; // Variable to hold temporary // decimal_t values int tempDeci; // Container to store // each chain ArrayList storeChain = new ArrayList(); // DFS algorithm depthFirst(i, graph, visited, storeChain); // Variable to hold each // chain size sizeChain = storeChain.Count; // Container to store values // of vertices of individual // chains int [] chainValues = new int [sizeChain + 1]; // Storing the values of // each chain for ( int j = 0; j < sizeChain; j++) { int temp = ( int )values[( int )storeChain[j] - 1]; chainValues[j] = temp; } // Function call to find // decimal_t equivalent tempDeci = decimal_t(chainValues, sizeChain); // Conditional to store maximum // value of decimal_t equivalent if (tempDeci > maxDeci) { maxDeci = tempDeci; } } } // Printing the decimal_t result // (global maxima) Console.WriteLine(maxDeci); } // Driver code public static void Main( string [] args) { // Initializing graph in the // form of adjacency list ArrayList graph = new ArrayList(); for ( int i = 0; i < 1001; i++) graph.Add( new ArrayList()); // Defining the number // of edges and vertices int V = 7; // Assigning the values for each // vertex of the undirected graph ArrayList values = new ArrayList(); values.Add(0); values.Add(1); values.Add(0); values.Add(0); values.Add(0); values.Add(1); values.Add(1); // Constructing the // undirected graph ((ArrayList)graph[1]).Add(2); ((ArrayList)graph[2]).Add(1); ((ArrayList)graph[3]).Add(4); ((ArrayList)graph[4]).Add(3); ((ArrayList)graph[4]).Add(5); ((ArrayList)graph[5]).Add(4); ((ArrayList)graph[6]).Add(7); ((ArrayList)graph[7]).Add(6); decimal_tValue(graph, V, values); } } // This code is contributed by pratham76 |
Javascript
// JavaScript Program to find // maximum decimal equivalent among // all connected components // Function to traverse the undirected // graph using the Depth first traversal function depthFirst(v, graph, visited, storeChain) { // Marking the visited // vertex as true visited[v] = true ; // Store the connected chain storeChain.push(v); graph[v].forEach((i) => { if (visited[i] === false ) { // Recursive call to // the DFS algorithm depthFirst(i, graph, visited, storeChain); } }); } // Function to return decimal // equivalent of each connected // component function decimal(arr, n) { var zeros = 0, ones = 0; // Storing the respective // counts of 1's and 0's for ( var i = 0; i < n; i++) { if (arr[i] === 0) zeros++; else ones++; } // If all are zero then // maximum decimal equivalent // is also zero if (zeros === n) return 0; var temp = n - ones; var dec = 0; // For all the 1's, calculate // the decimal equivalent by // appropriate multiplication // of power of 2's while (ones--) { dec += Math.pow(2, temp); temp++; } return dec; } // Function to find the maximum // decimal equivalent among all // connected components function decimalValue(graph, vertices, values) { // Initializing boolean array // to mark visited vertices var visited = Array(10001).fill( false ); // maxDeci stores the // maximum decimal value var maxDeci = Number.MIN_SAFE_INTEGER; // Following loop invokes // DFS algorithm for ( var i = 1; i <= vertices; i++) { if (visited[i] === false ) { // Variable to hold // temporary length var sizeChain; // Variable to hold temporary // Decimal values var tempDeci; // Container to store // each chain var storeChain = []; // DFS algorithm depthFirst(i, graph, visited, storeChain); // Variable to hold each // chain size sizeChain = storeChain.length; // Container to store values // of vertices of individual // chains var chainValues = Array(sizeChain + 1); // Storing the values of // each chain for ( var i = 0; i < sizeChain; i++) { var temp = values[storeChain[i] - 1]; chainValues[i] = temp; } // Function call to find // decimal equivalent tempDeci = decimal(chainValues, sizeChain); // Conditional to store maximum // value of decimal equivalent if (tempDeci > maxDeci) { maxDeci = tempDeci; } } } // Printing the decimal result // (global maxima) console.log(maxDeci); } // Driver code // Initializing graph in the // form of adjacency list var graph = Array(1001).fill(0).map(() => []); // Defining the number of // edges and vertices var E = 4; var V = 7; // Assigning the values // for each vertex of the // undirected graph var values = [0, 1, 0, 0, 0, 1, 1]; // Constructing the // undirected graph graph[1].push(2); graph[2].push(1); graph[3].push(4); graph[4].push(3); graph[4].push(5); graph[5].push(4); graph[6].push(7); graph[7].push(6); decimalValue(graph, V, values); // This code is contributed by prasad264 |
3
Complexity analysis:
Time Complexity: O(V2)
The DFS algorithm takes O(V + E) time to run, where V, E are the vertices and edges of the undirected graph. Further, the decimal equivalent is found at each iteration that takes an additional O(V) to compute and return the result. Hence, the overall complexity is O(V2)
Auxiliary Space: O(V)
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