There is a fence with n posts, each post can be painted with one of the k colors.

You have to paint all the posts such that no more than two adjacent fence posts have the same color.

Return the total number of ways you can paint the fence.

Note:

n and k are non-negative integers.

n and k are non-negative integers.

**Understand the problem:**

We can use a DP solution.

-- Define two DP arrays, diff[n] and same[i]. Where diff[i] means the number of ways for the fence i which has different color with fence i -1. same[i] means the number of ways if fence i has the same color with fence i - 1.

-- Initialization same[0] = 0, diff[0] = k.

-- same[i] = diff[i - 1].

-- diff[i] = (k - 1) * (same[i - 1] + diff[i - 1]);

-- Final state: same[n - 1] + diff[n - 1].

**Code (Java):**

public class Solution { public int numWays(int n, int k) { if (n <= 0 || k <= 0) { return 0; } if (n == 1) { return k; } // i -1 and i -2 with the same color int[] dp1 = new int[n]; // i - 1 and i - 2 with diff. color int[] dp2 = new int[n]; // Initializaiton dp1[0] = 0; dp2[0] = k; for (int i = 1; i < n; i++) { dp1[i] = dp2[i - 1]; dp2[i] = (k - 1) * (dp1[i - 1] + dp2[i - 1]); } // Final state return dp1[n - 1] + dp2[n - 1]; } }

**A Constant Space solution:**

public class Solution { public int numWays(int n, int k) { if (n <= 0 || k <= 0) { return 0; } if (n == 1) { return k; } int preSame = 0; int preDiff = k; for (int i = 1; i < n; i++) { int same = preDiff; int diff = (k - 1) * (preSame + preDiff); preSame = same; preDiff = diff; } return preSame + preDiff; } }

Two possibilities for DP[i]

ReplyDelete1. EndUpHavingSameColorAsPrevious

- Which means, the last two should be different colors.

- Number of ways last two can have different colors, makes it eligible for having same color.

- Number of ways last two can have different colors = "EndUpHavingDifferentColorAsPrevious"

- Since we choose same color as previous, then the number of possibilities is just carried over.

- EndUpHavingSameColorAsPrevious[i] = EndUpHavingDifferentColorAsPrevious[i-1]

2. EndUpHavingDifferentColorAsPrevious

- Case - 1 : The last two can have same colors - and we choose K-1 other colors for each. .. ie, EndUpHavingSameColorAsPrevious[i-1] * (K-1)

- Case - 2 : The last two can have different colors - and we choose K-1 colors for each.

- EndUpHavingDifferentColorAsPrevious[i-1] * (K-1)

- EndUpHavingDifferentColorAsPrevious[i] = above two cases;

TotalDP[i] = EndUpHavingSameColorAsPrevious[i] + EndUpHavingDifferentColorAsPrevious[i];

Thanks!,

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