K Inverse Pairs Array

This problem requires us to find the number of different arrays consisting of numbers from (1) to (n) that have exactly (k) inverse pairs. An inverse pair is a pair of integers ([i, j]) where (0 \leq i < j < \text{{nums.length}}) and (\text{{nums[i]}} > \text{{nums[j]}}).

We can use dynamic programming to solve this problem. Let’s define a 2D dp array where dp[i][j] will represent the number of arrays of length (i) having (j) inverse pairs.

Here’s the code to solve this problem:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19

class Solution:
    def kInversePairs(self, n: int, k: int) -> int:
        MOD = 10**9 + 7
        dp = [[0] * (k + 1) for _ in range(n + 1)]

        # Base case: there's only one way to have 0 inverse pairs with 0 numbers
        dp[0][0] = 1

        for i in range(1, n + 1):
            # Prefix sum array to optimize the transition
            prefix_sum = [0] * (k + 1)
            for j in range(k + 1):
                prefix_sum[j] = (prefix_sum[j - 1] + dp[i - 1][j]) % MOD

            for j in range(k + 1):
                # Transition: calculate dp[i][j] based on previous values
                dp[i][j] = (prefix_sum[j] - (prefix_sum[j - i] if j - i >= 0 else 0) + MOD) % MOD

        return dp[n][k]

Explanation:

  • We iterate through i from (1) to (n) and j from (0) to (k).
  • We use a prefix sum array to optimize the transition, which helps in calculating the sum of the previous row within a range.
  • The transition is calculated by taking the sum of previous values and subtracting the sum of the prefix sum array up to (j - i).
  • The answer will be the value in dp[n][k], representing the number of arrays of length (n) having exactly (k) inverse pairs.

The code returns the result modulo (10^9 + 7) and meets the given constraints.

Identifying Problem Isomorphism

“K Inverse Pairs Array” has an approximate isomorphic problem “Count Different Palindromic Subsequences”.

In “K Inverse Pairs Array”, you have to find the number of permutations of 1 to n so that exactly k of the permutations are inverse pairs. The crux of the problem involves understanding and formulating the dynamic programming relation, which accounts for different possibilities.

“Count Different Palindromic Subsequences” also requires a dynamic programming approach. In this problem, you’re required to find the number of distinct, non-empty palindromic subsequences in a given string. The dynamic programming relation must account for palindromes at different positions in the string and avoid counting duplicates.

Both of these problems require a solid understanding of dynamic programming. The solutions involve formulating a dynamic programming relation that accounts for different possibilities based on previous computations.

“Count Different Palindromic Subsequences” is simpler as it works with strings, which could be more intuitive than working with permutations and inverse pairs as in the “K Inverse Pairs Array” problem.

10 Prerequisite LeetCode Problems

“629. K Inverse Pairs Array” deals with combinatorics, dynamic programming, and modular arithmetic. Here are some simpler problems to prepare for this:

  1. LeetCode 70. Climbing Stairs

    • Basic dynamic programming problem for understanding the idea of overlapping subproblems.

  2. LeetCode 96. Unique Binary Search Trees

    • Helps you understand the concept of combinatorics in dynamic programming.

  3. LeetCode 198. House Robber

    • Simple dynamic programming problem for practicing state transitions.

  4. LeetCode 300. Longest Increasing Subsequence

    • This problem uses dynamic programming to solve a sequence problem.

  5. LeetCode 377. Combination Sum IV

    • This problem is about counting combinations, similar to the “K Inverse Pairs Array” problem.

  6. LeetCode 509. Fibonacci Number

    • Basic problem for understanding the idea of overlapping subproblems and how to solve it using dynamic programming.

  7. LeetCode 516. Longest Palindromic Subsequence

    • This problem uses dynamic programming to solve a sequence problem, similar to the “K Inverse Pairs Array” problem.

  8. LeetCode 688. Knight Probability in Chessboard

    • This problem is about computing probabilities, which are similar to counting problems like “K Inverse Pairs Array”.

  9. LeetCode 935. Knight Dialer

    • This problem uses dynamic programming to count valid sequences, similar to the “K Inverse Pairs Array” problem.

  10. LeetCode 1269. Number of Ways to Stay in the Same Place After Some Steps

    • This problem is about counting possibilities using dynamic programming, which is similar to the “K Inverse Pairs Array” problem.

By solving these problems, you will gain a good understanding of combinatorics and dynamic programming which are required to tackle the “629. K Inverse Pairs Array” problem.

The problem “629. K Inverse Pairs Array” belongs to the domain of dynamic programming. It deals with the number of permutations of the numbers from 1 to n so that exactly k inverse pairs are formed. An inverse pair is a pair (i, j) such that i < j and a[i] > a[j].

Here are some problems to prepare for tackling this problem:

  1. Problem 62. Unique Paths Another simple DP problem. The unique number of paths to any cell in the grid is the sum of paths to its left cell and the cell above it.

  2. Problem 300. Longest Increasing Subsequence A problem that introduces the concept of subsequence and uses DP to find the longest increasing subsequence in an array.

  3. Problem 91. Decode Ways This problem requires counting the number of ways to decode a number string. It is a good exercise for DP problem solving.

  4. Problem 322. Coin Change Another good problem for understanding DP that requires finding the minimum number of coins that make a certain amount.

  5. Problem 279. Perfect Squares This DP problem requires finding the least number of perfect square numbers that sum to a given number n.

  6. Problem 120. Triangle This problem requires finding the minimum path sum from top to bottom in a triangle, a good exercise for DP problem-solving skills.

  7. Problem 72. Edit Distance This problem introduces you to a more advanced concept in DP, calculating the minimum number of operations to convert one string to another.

  8. Problem 221. Maximal Square This problem requires finding the side length of the largest square containing only 1’s in a binary matrix, it is a good 2D DP problem to practice.

These problems will help you understand the concept of dynamic programming and build the skills necessary to tackle more complex DP problems like the K Inverse Pairs Array. Start with the simpler problems and gradually take on the more complex ones as you gain more confidence in your DP problem-solving skills.

For an integer array nums, an inverse pair is a pair of integers [i, j] where 0 <= i < j < nums.length and nums[i] > nums[j].

Given two integers n and k, return the number of different arrays consist of numbers from 1 to n such that there are exactly k inverse pairs. Since the answer can be huge, return it modulo 109 + 7.

Example 1:

Input: n = 3, k = 0 Output: 1 Explanation: Only the array [1,2,3] which consists of numbers from 1 to 3 has exactly 0 inverse pairs. Example 2:

Input: n = 3, k = 1 Output: 2 Explanation: The array [1,3,2] and [2,1,3] have exactly 1 inverse pair.

Constraints:

1 <= n <= 1000 0 <= k <= 1000

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

class Solution(object):
    def kInversePairs(self, n, k):
        """
        :type n: int
        :type k: int
        :rtype: int
        """
        dp = [[0] * (k+1) for _ in range(n+1)]
        dp[0][0] = 1

        for i in range(1, n+1):
            cumsum = 0
            for j in range(k+1):
                if j == 0:
                    dp[i][j] = 1
                    cumsum += 1
                else:
                    cumsum += dp[i-1][j]
                    if j-i >= 0:
                        cumsum -= dp[i-1][j-i]
                    dp[i][j] = cumsum % 1000000007
    
        return dp[n][k]

Problem Classification

The problem belongs to the domain of “Combinatorics and Dynamic Programming”.

‘What’ Components:

  1. Input - Two integers n and k.
  2. Output - The number of different arrays of length n, consisting of numbers from 1 to n, that have exactly k inverse pairs, modulo 10^9 + 7.
  3. Condition - An inverse pair is a pair of integers [i, j] where 0 <= i < j < nums.length and nums[i] > nums[j].

This is a “Counting and Arrangement” problem where we need to count the number of valid arrangements that satisfy the given conditions. It involves concepts from Combinatorics, as we are counting permutations of arrays. However, the direct application of combinatorial principles is not feasible due to the complexity of the problem (ensuring exactly k inverse pairs). Therefore, Dynamic Programming is also involved, as we need to build the solution incrementally, keeping track of previous computations.

Language Agnostic Coding Drills

  1. Understanding 2D arrays: The problem utilizes a 2D array (or a list of lists in Python), so the first concept to understand would be how to create, access, and manipulate 2D arrays.

  2. Nested Loops: The code makes use of nested loops, i.e., a loop inside another loop. Understanding the control flow and execution order within nested loops is essential.

  3. Modulus Arithmetic: The problem utilizes modulus arithmetic, specifically “modulo 1e9 + 7”, which is a common technique used in competitive programming to prevent integer overflow for large numbers.

  4. Cumulative Sum: Cumulative sum or prefix sum is a key technique used in this problem. The concept is to build an array where each element is the sum of the elements before it, which can help in quickly calculating the sum of any range of array elements.

  5. Conditional Statements: Understand how conditional statements like ‘if-else’ work. In this code, it is used to check if the index is out of range, and also to set the base case for dynamic programming.

  6. Dynamic Programming: The problem is solved using a dynamic programming approach, which involves breaking the problem down into smaller sub-problems, solving each sub-problem only once, and storing their results in case next time the same sub-problem occurs.

After understanding and implementing these drills separately, they can be combined to understand the solution to the problem.

The problem-solving approach starts by initializing a 2D array, dp, where dp[i][j] represents the number of permutations of the numbers from 1 to i with j inverse pairs. The outer loop iterates through 1 to n, and the inner loop iterates through 0 to k. For each pair (i, j), we update dp[i][j] based on dp[i-1][j] and dp[i-1][j-i], ensuring that the index j-i is not out of bounds. The result is stored in dp[n][k], and returned at the end.

Targeted Drills in Python

  1. Understanding 2D arrays

    Task: Create a 2D list, assign some values, and access these values.

    1
2

    matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
print(matrix[0][0])  # This should print '1'

  2. Nested Loops

    Task: Print a matrix using nested loops.

    1
2
3
4
5

    matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
for row in matrix:
    for element in row:
        print(element, end=' ')
    print()

  3. Modulus Arithmetic

    Task: Perform arithmetic operations and return the result mod 1000000007.

    1
2
3
4

    a = 1234567890
b = 9876543210
result = (a + b) % 1000000007
print(result)

  4. Cumulative Sum

    Task: Compute a cumulative sum array from a given list.

    1
2
3

    numbers = [1, 2, 3, 4, 5]
cumsum = [sum(numbers[:i+1]) for i in range(len(numbers))]
print(cumsum)  # Should print [1, 3, 6, 10, 15]

  5. Conditional Statements

    Task: Use if-else statements to print whether a number is even or odd.

    1
2
3
4
5

    num = 10
if num % 2 == 0:
    print("Even")
else:
    print("Odd")

  6. Dynamic Programming

    Task: Write a dynamic programming solution to find the nth Fibonacci number.

    1
2
3
4
5
6
7

    def fib(n):
    dp = [0, 1] + [0]*(n-1)
    for i in range(2, n+1):
        dp[i] = dp[i-1] + dp[i-2]
    return dp[n]

print(fib(10))  # Should print 55

After understanding and practicing these drills individually, you can use these concepts to understand and implement the solution to the problem.