Given an undirected and unweighted graph. The task is to find the product of the lengths of all cycles formed in it.
Example 1:
The above graph has two cycles of length 4 and 3, the product of cycle lengths is 12.
Example 2:
The above graph has two cycles of length 4 and 3, the product of cycle lengths is 12.
Approach: Using the graph coloring method, mark all the vertex of the different cycles with unique numbers. Once the graph traversal is completed, push all the similar marked numbers to an adjacency list and print the adjacency list accordingly.
Given below is the algorithm:
- Insert the edges into an adjacency list.
- Call the DFS function which uses the coloring method to mark the vertex.
- Whenever there is a partially visited vertex, backtrack till the current vertex is reached and mark all of them with cycle numbers. Once all the vertexes are marked, increase the cycle number.
- Once Dfs is completed, iterate for the edges and push the same marked number edges to another adjacency list.
- Iterate in the another adjacency list, and keep the count of number of vertex in a cycle using map and cycle numbers
- Iterate for cycle numbers, and multiply the lengths to get the final product which will be the answer.
Below is the implementation of the above approach:
// C++ program to find the // product of lengths of cycle #include <bits/stdc++.h> using namespace std;
const int N = 100000;
// variables to be used // in both functions vector< int > graph[N];
// Function to mark the vertex with // different colors for different cycles void dfs_cycle( int u, int p, int color[],
int mark[], int par[], int & cyclenumber)
{ // already (completely) visited vertex.
if (color[u] == 2) {
return ;
}
// seen vertex, but was not completely
// visited -> cycle detected.
// backtrack based on parents to find
// the complete cycle.
if (color[u] == 1) {
cyclenumber++;
int cur = p;
mark[cur] = cyclenumber;
// backtrack the vertex which are
// in the current cycle thats found
while (cur != u) {
cur = par[cur];
mark[cur] = cyclenumber;
}
return ;
}
par[u] = p;
// partially visited.
color[u] = 1;
// simple dfs on graph
for ( int v : graph[u]) {
// if it has not been visited previously
if (v == par[u]) {
continue ;
}
dfs_cycle(v, u, color, mark, par, cyclenumber);
}
// completely visited.
color[u] = 2;
} // add the edges to the graph void addEdge( int u, int v)
{ graph[u].push_back(v);
graph[v].push_back(u);
} // Function to print the cycles int productLength( int edges, int mark[], int & cyclenumber)
{ unordered_map< int , int > mp;
// push the edges that into the
// cycle adjacency list
for ( int i = 1; i <= edges; i++) {
if (mark[i] != 0)
mp[mark[i]]++;
}
int cnt = 1;
// product all the length of cycles
for ( int i = 1; i <= cyclenumber; i++) {
cnt = cnt * mp[i];
}
if (cyclenumber == 0)
cnt = 0;
return cnt;
} // Driver Code int main()
{ // add edges
addEdge(1, 2);
addEdge(2, 3);
addEdge(3, 4);
addEdge(4, 6);
addEdge(4, 7);
addEdge(5, 6);
addEdge(3, 5);
addEdge(7, 8);
addEdge(6, 10);
addEdge(5, 9);
addEdge(10, 11);
addEdge(11, 12);
addEdge(11, 13);
addEdge(12, 13);
// arrays required to color the
// graph, store the parent of node
int color[N];
int par[N];
// mark with unique numbers
int mark[N];
// store the numbers of cycle
int cyclenumber = 0;
int edges = 13;
// call DFS to mark the cycles
dfs_cycle(1, 0, color, mark, par, cyclenumber);
// function to print the cycles
cout << productLength(edges, mark, cyclenumber);
return 0;
} |
// Java program to find the // product of lengths of cycle import java.io.*;
import java.util.*;
class GFG
{ static int N = 100000 ;
static int cyclenumber;
// variables to be used
// in both functions
//@SuppressWarnings("unchecked")
static Vector<Integer>[] graph = new Vector[N];
// This static block is used to initialize
// array of Vector, otherwise it will throw
// NullPointerException
static
{
for ( int i = 0 ; i < N; i++)
graph[i] = new Vector<>();
}
// Function to mark the vertex with
// different colors for different cycles
static void dfs_cycle( int u, int p, int [] color,
int [] mark, int [] par)
{
// already (completely) visited vertex.
if (color[u] == 2 )
return ;
// seen vertex, but was not completely
// visited -> cycle detected.
// backtrack based on parents to find
// the complete cycle.
if (color[u] == 1 )
{
cyclenumber++;
int cur = p;
mark[cur] = cyclenumber;
// backtrack the vertex which are
// in the current cycle thats found
while (cur != u)
{
cur = par[cur];
mark[cur] = cyclenumber;
}
return ;
}
par[u] = p;
// partially visited.
color[u] = 1 ;
// simple dfs on graph
for ( int v : graph[u])
{
// if it has not been visited previously
if (v == par[u])
{
continue ;
}
dfs_cycle(v, u, color, mark, par);
}
// completely visited.
color[u] = 2 ;
}
// add the edges to the graph
static void addEdge( int u, int v)
{
graph[u].add(v);
graph[v].add(u);
}
// Function to print the cycles
static int productLength( int edges, int [] mark)
{
HashMap<Integer,
Integer> mp = new HashMap<>();
// push the edges that into the
// cycle adjacency list
for ( int i = 1 ; i <= edges; i++)
{
if (mark[i] != 0 )
{
mp.put(mark[i], mp.get(mark[i]) == null ?
1 : mp.get(mark[i]) + 1 );
}
}
int cnt = 1 ;
// product all the length of cycles
for ( int i = 1 ; i <= cyclenumber; i++)
{
cnt = cnt * mp.get(i);
}
if (cyclenumber == 0 )
cnt = 0 ;
return cnt;
}
// Driver Code
public static void main(String[] args) throws IOException
{
// add edges
addEdge( 1 , 2 );
addEdge( 2 , 3 );
addEdge( 3 , 4 );
addEdge( 4 , 6 );
addEdge( 4 , 7 );
addEdge( 5 , 6 );
addEdge( 3 , 5 );
addEdge( 7 , 8 );
addEdge( 6 , 10 );
addEdge( 5 , 9 );
addEdge( 10 , 11 );
addEdge( 11 , 12 );
addEdge( 11 , 13 );
addEdge( 12 , 13 );
// arrays required to color the
// graph, store the parent of node
int [] color = new int [N];
int [] par = new int [N];
// mark with unique numbers
int [] mark = new int [N];
// store the numbers of cycle
cyclenumber = 0 ;
int edges = 13 ;
// call DFS to mark the cycles
dfs_cycle( 1 , 0 , color, mark, par);
// function to print the cycles
System.out.println(productLength(edges, mark));
}
} // This code is contributed by // sanjeev2552 |
# Python3 program to find the # product of lengths of cycle from collections import defaultdict
# Function to mark the vertex with # different colors for different cycles def dfs_cycle(u, p, color, mark, par):
global cyclenumber
# already (completely) visited vertex.
if color[u] = = 2 :
return
# seen vertex, but was not completely
# visited -> cycle detected.
# backtrack based on parents to find
# the complete cycle.
if color[u] = = 1 :
cyclenumber + = 1
cur = p
mark[cur] = cyclenumber
# backtrack the vertex which are
# in the current cycle thats found
while cur ! = u:
cur = par[cur]
mark[cur] = cyclenumber
return
par[u] = p
# partially visited.
color[u] = 1
# simple dfs on graph
for v in graph[u]:
# if it has not been visited previously
if v = = par[u]:
continue
dfs_cycle(v, u, color, mark, par)
# completely visited.
color[u] = 2
# add the edges to the graph def addEdge(u, v):
graph[u].append(v)
graph[v].append(u)
# Function to print the cycles def productLength(edges, mark, cyclenumber):
mp = defaultdict( lambda : 0 )
# push the edges that into the
# cycle adjacency list
for i in range ( 1 , edges + 1 ):
if mark[i] ! = 0 :
mp[mark[i]] + = 1
cnt = 1
# product all the length of cycles
for i in range ( 1 , cyclenumber + 1 ):
cnt = cnt * mp[i]
if cyclenumber = = 0 :
cnt = 0
return cnt
# Driver Code if __name__ = = "__main__" :
N = 100000
graph = [[] for i in range (N)]
# add edges
addEdge( 1 , 2 )
addEdge( 2 , 3 )
addEdge( 3 , 4 )
addEdge( 4 , 6 )
addEdge( 4 , 7 )
addEdge( 5 , 6 )
addEdge( 3 , 5 )
addEdge( 7 , 8 )
addEdge( 6 , 10 )
addEdge( 5 , 9 )
addEdge( 10 , 11 )
addEdge( 11 , 12 )
addEdge( 11 , 13 )
addEdge( 12 , 13 )
# arrays required to color the
# graph, store the parent of node
color, par = [ None ] * N, [ None ] * N
# mark with unique numbers
mark = [ None ] * N
# store the numbers of cycle
cyclenumber, edges = 0 , 13
# call DFS to mark the cycles
dfs_cycle( 1 , 0 , color, mark, par)
# function to print the cycles
print (productLength(edges, mark,
cyclenumber))
# This code is contributed by Rituraj Jain |
// C# program to find the // product of lengths of cycle using System;
using System.Collections.Generic;
class GFG
{ static int N = 100000;
static int cyclenumber;
// variables to be used
// in both functions
//@SuppressWarnings("unchecked")
static List< int >[] graph = new List< int >[N];
// This static block is used to initialize
// array of List, otherwise it will throw
// NullPointerException
// Function to mark the vertex with
// different colors for different cycles
static void dfs_cycle( int u, int p, int [] color,
int [] mark, int [] par)
{
// already (completely) visited vertex.
if (color[u] == 2)
return ;
// seen vertex, but was not completely
// visited -> cycle detected.
// backtrack based on parents to find
// the complete cycle.
if (color[u] == 1)
{
cyclenumber++;
int cur = p;
mark[cur] = cyclenumber;
// backtrack the vertex which are
// in the current cycle thats found
while (cur != u)
{
cur = par[cur];
mark[cur] = cyclenumber;
}
return ;
}
par[u] = p;
// partially visited.
color[u] = 1;
// simple dfs on graph
foreach ( int v in graph[u])
{
// if it has not been visited previously
if (v == par[u])
{
continue ;
}
dfs_cycle(v, u, color, mark, par);
}
// completely visited.
color[u] = 2;
}
// add the edges to the graph
static void addEdge( int u, int v)
{
graph[u].Add(v);
graph[v].Add(u);
}
// Function to print the cycles
static int productLength( int edges, int [] mark)
{
Dictionary< int , int > mp = new Dictionary< int , int >();
// push the edges that into the
// cycle adjacency list
for ( int i = 1; i <= edges; i++)
{
if (mark[i] != 0)
{
if (mp.ContainsKey(mark[i]))
mp[mark[i]] = mp[mark[i]] + 1;
else
mp.Add(mark[i], 1);
}
}
int cnt = 1;
// product all the length of cycles
for ( int i = 1; i <= cyclenumber; i++)
{
cnt = cnt * mp[i];
}
if (cyclenumber == 0)
cnt = 0;
return cnt;
}
// Driver Code
public static void Main(String[] args)
{
for ( int i = 0; i < N; i++)
graph[i] = new List< int >();
// add edges
addEdge(1, 2);
addEdge(2, 3);
addEdge(3, 4);
addEdge(4, 6);
addEdge(4, 7);
addEdge(5, 6);
addEdge(3, 5);
addEdge(7, 8);
addEdge(6, 10);
addEdge(5, 9);
addEdge(10, 11);
addEdge(11, 12);
addEdge(11, 13);
addEdge(12, 13);
// arrays required to color the
// graph, store the parent of node
int [] color = new int [N];
int [] par = new int [N];
// mark with unique numbers
int [] mark = new int [N];
// store the numbers of cycle
cyclenumber = 0;
int edges = 13;
// call DFS to mark the cycles
dfs_cycle(1, 0, color, mark, par);
// function to print the cycles
Console.WriteLine(productLength(edges, mark));
}
} // This code is contributed by 29AjayKumar |
<script> // JavaScript program to find the // product of lengths of cycle var N = 100000;
var cyclenumber;
// variables to be used // in both functions //@SuppressWarnings("unchecked") var graph = Array.from(Array(N), ()=>Array());
// This static block is used to initialize // array of List, otherwise it will throw // NullPointerException // Function to mark the vertex with // different colors for different cycles function dfs_cycle(u, p, color, mark, par)
{ // already (completely) visited vertex.
if (color[u] == 2)
return ;
// seen vertex, but was not completely
// visited -> cycle detected.
// backtrack based on parents to find
// the complete cycle.
if (color[u] == 1)
{
cyclenumber++;
var cur = p;
mark[cur] = cyclenumber;
// backtrack the vertex which are
// in the current cycle thats found
while (cur != u)
{
cur = par[cur];
mark[cur] = cyclenumber;
}
return ;
}
par[u] = p;
// partially visited.
color[u] = 1;
// simple dfs on graph
for ( var v of graph[u])
{
// if it has not been visited previously
if (v == par[u])
{
continue ;
}
dfs_cycle(v, u, color, mark, par);
}
// completely visited.
color[u] = 2;
} // add the edges to the graph function addEdge(u, v)
{ graph[u].push(v);
graph[v].push(u);
} // Function to print the cycles function productLength(edges, mark)
{ var mp = new Map();
// push the edges that into the
// cycle adjacency list
for ( var i = 1; i <= edges; i++)
{
if (mark[i] != 0)
{
if (mp.has(mark[i]))
mp.set(mark[i], mp.get(mark[i])+1);
else
mp.set(mark[i], 1);
}
}
var cnt = 1;
// product all the length of cycles
for ( var i = 1; i <= cyclenumber; i++)
{
cnt = cnt * mp.get(i);
}
if (cyclenumber == 0)
cnt = 0;
return cnt;
} // Driver Code // add edges addEdge(1, 2); addEdge(2, 3); addEdge(3, 4); addEdge(4, 6); addEdge(4, 7); addEdge(5, 6); addEdge(3, 5); addEdge(7, 8); addEdge(6, 10); addEdge(5, 9); addEdge(10, 11); addEdge(11, 12); addEdge(11, 13); addEdge(12, 13); // arrays required to color the // graph, store the parent of node var color = Array(N).fill(0);
var par = Array(N).fill(0);
// mark with unique numbers var mark = Array(N).fill(0);
// store the numbers of cycle cyclenumber = 0; var edges = 13;
// call DFS to mark the cycles dfs_cycle(1, 0, color, mark, par); // function to print the cycles document.write(productLength(edges, mark)); </script> |
12
Time Complexity: O(N), where N is the number of nodes in the graph.