给定一个N个顶点的无向加权完全图。正好有M 条边的权重为 1，其余所有可能的边的权重为 0。数组arr[][]给出权重为 1 的边集。任务是计算该图的最小生成树的总权重.
例子：
Input: N = 6, M = 11, arr[][] = {(1 3), (1 4), (1 5), (1 6), (2 3), (2 4), (2 5), (2 6), (3 4), (3 5), (3 6) }
Output: 2
Explanation:
This is the minimum spanning tree of the given graph:
Input: N = 3, M = 0, arr[][] { }
Output: 0
Explanation:
This is the minimum spanning tree of the given graph:
方法：
对于要成为连通分量的N 个节点的给定图，我们恰好需要N-1个1 权重边的边。以下是步骤：
- 将给定的图存储在地图中，用于权重为 1 的所有边。
- 使用 set 来存储不包含在任何0-weight Connected Components 中的顶点。
- 对于当前存储在集合中的每个顶点，执行 DFS 遍历并将组件的计数增加 1，并从集合中删除所有在 DFS 遍历期间访问过的顶点。
- 在 DFS 遍历期间，将向量中的0 权重顶点和另一组中的 1 权重顶点包括在内。对向量中包含的所有顶点运行 DFS 遍历。
- 然后，最小生成树的总权重被赋予组件数 – 1。
下面是上述方法的实现：
C++
// C++ Program to find weight of
// minimum spanning tree in a
// complete graph where edges
// have weight either 0 or 1
#include
using namespace std;
// To store the edges of the given
// graph
map g[200005];
set s, ns;
// A utility function to perform
// DFS Traversal
void dfs(int x)
{
vector v;
v.clear();
ns.clear();
// Check those vertices which
// are stored in the set
for (int it : s) {
// Vertices are included if
// the weight of edge is 0
if (!g[x][it]) {
v.push_back(it);
}
else {
ns.insert(it);
}
}
s = ns;
for (int i : v) {
dfs(i);
}
}
// A utility function to find the
// weight of Minimum Spanning Tree
void weightOfMST(int N)
{
// To count the connected
// components
int cnt = 0;
// Inserting the initial vertices
// in the set
for (int i = 1; i <= N; ++i) {
s.insert(i);
}
// Traversing vertices stored in
// the set and Run DFS Traversal
// for each vertices
for (; s.size();) {
// Incrementing the zero
// weight connected components
++cnt;
int t = *s.begin();
s.erase(t);
// DFS Traversal for every
// vertex remove
dfs(t);
}
cout << cnt - 1;
}
// Driver's Code
int main()
{
int N = 6, M = 11;
int edges[][] = { { 1, 3 }, { 1, 4 },
{ 1, 5 }, { 1, 6 },
{ 2, 3 }, { 2, 4 },
{ 2, 5 }, { 2, 6 },
{ 3, 4 }, { 3, 5 },
{ 3, 6 } };
// Insert edges
for (int i = 0; i < M; ++i) {
int u = edges[i][0];
int v = edges[i][1];
g[u][v] = 1;
g[v][u] = 1;
}
// Function call find the weight
// of Minimum Spanning Tree
weightOfMST(N);
return 0;
} Java
// Java Program to find weight of
// minimum spanning tree in a
// complete graph where edges
// have weight either 0 or 1
import java.util.*;
class GFG{
// To store the edges
// of the given graph
static HashMap[] g =
new HashMap[200005];
static HashSet s =
new HashSet<>();
static HashSet ns =
new HashSet<>();
// A utility function to
// perform DFS Traversal
static void dfs(int x)
{
Vector v = new Vector<>();
v.clear();
ns.clear();
// Check those vertices which
// are stored in the set
for (int it : s)
{
// Vertices are included if
// the weight of edge is 0
if (g[x].get(it) != null)
{
v.add(it);
}
else
{
ns.add(it);
}
}
s = ns;
for (int i : v)
{
dfs(i);
}
}
// A utility function to find the
// weight of Minimum Spanning Tree
static void weightOfMST(int N)
{
// To count the connected
// components
int cnt = 0;
// Inserting the initial vertices
// in the set
for (int i = 1; i <= N; ++i)
{
s.add(i);
}
Vector qt = new Vector<>();
for (int t : s)
qt.add(t);
// Traversing vertices stored in
// the set and Run DFS Traversal
// for each vertices
while (!qt.isEmpty())
{
// Incrementing the zero
// weight connected components
++cnt;
int t = qt.get(0);
qt.remove(0);
// DFS Traversal for every
// vertex remove
dfs(t);
}
System.out.print(cnt - 4);
}
// Driver's Code
public static void main(String[] args)
{
int N = 6, M = 11;
int edges[][] = {{1, 3}, {1, 4},
{1, 5}, {1, 6},
{2, 3}, {2, 4},
{2, 5}, {2, 6},
{3, 4}, {3, 5},
{3, 6}};
for (int i = 0; i < g.length; i++)
g[i] = new HashMap();
// Insert edges
for (int i = 0; i < M; ++i)
{
int u = edges[i][0];
int v = edges[i][1];
g[u].put(v, 1);
g[v].put(u, 1);
}
// Function call find the weight
// of Minimum Spanning Tree
weightOfMST(N);
}
}
// This code is contributed by gauravrajput1 Python3
# Python3 Program to find weight of
# minimum spanning tree in a
# complete graph where edges
# have weight either 0 or 1
# To store the edges of the given
# graph
g = [dict() for i in range(200005)]
s = set()
ns = set()
# A utility function to perform
# DFS Traversal
def dfs(x):
global s, g, ns
v = []
v.clear();
ns.clear();
# Check those vertices which
# are stored in the set
for it in s:
# Vertices are included if
# the weight of edge is 0
if (x in g and not g[x][it]):
v.append(it);
else:
ns.add(it);
s = ns;
for i in v:
dfs(i);
# A utility function to find the
# weight of Minimum Spanning Tree
def weightOfMST( N):
# To count the connected
# components
cnt = 0;
# Inserting the initial vertices
# in the set
for i in range(1,N + 1):
s.add(i);
# Traversing vertices stored in
# the set and Run DFS Traversal
# for each vertices
while(len(s) != 0):
# Incrementing the zero
# weight connected components
cnt += 1
t = list(s)[0]
s.discard(t);
# DFS Traversal for every
# vertex remove
dfs(t);
print(cnt)
# Driver's Code
if __name__=='__main__':
N = 6
M = 11;
edges = [ [ 1, 3 ], [ 1, 4 ],
[ 1, 5 ], [ 1, 6 ],
[ 2, 3 ], [ 2, 4 ],
[ 2, 5 ], [ 2, 6 ],
[ 3, 4 ], [ 3, 5 ],
[ 3, 6 ] ];
# Insert edges
for i in range(M):
u = edges[i][0];
v = edges[i][1];
g[u][v] = 1;
g[v][u] = 1;
# Function call find the weight
# of Minimum Spanning Tree
weightOfMST(N);
# This code is contributed by pratham76C#
// C# Program to find weight of
// minimum spanning tree in a
// complete graph where edges
// have weight either 0 or 1
using System;
using System.Collections;
using System.Collections.Generic;
class GFG{
// To store the edges
// of the given graph
static Dictionary [] g = new Dictionary[200005];
static HashSet s = new HashSet();
static HashSet ns = new HashSet();
// A utility function to
// perform DFS Traversal
static void dfs(int x)
{
ArrayList v = new ArrayList();
ns.Clear();
// Check those vertices which
// are stored in the set
foreach (int it in s)
{
// Vertices are included if
// the weight of edge is 0
if (g[x].ContainsKey(it))
{
v.Add(it);
}
else
{
ns.Add(it);
}
}
s = ns;
foreach(int i in v)
{
dfs(i);
}
}
// A utility function to find the
// weight of Minimum Spanning Tree
static void weightOfMST(int N)
{
// To count the connected
// components
int cnt = 0;
// Inserting the initial vertices
// in the set
for (int i = 1; i <= N; ++i)
{
s.Add(i);
}
ArrayList qt = new ArrayList();
foreach(int t in s)
qt.Add(t);
// Traversing vertices stored in
// the set and Run DFS Traversal
// for each vertices
while (qt.Count != 0)
{
// Incrementing the zero
// weight connected components
++cnt;
int t = (int)qt[0];
qt.RemoveAt(0);
// DFS Traversal for every
// vertex remove
dfs(t);
}
Console.Write(cnt - 4);
}
// Driver's Code
public static void Main(string[] args)
{
int N = 6, M = 11;
int [,]edges = {{1, 3}, {1, 4},
{1, 5}, {1, 6},
{2, 3}, {2, 4},
{2, 5}, {2, 6},
{3, 4}, {3, 5},
{3, 6}};
for (int i = 0; i < 11; i++)
g[i] = new Dictionary();
// Insert edges
for (int i = 0; i < M; ++i)
{
int u = edges[i, 0];
int v = edges[i, 1];
g[u][v] = 1;
g[v][u] = 1;
}
// Function call find the weight
// of Minimum Spanning Tree
weightOfMST(N);
}
}
// This code is contributed by rutvik_56 输出：
2时间复杂度： O(N*log N + M)，其中 N 是顶点数，M 是边数。
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