Divide Array Into Increasing Sequences

Divide Array Into Increasing Sequences

What is this problem about?

The Divide Array Into Increasing Sequences coding problem asks you to partition a sorted array of positive integers into one or more "strictly increasing" sequences, each having a length of at least k. You need to determine if such a partition is possible. Since the array is already sorted, the challenge is dealing with duplicate elements that must be placed into different sequences.

Why is this asked in interviews?

Google uses this math and array interview pattern to test a candidate's ability to identify the "bottleneck" in a problem. The core observation is that no two identical elements can be in the same strictly increasing sequence. Therefore, the number of sequences you need is at least the frequency of the most common element. It evaluates if you can translate a complex partitioning requirement into a simple frequency constraint.

Algorithmic pattern used

The problem is solved using Frequency Counting and Constraint Analysis.

1. Find Max Frequency: Let maxFreq be the maximum number of times any element appears in the array.
2. Calculate Minimum Elements Needed: To satisfy maxFreq sequences each with length k, you need at least maxFreq * k total elements.
3. Check: If n >= maxFreq * k, then it is mathematically possible to form the sequences. Otherwise, it is not.

Example explanation

Array: [1, 2, 2, 3, 3, 4], $k = 3$.

1. Max frequency is 2 (two 2s, two 3s). So we need at least 2 sequences.
2. Total elements needed: $2 \times 3 = 6$.
3. Total elements available: 6. Result: True. (Sequences: [1, 2, 3] and [2, 3, 4]).

Array: [5, 5, 5, 5], $k = 2$.

1. Max frequency is 4. Need 4 sequences.
2. Total elements needed: $4 \times 2 = 8$.
3. Available: 4. Result: False.

Common mistakes candidates make

• Greedy Simulation: Trying to actually construct the sequences, which is $O(n)$ but requires much more complex code than the simple frequency check.
• Ignoring the Sorted Property: Not realizing that since the input is sorted, you can find maxFreq in a single pass without a Hash Map.
• Miscalculating the bottleneck: Thinking the total sum or the range of numbers matters more than the frequency of duplicates.

Interview preparation tip

For partitioning problems, always look for the "Pigeonhole Principle." If you have $X$ copies of a number, you must have at least $X$ different sets to put them in. This observation often reduces the problem to a simple $O(n)$ check.