Is Constructing Two Increasing Arrays easy or hard?

Constructing Two Increasing Arrays is classified as a Hard problem because it requires careful dynamic programming state design and transition optimization. Recognizing that valid previous states form a continuous range and applying prefix sums to reduce complexity is the key challenge.

Constructing Two Increasing Arrays Python/Java solution

The typical implementation builds a DP array representing valid values for a[i]. For each index, iterate through possible splits, enforce increasing constraints, and use prefix sums to accumulate counts efficiently. The same logic translates cleanly across Python, Java, C++, Go, and TypeScript using arrays and modular arithmetic.

How to solve Constructing Two Increasing Arrays in O(n)?

Strict O(n) is generally not achievable because each index may require evaluating multiple candidate values for the split. The practical optimal solution is O(n · m) using dynamic programming and prefix sums, where m is the maximum allowed value for a[i]. Prefix sums eliminate nested transitions and keep each state update constant time.

What is the best approach for Constructing Two Increasing Arrays?

The best approach uses dynamic programming with prefix sum optimization. Define a DP state for the value chosen in the first array at each index and derive the second array value from nums[i] - a[i]. Using prefix sums allows you to aggregate valid previous states efficiently, reducing the transition cost and achieving O(n · m) time complexity.

Is Constructing Two Increasing Arrays asked at Google/Amazon/Meta?

Problems involving counting valid constructions with monotonic constraints commonly appear in interviews at companies like Google, Amazon, and Meta. Variants that combine array constraints with dynamic programming transitions are particularly common in senior-level algorithm rounds.

What data structure is used in Constructing Two Increasing Arrays?

The main technique is dynamic programming implemented with arrays to store DP states. Prefix sum arrays are used to speed up transitions by quickly summing ranges of valid previous states. The algorithm primarily relies on array iteration and DP state transitions.

What is the time complexity of Constructing Two Increasing Arrays?

The optimized dynamic programming solution runs in O(n · m) time, where n is the length of nums and m is the maximum possible value considered for a[i]. Space complexity is O(m) by storing only the previous DP row. A naive DP approach without prefix sums can take O(n · m^2).

Description Hints 2 Solution

3269. Constructing Two Increasing Arrays

Hard61.8% acceptancePremium elsewhere, free here

Given 2 integer arrays nums1 and nums2 consisting only of 0 and 1, your task is to calculate the minimum possible largest number in arrays nums1 and nums2, after doing the following.

Replace every 0 with an even positive integer and every 1 with an odd positive integer. After replacement, both arrays should be increasing and each integer should be used at most once.

Return the minimum possible largest number after applying the changes.

Example 1:

Input: nums1 = [], nums2 = [1,0,1,1]

Output: 5

Explanation:

After replacing, nums1 = [], and nums2 = [1, 2, 3, 5].

Example 2:

Input: nums1 = [0,1,0,1], nums2 = [1,0,0,1]

Output: 9

Explanation:

One way to replace, having 9 as the largest element is nums1 = [2, 3, 8, 9], and nums2 = [1, 4, 6, 7].

Example 3:

Input: nums1 = [0,1,0,0,1], nums2 = [0,0,0,1]

Output: 13

Explanation:

One way to replace, having 13 as the largest element is nums1 = [2, 3, 4, 6, 7], and nums2 = [8, 10, 12, 13].

Constraints:

0 <= nums1.length <= 1000
1 <= nums2.length <= 1000
nums1 and nums2 consist only of 0 and 1.

Topics

Array Dynamic Programming

Hints open one at a time

Hint 1

Think of dynamic programming.

Hint 2

Define dp[i][j][flag] as the answer to the problem if we only consider the first i elements of the first array, the first j elements of the second array, and flag is either 0 or 1 indicating which array contains the largest element.

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Examples

Case 1Case 2Case 3

Input

[]
[1,0,1,1]

Input

[0,1,0,1]
[1,0,0,1]

Input

[0,1,0,0,1]
[0,0,0,1]

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