#801

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Swaps To Make Sequences Increasing

This problem involves finding the minimum number of swaps needed to make two sequences strictly increasing using dynamic programming.

array dynamic programming

Problem Statement

Given two integer arrays, nums1 and nums2, of the same length, you can swap any corresponding elements nums1[i] and nums2[i] to make both sequences strictly increasing. The goal is to find the minimum number of operations required to achieve this.

A strictly increasing sequence means that each element must be smaller than the one that follows it. Your task is to determine the minimum number of swaps needed to ensure that both nums1 and nums2 become strictly increasing by performing the fewest swaps.

Examples

Example 1

Input: nums1 = [1,3,5,4], nums2 = [1,2,3,7]

Output: 1

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

Example details omitted.

Constraints

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

Solution Approach

State Transition Dynamic Programming

The problem can be approached using dynamic programming, where the states represent whether an element in nums1 or nums2 has been swapped at each position. The key idea is to track the minimum number of swaps needed to maintain the strictly increasing order as you iterate through the arrays.

Transition Between States

For each index, consider both possibilities: keeping the elements in their current positions or swapping them. Update the dynamic programming state based on these transitions, considering the minimum swaps required to maintain increasing sequences.

Final Optimal Result

After processing all elements, the final answer will be the minimum swaps needed to achieve a valid strictly increasing sequence for both arrays, considering the best possible transitions through the dynamic states.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the specific dynamic programming implementation. If a solution involves iterating over the arrays multiple times with a small constant factor, the time complexity is typically O(n), where n is the length of the arrays. The space complexity is also O(n) to store the dynamic programming states.

What Interviewers Usually Probe

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

Common Pitfalls or Variants

Common pitfalls

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

Follow-up variants

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.
Consideration of other types of operations besides swapping, such as sorting or shifting elements, which would change the dynamic programming approach.
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.

FAQ

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.

terminal

Solution

Solution 1: Dynamic Programming

Define $a$ and $b$ to represent the minimum number of swaps needed to make the element sequences strictly increasing up to index $[0..i]$, with the $i$-th element not swapped and swapped, respectively. The index starts from $0$.

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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)

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