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We can use dynamic programming to solve this problem by maintaining a 3D array where dp[i][j][k] represents the number of ways to construct an array of length i, with values up to j and exactly k comparisons.
Recurrence relations can be defined by considering whether you want to extend the array by a value that does or doesn't increase the number of comparisons. Use modulo operations to manage large numbers.
Time Complexity: O(n * m^2 * k). Space Complexity: O(n * m * k).
1MOD = 10**9 + 7
2
3def numOfArrays(n, m, k):
4 dp = [[[0]*(k+1) for _ in range(m+1)] for _ in range(n+1)]
5 for j in range(1, m + 1):
6 dp[1][j][1] = 1
7 for i in range(2, n + 1):
8 for j in range(1, m + 1):
9 for z in range(1, k + 1):
10 dp[i][j][z] = (dp[i-1][j][z] * j) % MOD
11 for p in range(1, j):
12 dp[i][j][z] = (dp[i][j][z] + dp[i-1][p][z-1]) % MOD
13 return sum(dp[n][j][k] for j in range(1, m+1)) % MOD
14The initial step is to set the dp array for arrays of length 1 with exactly one comparison. It can also reach up to any value from 1 to m. As you build arrays, you either keep the maximum unchanged or increase it with a new maximum. The complexity arises from trying to keep a 3D table of results for all possible combinations of parameters and then aggregating those results.
This approach uses recursion with memoization to avoid recomputing the solutions for subproblems. The recursive function attempts to build the array incrementally by deciding, step by step, which value to add, and whether it will increase the maximum so far. By caching previously computed results, it reduces repeated calculations.
Time Complexity: O(n * m * k). Space Complexity: O(n * m * k) due to recursion stack and memoization storage.
1import java.util.*;
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3
The Java solution initializes a 3D memoization table filled with -1 (uncomputed states) and recursively defines valid states for the dynamic structure of the array. It traverses through not only possible values to append but also considers whether adding the value modifies the comparison metric used (k comparisons).