# Painting Fence Algorithm

Given a fence with n posts and k colors, find out the number of ways of painting the fence such that at most 2 adjacent posts have the same color. Since answer can be large return it modulo 10^9 + 7.

**Examples:**

Input : n = 2 k = 4 Output : 16 We have 4 colors and 2 posts. Ways when both posts have same color : 4 Ways when both posts have diff color : 4*(choices for 1st post) * 3(choices for 2nd post) = 12 Input : n = 3 k = 2 Output : 6

Following image depicts the 6 possible ways of painting 3 posts with 2 colors:

Consider the following image in which c, c’ and c” are respective colors of posts i, i-1 and i -2.

According to the constraint of the problem, c = c’ = c” is not possible simultaneously, so either c’ != c or c” != c or both. There are k – 1 possibilities for c’ != c and k – 1 for c” != c.

diff = no of ways when color of last two posts is different same = no of ways when color of last two posts is same total ways = diff + sum for n = 1 diff = k, same = 0 total = k for n = 2 diff = k * (k-1) //k choices for first post, k-1 for next same = k //k choices for common color of two posts total = k + k * (k-1) for n = 3 diff = [k + k * (k-1)] * (k-1) (k-1) choices for 3rd post to not have color of 2nd post. same = k * (k-1) c'' != c, (k-1) choices for it Hence we deduce that, total[i] = same[i] + diff[i] same[i] = diff[i-1] diff[i] = (diff[i-1] + diff[i-2]) * (k-1) = total[i-1] * (k-1)

Below is the implementation of the problem:

## C++

`// C++ program for Painting Fence Algorithm ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Returns count of ways to color k posts ` `// using k colors ` `long` `countWays(` `int` `n, ` `int` `k) ` `{ ` ` ` `// To store results for subproblems ` ` ` `long` `dp[n + 1]; ` ` ` `memset` `(dp, 0, ` `sizeof` `(dp)); ` ` ` `int` `mod = 1000000007; ` ` ` ` ` `// There are k ways to color first post ` ` ` `dp[1] = k; ` ` ` ` ` `// There are 0 ways for single post to ` ` ` `// violate (same color_ and k ways to ` ` ` `// not violate (different color) ` ` ` `int` `same = 0, diff = k; ` ` ` ` ` `// Fill for 2 posts onwards ` ` ` `for` `(` `int` `i = 2; i <= n; i++) ` ` ` `{ ` ` ` `// Current same is same as previous diff ` ` ` `same = diff; ` ` ` ` ` `// We always have k-1 choices for next post ` ` ` `diff = dp[i-1] * (k-1); ` ` ` `diff = diff % mod; ` ` ` ` ` `// Total choices till i. ` ` ` `dp[i] = (same + diff) % mod; ` ` ` `} ` ` ` ` ` `return` `dp[n]; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 3, k = 2; ` ` ` `cout << countWays(n, k) << endl; ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// Java program for Painting Fence Algorithm ` `import` `java.util.*; ` ` ` `class` `GfG ` `{ ` ` ` `// Returns count of ways to color k posts ` `// using k colors ` `static` `long` `countWays(` `int` `n, ` `int` `k) ` `{ ` ` ` `// To store results for subproblems ` ` ` `long` `dp[] = ` `new` `long` `[n + ` `1` `]; ` ` ` `Arrays.fill(dp, ` `0` `); ` ` ` `int` `mod = ` `1000000007` `; ` ` ` ` ` `// There are k ways to color first post ` ` ` `dp[` `1` `] = k; ` ` ` ` ` `// There are 0 ways for single post to ` ` ` `// violate (same color_ and k ways to ` ` ` `// not violate (different color) ` ` ` `int` `same = ` `0` `, diff = k; ` ` ` ` ` `// Fill for 2 posts onwards ` ` ` `for` `(` `int` `i = ` `2` `; i <= n; i++) ` ` ` `{ ` ` ` `// Current same is same as previous diff ` ` ` `same = diff; ` ` ` ` ` `// We always have k-1 choices for next post ` ` ` `diff = (` `int` `) (dp[i-` `1` `] * (k-` `1` `)); ` ` ` `diff = diff % mod; ` ` ` ` ` `// Total choices till i. ` ` ` `dp[i] = (same + diff) % mod; ` ` ` `} ` ` ` ` ` `return` `dp[n]; ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{ ` ` ` `int` `n = ` `3` `, k = ` `2` `; ` ` ` `System.out.println(countWays(n, k)); ` ` ` `} ` `} ` ` ` `// This code contributed by Rajput-Ji ` |

*chevron_right*

*filter_none*

**Output:**

6

**Space optimization : **

We can optimize above solution to use one variable instead of a table.

Below is the the implementation of the problem:

## C++

`// C++ program for Painting Fence Algorithm ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Returns count of ways to color k posts ` `// using k colors ` `long` `countWays(` `int` `n, ` `int` `k) ` `{ ` ` ` `// There are k ways to color first post ` ` ` `long` `total = k; ` ` ` `int` `mod = 1000000007; ` ` ` ` ` `// There are 0 ways for single post to ` ` ` `// violate (same color_ and k ways to ` ` ` `// not violate (different color) ` ` ` `int` `same = 0, diff = k; ` ` ` ` ` `// Fill for 2 posts onwards ` ` ` `for` `(` `int` `i = 2; i <= n; i++) ` ` ` `{ ` ` ` `// Current same is same as previous diff ` ` ` `same = diff; ` ` ` ` ` `// We always have k-1 choices for next post ` ` ` `diff = total * (k-1); ` ` ` `diff = diff % mod; ` ` ` ` ` `// Total choices till i. ` ` ` `total = (same + diff) % mod; ` ` ` `} ` ` ` ` ` `return` `total; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 3, k = 2; ` ` ` `cout << countWays(n, k) << endl; ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// Java program for Painting Fence Algorithm ` `class` `GFG ` `{ ` ` ` `// Returns count of ways to color k posts ` `// using k colors ` `static` `long` `countWays(` `int` `n, ` `int` `k) ` `{ ` ` ` `// There are k ways to color first post ` ` ` `long` `total = k; ` ` ` `int` `mod = ` `1000000007` `; ` ` ` ` ` `// There are 0 ways for single post to ` ` ` `// violate (same color_ and k ways to ` ` ` `// not violate (different color) ` ` ` `int` `same = ` `0` `, diff = k; ` ` ` ` ` `// Fill for 2 posts onwards ` ` ` `for` `(` `int` `i = ` `2` `; i <= n; i++) ` ` ` `{ ` ` ` `// Current same is same as previous diff ` ` ` `same = diff; ` ` ` ` ` `// We always have k-1 choices for next post ` ` ` `diff = (` `int` `)total * (k - ` `1` `); ` ` ` `diff = diff % mod; ` ` ` ` ` `// Total choices till i. ` ` ` `total = (same + diff) % mod; ` ` ` `} ` ` ` `return` `total; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` ` ` `int` `n = ` `3` `, k = ` `2` `; ` ` ` `System.out.println(countWays(n, k)); ` `} ` `} ` ` ` `//This code is contributed by Mukul Singh ` |

*chevron_right*

*filter_none*

## Python3

`# Python3 program for Painting ` `# Fence Algorithm ` ` ` `# Returns count of ways to color ` `# k posts using k colors ` `def` `countWays(n, k) : ` ` ` ` ` `# There are k ways to color first post ` ` ` `total ` `=` `k ` ` ` `mod ` `=` `1000000007` ` ` ` ` `# There are 0 ways for single post to ` ` ` `# violate (same color_ and k ways to ` ` ` `# not violate (different color) ` ` ` `same, diff ` `=` `0` `, k ` ` ` ` ` `# Fill for 2 posts onwards ` ` ` `for` `i ` `in` `range` `(` `2` `, n ` `+` `1` `) : ` ` ` ` ` `# Current same is same as ` ` ` `# previous diff ` ` ` `same ` `=` `diff ` ` ` ` ` `# We always have k-1 choices ` ` ` `# for next post ` ` ` `diff ` `=` `total ` `*` `(k ` `-` `1` `) ` ` ` `diff ` `=` `diff ` `%` `mod ` ` ` ` ` `# Total choices till i. ` ` ` `total ` `=` `(same ` `+` `diff) ` `%` `mod ` ` ` ` ` `return` `total ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `n, k ` `=` `3` `, ` `2` ` ` `print` `(countWays(n, k)) ` ` ` `# This code is contributed by Ryuga ` |

*chevron_right*

*filter_none*

## C#

`// C# program for Painting Fence Algorithm ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `// Returns count of ways to color k posts ` ` ` `// using k colors ` ` ` `static` `long` `countWays(` `int` `n, ` `int` `k) ` ` ` `{ ` ` ` `// There are k ways to color first post ` ` ` `long` `total = k; ` ` ` `int` `mod = 1000000007; ` ` ` ` ` `// There are 0 ways for single post to ` ` ` `// violate (same color_ and k ways to ` ` ` `// not violate (different color) ` ` ` `long` `same = 0, diff = k; ` ` ` ` ` `// Fill for 2 posts onwards ` ` ` `for` `(` `int` `i = 2; i <= n; i++) ` ` ` `{ ` ` ` `// Current same is same as previous diff ` ` ` `same = diff; ` ` ` ` ` `// We always have k-1 choices for next post ` ` ` `diff = total * (k - 1); ` ` ` `diff = diff % mod; ` ` ` ` ` `// Total choices till i. ` ` ` `total = (same + diff) % mod; ` ` ` `} ` ` ` ` ` `return` `total; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `static` `void` `Main() ` ` ` `{ ` ` ` `int` `n = 3, k = 2; ` ` ` `Console.Write(countWays(n, k)); ` ` ` `} ` `} ` ` ` `//This code is contributed by DrRoot_ ` |

*chevron_right*

*filter_none*

## PHP

`<?php ` `// PHP program for Painting Fence Algorithm ` ` ` `// Returns count of ways to color k ` `// posts using k colors ` `function` `countWays(` `$n` `, ` `$k` `) ` `{ ` ` ` `// There are k ways to color first post ` ` ` `$total` `= ` `$k` `; ` ` ` `$mod` `= 1000000007; ` ` ` ` ` `// There are 0 ways for single post to ` ` ` `// violate (same color_ and k ways to ` ` ` `// not violate (different color) ` ` ` `$same` `= 0; ` ` ` `$diff` `= ` `$k` `; ` ` ` ` ` `// Fill for 2 posts onwards ` ` ` `for` `(` `$i` `= 2; ` `$i` `<= ` `$n` `; ` `$i` `++) ` ` ` `{ ` ` ` `// Current same is same as previous diff ` ` ` `$same` `= ` `$diff` `; ` ` ` ` ` `// We always have k-1 choices for next post ` ` ` `$diff` `= ` `$total` `* (` `$k` `- 1); ` ` ` `$diff` `= ` `$diff` `% ` `$mod` `; ` ` ` ` ` `// Total choices till i. ` ` ` `$total` `= (` `$same` `+ ` `$diff` `) % ` `$mod` `; ` ` ` `} ` ` ` ` ` `return` `$total` `; ` `} ` ` ` `// Driver code ` `$n` `= 3; ` `$k` `= 2; ` `echo` `countWays(` `$n` `, ` `$k` `) . ` `"\n"` `; ` ` ` `// This code is contributed by ita_c ` `?> ` |

*chevron_right*

*filter_none*

**Output:**

6

This article is contributed by **Aditi Sharma**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

## Recommended Posts:

- In-Place Algorithm
- Jump Pointer Algorithm
- LRU Approximation (Second Chance Algorithm)
- Floyd-Rivest Algorithm
- Bellman–Ford Algorithm | DP-23
- Floyd Warshall Algorithm | DP-16
- Distinct elements in subarray using Mo's Algorithm
- Reversal algorithm for array rotation
- Reversal algorithm for right rotation of an array
- Bellman Ford Algorithm (Simple Implementation)
- Welsh Powell Graph colouring Algorithm
- C Program for Reversal algorithm for array rotation
- Block swap algorithm for array rotation
- Maximum Subarray Sum using Divide and Conquer algorithm
- Java Program for Reversal algorithm for array rotation