#1420

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Build Array Where You Can Find The Maximum Exactly K Comparisons

This problem asks to build an array where the maximum element is found using exactly K comparisons, with dynamic programming as the primary approach.

dynamic programming prefix sum

Problem Statement

You are given three integers n, m, and k. Your task is to build an array arr of length n where the maximum element appears in exactly k comparisons. The array contains only positive integers less than or equal to m.

Return the number of possible ways to build such an array. Since the result can be large, compute the answer modulo 10^9 + 7.

Examples

Example 1

Input: n = 2, m = 3, k = 1

Output: 6

The possible arrays are [1, 1], [2, 1], [2, 2], [3, 1], [3, 2] [3, 3]

Example 2

Input: n = 5, m = 2, k = 3

Output: 0

There are no possible arrays that satisfy the mentioned conditions.

Example 3

Input: n = 9, m = 1, k = 1

Output: 1

The only possible array is [1, 1, 1, 1, 1, 1, 1, 1, 1]

Constraints

1 <= n <= 50
1 <= m <= 100
0 <= k <= n

Solution Approach

Dynamic Programming Approach

We use a dynamic programming approach to solve this problem. We define a DP table dp[a][b][c] where a represents the length of the array, b is the maximum value in the array, and c represents the number of times the maximum element appears. The goal is to compute the number of valid arrays where the maximum element appears exactly k times.

State Transition and Recursion

The state transition is based on whether the current value being placed in the array is the maximum element or not. When the current element is the maximum value, we increase the count of comparisons with the maximum element. The recursive nature of this problem ensures that we build the solution iteratively while checking the constraints on comparisons.

Prefix Sum Optimization

To optimize the solution, we use prefix sum techniques to quickly calculate the number of valid configurations. By maintaining a running sum of previously computed values, we can avoid recomputing redundant states, which speeds up the solution and ensures that it runs within time limits.

Complexity Analysis

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

The time and space complexity depend on the approach taken for filling the DP table. With dynamic programming and prefix sum optimization, the time complexity is typically O(n * m * k), and space complexity depends on the DP table structure. The use of modulo ensures that the result fits within the required bounds.

What Interviewers Usually Probe

Can the candidate break down the dynamic programming transition states?
How well does the candidate optimize the solution using prefix sums?
Is the candidate aware of the need to use modulo operations to prevent overflow?

Common Pitfalls or Variants

Common pitfalls

Overlooking the constraints and not managing the dynamic programming transitions correctly.
Failing to use prefix sum optimization, leading to excessive computation time.
Not correctly handling the modulo operation, which can lead to overflow or incorrect results.

Follow-up variants

Change the value of k to test the candidate's ability to adapt the DP table to different constraints.
Vary the range of possible values for m to test how the solution scales.
Increase the size of the array n and ask for optimizations to handle larger inputs efficiently.

FAQ

How does dynamic programming help in solving the Build Array Where You Can Find The Maximum Exactly K Comparisons problem?

Dynamic programming helps by breaking down the problem into subproblems. It tracks possible states for array length, maximum element, and number of comparisons, ensuring an optimal solution.

What is the role of prefix sums in solving this problem?

Prefix sums optimize the DP solution by allowing efficient calculation of valid configurations, reducing redundant computations and improving runtime.

What is the time complexity of the Build Array Where You Can Find The Maximum Exactly K Comparisons problem?

The time complexity is O(n * m * k) due to the need to compute the DP table with multiple states for array length, maximum element, and comparisons.

How does the modulo operation affect the solution to this problem?

The modulo operation ensures that the solution stays within the bounds of 10^9 + 7, preventing overflow and ensuring correctness in large inputs.

What is the most common mistake when solving the Build Array Where You Can Find The Maximum Exactly K Comparisons problem?

A common mistake is not managing the DP transitions correctly or failing to apply optimizations like prefix sums, leading to inefficient solutions.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def numOfArrays(self, n: int, m: int, k: int) -> int:
        if k == 0:
            return 0
        dp = [[[0] * (m + 1) for _ in range(k + 1)] for _ in range(n + 1)]
        mod = 10**9 + 7
        for i in range(1, m + 1):
            dp[1][1][i] = 1
        for i in range(2, n + 1):
            for c in range(1, min(k + 1, i + 1)):
                for j in range(1, m + 1):
                    dp[i][c][j] = dp[i - 1][c][j] * j
                    for j0 in range(1, j):
                        dp[i][c][j] += dp[i - 1][c - 1][j0]
                        dp[i][c][j] %= mod
        ans = 0
        for i in range(1, m + 1):
            ans += dp[n][k][i]
            ans %= mod
        return ans

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