Build Array Where You Can Find The Maximum Exactly K Comparisons

Build Array Where You Can Find The Maximum Exactly K Comparisons

What is this problem about?

The "Build Array Where You Can Find The Maximum Exactly K Comparisons interview question" is a complex combinatorial problem. You are given three integers: $n$ (array length), $m$ (maximum value allowed), and $k$ (search cost). You need to count how many arrays of length $n$ can be built using numbers from $1$ to $m$ such that applying a specific "find maximum" algorithm results in exactly $k$ updates to the maximum value.

Why is this asked in interviews?

Google and Adobe ask the "Build Array Where You Can Find The Maximum Exactly K Comparisons coding problem" to test a candidate's mastery of Dynamic Programming. It is a high-level problem that requires defining a multi-dimensional state and optimizing transitions. It evaluates the ability to translate an algorithmic process (finding a maximum) into a mathematical counting problem.

Algorithmic pattern used

This problem follows the Dynamic Programming and Prefix Sum patterns.

1. State Definition: dp[i][j][l] represents the number of arrays of length i, where the current maximum is j, and the search cost is l.
2. Transitions:
   • No update to max: The next element is $\leq j$. There are $j$ such choices.
   • Update to max: The next element $x$ is $> j$. This increases the cost to $l+1$.
3. Base Case: $i=1$, cost $l=1$ for any $j \in [1, m]$.
4. Optimization: Use prefix sums to calculate the "Update to max" transitions in $O(1)$ time, reducing the overall complexity.

Example explanation

$n=2, m=3, k=1$ Arrays of length 2 where max is updated once:

• The first element is the max. The second element must be $\leq$ the first.
• Possible: [1,1], [2,1], [2,2], [3,1], [3,2], [3,3]. Total = 6 arrays. The search cost is 1 because the first element sets the initial max, and it never changes.

Common mistakes candidates make

• Wrong state dimensions: Forgetting to include the "current maximum" in the DP state.
• Redundant loops: Calculating the sum of previous DP states in every step instead of using prefix sums, leading to a Time Limit Exceeded (TLE).
• Modulo errors: Forgetting to apply modulo $10^9 + 7$ at every addition.

Interview preparation tip

For counting problems with multiple constraints, always start by defining the smallest sub-problem. This "Dynamic Programming interview pattern" is essential for solving "Hard" category questions.