给定一个具有V个顶点和E 个边的二进制值无向图，任务是在该图的所有连通分量中找到最大的十进制等效值。一个二进制值图可以被认为只有二进制数（0 或 1）作为顶点值。
例子：
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 = 3
Input: 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
方法：
- 这个想法是使用深度优先搜索遍历来跟踪无向图中的连接组件，如本文所述。
- 对于每个连接的组件，存储二进制字符串并计算等效的十进制值。
- 设置全局最大值，将其与每次迭代后获得的最大十进制等效值进行比较以获得最终结果。
下面是上述方法的实现：
C++
// C++ Program to find
// maximum decimal equivalent among
// all connected components
#include
using namespace std;
// Function to traverse the undirected
// graph using the Depth first traversal
void depthFirst(int v, vector graph[],
vector& visited,
vector& 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 graph[], int vertices,
vector values)
{
// Initializing boolean array
// to mark visited vertices
vector 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 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 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 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> graph,
boolean[] visited,
List 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> graph,
int vertices,
List 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 storeChain = new ArrayList();
// 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> graph = new ArrayList();
for(int i = 0; i < 1001; i++)
graph.add(new ArrayList());
// Defining the number
// of edges and vertices
int E = 4, V = 7;
// Assigning the values for each
// vertex of the undirected graph
List 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
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_56C#
// 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输出：
3复杂度分析：
时间复杂度： O(V 2 )
DFS 算法运行时间为 O(V + E)，其中 V、E 是无向图的顶点和边。此外，在每次迭代时都会找到十进制等效值，这需要额外的 O(V) 来计算和返回结果。因此，整体复杂度为O(V 2 )
辅助空间： O(V)