801 Minimum Swaps To Make Sequences Increasing

801. Minimum Swaps To Make Sequences Increasing

1. Question

We have two integer sequencesAandBof the same non-zero length.

We are allowed to swap elementsA[i]andB[i]. Note that both elements are in the same index position in their respective sequences.

At the end of some number of swaps,AandBare both strictly increasing. (A sequence is_strictly increasing_if and only ifA[0] < A[1] < A[2] < ... < A[A.length - 1].)

Given A and B, return the minimum number of swaps to make both sequences strictly increasing. It is guaranteed that the given input always makes it possible.

Example:
Input: A = [1,3,5,4], B = [1,2,3,7]

Output: 1

Explanation: 
Swap A[3] and B[3].  Then the sequences are:
A = [1, 3, 5, 7] and B = [1, 2, 3, 4]
which are both strictly increasing.

Note:

• A, Bare arrays with the same length, and that length will be in the range[1, 1000].

• A[i], B[i]are integer values in the range[0, 2000].

2. Implementation

(1) DP

思路: 对于数组上的每个位置我们只有两种操作: swap, no swap。所以我们建立一个2二维数组dp, dp[i][0]表示我们不在第i个位置swap, dp[i][1]表示在第i个位置swap

有以下几种情况影响我们的状态转移方程:

1.如果A[i - 1] < A[i] && B[i - 1] < B[i] && A[i - 1] < B[i] && B[i - 1] < A[i], 这个时候我们在第i个位置既可以swap也可以不swap

1. 如果A[i - 1] < A[i] && B[i - 1] < B[i] && (A[i - 1] > B[i] || B[i - 1] > A[i]), 这个时候我们在第i个位置的操作和第i - 1个位置的操作必须一致

2. 如果(A[i - 1] > A[i] || B[i - 1] > B[i]) && (A[i - 1] < B[i] && B[i - 1] < A[i]), 这个时候在第i个位置的操作和在第i - 1个位置的操作相反

class Solution {
    public int minSwap(int[] A, int[] B) {
        int n = A.length;
        int[][] dp = new int[n][2];
        // dp[i][0] means we don't swap at ith position
        // dp[i][1] means we swap at ith position
        dp[0][0] = 0;
        dp[0][1] = 1;

        for (int i = 1; i < n; i++) {
            // Case 1: In this case we can either swap or stay the same at ith poition
            if (A[i - 1] < A[i] && B[i - 1] < B[i] && A[i - 1] < B[i] && B[i - 1] < A[i]) {
                dp[i][0] = Math.min(dp[i - 1][0], dp[i - 1][1]);
                dp[i][1] = Math.min(dp[i - 1][0], dp[i - 1][1]) + 1;
            }
            // Case 2: If we swap at (i - 1)th position, we should swap at ith position
            // If we don't sway at (i - 1)th position, we keep the same operation at ith position
            else if (A[i - 1] < A[i] && B[i - 1] < B[i]) {
                dp[i][0] = dp[i - 1][0];
                dp[i][1] = dp[i - 1][1] + 1;
            }
            // Case 3: If we swap at (i - 1)th position, we don't swap at ith position
            // If we don't swap at (i - 1)th position, we swap at ith position
            else {
                dp[i][0] = dp[i - 1][1];
                dp[i][1] = dp[i - 1][0] + 1;
            }
        }
        return Math.min(dp[n - 1][0], dp[n - 1][1]);
    }
}

3. Time & Space Complexity

DP: 时间复杂度O(n), 空间复杂度O(n)