K Inverse Pairs Array

K Inverse Pairs Array

1. What is this problem about?

The K Inverse Pairs Array interview question is a difficult combinatorial problem. An inverse pair in a permutation p is a pair $(i, j)$ such that $i < j$ and $p[i] > p[j]$. You are given two integers $n$ and $k$, and you need to find the total number of permutations of $1 \dots n$ that have exactly $k$ inverse pairs.

2. Why is this asked in interviews?

Tech giants like Microsoft and Google ask the K Inverse Pairs coding problem to test a candidate's mastery of Dynamic Programming and optimization. It requires deriving a complex recurrence relation and then optimizing it using Prefix Sums to avoid $O(N \cdot K^2)$ complexity. It’s a top-tier Math interview pattern.

3. Algorithmic pattern used

This problem is solved using DP with Prefix Sum Optimization.

1. DP State: dp[i][j] is the number of permutations of length i with j inverse pairs.
2. Transition: When adding the $i^{th}$ number to a permutation of size $i-1$, you can create between 0 and $i-1$ new inverse pairs.
   • dp[i][j] = sum(dp[i-1][j-m]) for $0 \leq m < i$.
3. Optimization: The sum is over a range of the previous row. Use a prefix sum of the previous row to calculate this in $O(1)$.
4. Complexity: $O(N \cdot K)$ time and $O(K)$ space.

4. Example explanation

$n = 3, k = 1$.

• dp[2][0] = 1 ([1, 2]), dp[2][1] = 1 ([2, 1]).
• To get 1 pair in $n=3$:
  • Add '3' at the end (0 new pairs): need 1 pair from $n=2$. dp[2][1] = 1.
  • Add '3' in the middle (1 new pair): need 0 pairs from $n=2$. dp[2][0] = 1. Total: 2. ([1, 3, 2] and [2, 1, 3]).

5. Common mistakes candidates make

• $O(N \cdot K^2)$ approach: Failing to use prefix sums for the range addition, leading to a Time Limit Exceeded.
• Modulo errors: Forgetting to handle negative results when using prefix sum subtraction: (S[j] - S[j-i] + MOD) % MOD.
• Space Complexity: Using a full 2D table when only the previous row is needed.

6. Interview preparation tip

Combinatorial DP problems often involve "transitions based on position." For permutations, adding an element at position $X$ always adds $N - 1 - X$ inversions. This visualization is the key to many "Hard" Dynamic Programming interview patterns.