How do I solve Minimum Swaps To Make Sequences Increasing?

Use dynamic programming to track the minimum number of swaps needed at each index while ensuring both arrays remain strictly increasing.

What is the dynamic programming approach for this problem?

The dynamic programming approach involves maintaining states representing whether an element has been swapped, and transitioning between states based on maintaining the increasing order of both arrays.

How do state transitions work in Minimum Swaps To Make Sequences Increasing?

State transitions involve considering both keeping the elements as is and swapping them, then updating the state based on which option minimizes the swap count while preserving increasing order.

What are the common pitfalls to avoid in this problem?

Avoid neglecting valid state transitions, overlooking boundary conditions, and misunderstanding the problem's constraints when implementing the solution.

How can GhostInterview help me with this problem?

GhostInterview helps by providing detailed guidance on dynamic programming techniques, optimizing the solution for time and space complexity, and highlighting failure modes to prevent mistakes.

#801

Hard

auto_awesome状态·转移·动态规划

LeetCode 题解工作台

使序列递增的最小交换次数

我们有两个长度相等且不为空的整型数组 nums1 和 nums2 。在一次操作中，我们可以交换 nums1[i] 和 nums2[i] 的元素。例如，如果 nums1 = [1,2,3, 8 ] ， nums2 =[5,6,7, 4 ] ，你可以交换 i = 3 处的元素，得到 nums1 =[1…

数组动态规划

题目描述

我们有两个长度相等且不为空的整型数组 nums1 和 nums2 。在一次操作中，我们可以交换 nums1[i] 和 nums2[i]的元素。

例如，如果 nums1 = [1,2,3,8] ， nums2 =[5,6,7,4] ，你可以交换 i = 3 处的元素，得到 nums1 =[1,2,3,4] 和 nums2 =[5,6,7,8] 。

返回 使 nums1 和 nums2 严格递增 所需操作的最小次数 。

数组 arr 严格递增 且 arr[0] < arr[1] < arr[2] < ... < arr[arr.length - 1] 。

注意：

用例保证可以实现操作。

示例 1:

输入: nums1 = [1,3,5,4], nums2 = [1,2,3,7]
输出: 1
解释: 
交换 A[3] 和 B[3] 后，两个数组如下:
A = [1, 3, 5, 7] ， B = [1, 2, 3, 4]
两个数组均为严格递增的。

示例 2:

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

提示:

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

lightbulb

解题思路

方法一：动态规划

定义 $a$ , $b$ 分别表示使得下标 $[0..i]$ 的元素序列严格递增，且第 $i$ 个元素不交换、交换的最小交换次数。下标从 $0$ 开始。

当 $i=0$ 时，有 $a = 0$ , $b=1$ 。

当 $i\gt 0$ 时，我们先将此前 $a$ , $b$ 的值保存在 $x$ , $y$ 中，然后分情况讨论：

如果 $nums1[i - 1] \ge nums1[i]$ 或者 $nums2[i - 1] \ge nums2[i]$ ，为了使得两个序列均严格递增，下标 $i-1$ 和 $i$ 对应的元素的相对位置必须发生变化。也就是说，如果前一个位置交换了，那么当前位置不交换，因此有 $a = y$ ；如果前一个位置没有交换，那么当前位置必须交换，因此有 $b = x + 1$ 。

否则，下标 $i-1$ 和 $i$ 对应的元素的相对位置可以不发生变化，那么有 $b = y + 1$ 。另外，如果满足 $nums1[i - 1] \lt nums2[i]$ 并且 $nums2[i - 1] \lt nums1[i]$ ，那么下标 $i-1$ 和 $i$ 对应的元素的相对位置可以发生变化，此时 $a$ 和 $b$ 可以取较小值，因此有 $a = \min(a, y)$ 和 $b = \min(b, x + 1)$ 。

最后，返回 $a$ 和 $b$ 中较小值即可。

时间复杂度 $O(n)$ ，空间复杂度 $O(1)$ 。

class Solution:
    def minSwap(self, nums1: List[int], nums2: List[int]) -> int:
        a, b = 0, 1
        for i in range(1, len(nums1)):
            x, y = a, b
            if nums1[i - 1] >= nums1[i] or nums2[i - 1] >= nums2[i]:
                a, b = y, x + 1
            else:
                b = y + 1
                if nums1[i - 1] < nums2[i] and nums2[i - 1] < nums1[i]:
                    a, b = min(a, y), min(b, x + 1)
        return min(a, b)

speed

复杂度分析

指标	值
时间	Depends on the final approach
空间	Depends on the final approach

psychology

面试官常问的追问

外企场景

question_mark
Focus on the candidate's ability to implement dynamic programming solutions and handle state transitions efficiently.
question_mark
Watch for how well the candidate optimizes space and time complexity in dynamic programming approaches.
question_mark
Evaluate if the candidate can clearly explain how their solution ensures that both sequences are strictly increasing after the minimum number of swaps.

warning

常见陷阱

外企场景

error
Not considering all valid transitions between states, leading to incorrect or suboptimal solutions.
error
Forgetting to account for boundary conditions, such as checking the relationship between consecutive elements in both sequences.
error
Misunderstanding the problem's constraints, such as assuming swapping elements at certain indices is always beneficial without verifying the effect on subsequent positions.

swap_horiz

进阶变体

外企场景

arrow_right_alt
What if the problem allows for more than two arrays to be considered? This would introduce complexity in managing multiple sequences and maintaining the increasing order.
arrow_right_alt
Consideration of other types of operations besides swapping, such as sorting or shifting elements, which would change the dynamic programming approach.
arrow_right_alt
What if the sequence lengths are significantly larger, such as reaching up to 10^6 elements? This would require optimizing both time and space complexity to handle large inputs.

help

常见问题

外企场景

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