Minimum Swaps To Make Sequences Increasing

Description

You are given two integer arrays of the same length nums1 and nums2. In one operation, you are allowed to swap nums1[i] with nums2[i].

• For example, if nums1 = [1,2,3,8], and nums2 = [5,6,7,4], you can swap the element at i = 3 to obtain nums1 = [1,2,3,4] and nums2 = [5,6,7,8].

Return the minimum number of needed operations to make nums1 and nums2 strictly increasing. The test cases are generated so that the given input always makes it possible.

An array arr is strictly increasing if and only if arr[0] < arr[1] < arr[2] < ... < arr[arr.length - 1].

Example 1:

Input: nums1 = [1,3,5,4], nums2 = [1,2,3,7] Output: 1 Explanation: Swap nums1[3] and nums2[3]. Then the sequences are: nums1 = [1, 3, 5, 7] and nums2 = [1, 2, 3, 4] which are both strictly increasing.

Example 2:

Input: nums1 = [0,3,5,8,9], nums2 = [2,1,4,6,9] Output: 1

Constraints:

• 2 <= nums1.length <= 105
• nums2.length == nums1.length
• 0 <= nums1[i], nums2[i] <= 2 * 105

Code

DP with memoization

Time Complexity: $O(n)$, Space Complexity: $O(n)$

Intuition 是：對於每個 index，都只有 swap or do not swap 兩種選擇。和 Shopping Offers 類似。

class Solution {
public:
    int minSwap(vector<int>& nums1, vector<int>& nums2) {
        int n = nums1.size();
        vector<vector<int>> memo(n + 1, vector<int>(2, -1));
        return solve(0, nums1, nums2, -1, -1, 0, memo);
    }
 
    int solve(int idx, vector<int>& nums1, vector<int>& nums2, int pre1, int pre2, int swapped, vector<vector<int>>& memo) {
        if(idx == nums1.size()) {
            return 0;
        }
 
        if(memo[idx][swapped] != -1) return memo[idx][swapped];
        
        int res = 100001;
 
        if(nums1[idx] > pre1 && nums2[idx] > pre2)
            res = min(res, solve(idx + 1, nums1, nums2, nums1[idx], nums2[idx], 0, memo));
 
        swap(nums1[idx], nums2[idx]);
        if(nums1[idx] > pre1 && nums2[idx] > pre2)
            res = min(res, 1 + solve(idx + 1, nums1, nums2, nums1[idx], nums2[idx], 1, memo));
 
        swap(nums1[idx], nums2[idx]);
        
        return memo[idx][swapped] = res;
    }
};

Source

• Minimum Swaps To Make Sequences Increasing - LeetCode