# Count ways to change direction of edges such that graph becomes acyclic

Given a directed and unweighted graph consisting of **N** vertices and an array **arr[]** where **ith** vertex have a directed edge to **arr[i]**. The task is to find the number of ways to change the direction of edges such that the given graph is acyclic.

**Examples:**

Input:N = 3, arr[] = {2, 3, 1}

The directed graph form by the given infromation is:

Output:6

Explanation:

There are 6 possible ways to change the direction of edges to make grpah acyclic:

**Approach:** The idea is to check whether the Connected Components form a cycle or not.

- If the component is a path, then however we orient the edges we won’t form a cycle.
- If the component has a cycle with N edges, then there are
**2**ways to arrange all the edges out of which only 2 ways are going to form a cycle. So there are^{N}**(2**ways to change the edges so that graph becomes acyclic.^{N}– 2)

**Steps**:

- Using Depth First Search(DFS) traversal find the cycles in the given graph and number of vertices associated with each cycle.
- After DFS traversal, the total number of ways to change the direction of edges is the product of the following:
- Number of ways form by each cycle of
**X**vertices is given by**(2**.^{X}– 2) - Number of ways form by each path of
**Y**vertices is given by**(2**.^{Y})

- Number of ways form by each cycle of

Below is the implementation of the above approach:

## C++

`// C++ program to count the ` `// number of ways to change ` `// the direction of edges ` `// such that no cycle is ` `// present in the graph ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Vector cycles[] to store ` `// the cycle with vertices ` `// associated with each cycle ` `vector<` `int` `> cycles; ` ` ` `// Count of cycle ` `int` `cyclecnt; ` ` ` `// Function to count the ` `// number of vertices in the ` `// current cycle ` `void` `DFSUtil(` `int` `u, ` `int` `arr[], ` `int` `vis[]) ` `{ ` ` ` `cycles[cyclecnt]++; ` ` ` `vis[u] = 3; ` ` ` ` ` `// Returns when the same ` ` ` `// initial vertex is found ` ` ` `if` `(vis[arr[u]] == 3) { ` ` ` `return` `; ` ` ` `} ` ` ` ` ` `// Recurr for next vertex ` ` ` `DFSUtil(arr[u], arr, vis); ` `} ` ` ` `// DFS traversal to detect ` `// the cycle in graph ` `void` `DFS(` `int` `u, ` `int` `arr[], ` `int` `vis[]) ` `{ ` ` ` `// Marke vis[u] to 2 to ` ` ` `// check for any cycle form ` ` ` `vis[u] = 2; ` ` ` ` ` `// If the vertex arr[u] ` ` ` `// is not visited ` ` ` `if` `(vis[arr[u]] == 0) { ` ` ` `// Call DFS ` ` ` `DFS(arr[u], arr, vis); ` ` ` `} ` ` ` ` ` `// If current node is ` ` ` `// processed ` ` ` `else` `if` `(vis[arr[u]] == 1) { ` ` ` `vis[u] = 1; ` ` ` `return` `; ` ` ` `} ` ` ` ` ` `// Cycle found, call DFSUtil ` ` ` `// to count the number of ` ` ` `// vertices in the current ` ` ` `// cycle ` ` ` `else` `{ ` ` ` `cycles.push_back(0); ` ` ` ` ` `// Count number of ` ` ` `// vertices in cycle ` ` ` `DFSUtil(u, arr, vis); ` ` ` `cyclecnt++; ` ` ` `} ` ` ` ` ` `// Current Node is processed ` ` ` `vis[u] = 1; ` `} ` ` ` `// Function to count the ` `// number of ways ` `int` `countWays(` `int` `arr[], ` `int` `N) ` `{ ` ` ` ` ` `int` `i, ans = 1; ` ` ` ` ` `// To precompute the power ` ` ` `// of 2 ` ` ` `int` `dp[N + 1]; ` ` ` `dp[0] = 1; ` ` ` ` ` `// Storing power of 2 ` ` ` `for` `(` `int` `i = 1; i <= N; i++) { ` ` ` `dp[i] = (dp[i - 1] * 2); ` ` ` `} ` ` ` ` ` `// Array vis[] created for ` ` ` `// DFS traversal ` ` ` `int` `vis[N + 1] = { 0 }; ` ` ` ` ` `// DFS traversal from Node 1 ` ` ` `for` `(` `int` `i = 1; i <= N; i++) { ` ` ` `if` `(vis[i] == 0) { ` ` ` ` ` `// Calling DFS ` ` ` `DFS(i, arr, vis); ` ` ` `} ` ` ` `} ` ` ` ` ` `int` `cnt = N; ` ` ` ` ` `// Traverse the cycles array ` ` ` `for` `(i = 0; i < cycles.size(); i++) { ` ` ` ` ` `// Remove the vertices ` ` ` `// which are part of a ` ` ` `// cycle ` ` ` `cnt -= cycles[i]; ` ` ` ` ` `// Count form by number ` ` ` `// vertices form cycle ` ` ` `ans *= dp[cycles[i]] - 2; ` ` ` `} ` ` ` ` ` `// Count form by number of ` ` ` `// vertices not forming ` ` ` `// cycle ` ` ` `ans = (ans * dp[cnt]); ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Driver's Code ` `int` `main() ` `{ ` ` ` `int` `N = 3; ` ` ` `int` `arr[] = { 0, 2, 3, 1 }; ` ` ` ` ` `// Function to count ways ` ` ` `cout << countWays(arr, N); ` ` ` `return` `0; ` `} ` |

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## Python3

`# Python 3 program to count the ` `# number of ways to change ` `# the direction of edges ` `# such that no cycle is ` `# present in the graph ` ` ` `# List cycles[] to store ` `# the cycle with vertices ` `# associated with each cycle ` `cycles ` `=` `[] ` ` ` `# Function to count the ` `# number of vertices in the ` `# current cycle ` `def` `DFSUtil(u, arr, vis, cyclecnt): ` ` ` ` ` `cycles[cyclecnt] ` `+` `=` `1` ` ` `vis[u] ` `=` `3` ` ` ` ` `# Returns when the same ` ` ` `# initial vertex is found ` ` ` `if` `(vis[arr[u]] ` `=` `=` `3` `) : ` ` ` `return` ` ` ` ` `# Recurr for next vertex ` ` ` `DFSUtil(arr[u], arr, vis, cyclecnt) ` ` ` `# DFS traversal to detect ` `# the cycle in graph ` `def` `DFS( u, arr, vis, cyclecnt): ` ` ` ` ` `# Marke vis[u] to 2 to ` ` ` `# check for any cycle form ` ` ` `vis[u] ` `=` `2` ` ` ` ` `# If the vertex arr[u] ` ` ` `# is not visited ` ` ` `if` `(vis[arr[u]] ` `=` `=` `0` `) : ` ` ` ` ` `# Call DFS ` ` ` `DFS(arr[u], arr, vis, cyclecnt) ` ` ` ` ` `# If current node is ` ` ` `# processed ` ` ` `elif` `(vis[arr[u]] ` `=` `=` `1` `): ` ` ` `vis[u] ` `=` `1` ` ` `return` ` ` ` ` `# Cycle found, call DFSUtil ` ` ` `# to count the number of ` ` ` `# vertices in the current ` ` ` `# cycle ` ` ` `else` `: ` ` ` `cycles.append(` `0` `) ` ` ` ` ` `# Count number of ` ` ` `# vertices in cycle ` ` ` `DFSUtil(u, arr, vis,cyclecnt) ` ` ` `cyclecnt ` `+` `=` `1` ` ` ` ` `# Current Node is processed ` ` ` `vis[u] ` `=` `1` ` ` `# Function to count the ` `# number of ways ` `def` `countWays(arr, N,cyclecnt): ` ` ` ` ` `ans ` `=` `1` ` ` ` ` `# To precompute the power ` ` ` `# of 2 ` ` ` `dp ` `=` `[` `0` `]` `*` `(N ` `+` `1` `) ` ` ` `dp[` `0` `] ` `=` `1` ` ` ` ` `# Storing power of 2 ` ` ` `for` `i ` `in` `range` `(` `1` `, N ` `+` `1` `): ` ` ` `dp[i] ` `=` `(dp[i ` `-` `1` `] ` `*` `2` `) ` ` ` ` ` `# Array vis[] created for ` ` ` `# DFS traversal ` ` ` `vis ` `=` `[` `0` `]` `*` `(N ` `+` `1` `) ` ` ` ` ` `# DFS traversal from Node 1 ` ` ` `for` `i ` `in` `range` `(` `1` `, N ` `+` `1` `) : ` ` ` `if` `(vis[i] ` `=` `=` `0` `) : ` ` ` ` ` `# Calling DFS ` ` ` `DFS(i, arr, vis, cyclecnt) ` ` ` ` ` `cnt ` `=` `N ` ` ` ` ` `# Traverse the cycles array ` ` ` `for` `i ` `in` `range` `(` `len` `(cycles)) : ` ` ` ` ` `# Remove the vertices ` ` ` `# which are part of a ` ` ` `# cycle ` ` ` `cnt ` `-` `=` `cycles[i] ` ` ` ` ` `# Count form by number ` ` ` `# vertices form cycle ` ` ` `ans ` `*` `=` `dp[cycles[i]] ` `-` `2` ` ` ` ` `# Count form by number of ` ` ` `# vertices not forming ` ` ` `# cycle ` ` ` `ans ` `=` `(ans ` `*` `dp[cnt]) ` ` ` ` ` `return` `ans ` ` ` `# Driver's Code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `N ` `=` `3` ` ` `cyclecnt ` `=` `0` ` ` `arr ` `=` `[ ` `0` `, ` `2` `, ` `3` `, ` `1` `] ` ` ` ` ` `# Function to count ways ` ` ` `print` `(countWays(arr, N,cyclecnt)) ` ` ` `# This code is contributed by chitranayal ` |

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**Output:**

6

**Time Complexity :** O(V + E)

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