Problem Description

The problem presents an integer array nums and a positive integer k. We have to determine the number of indices in the array that satisfy a certain "k-big" property. An index i is considered "k-big" if there are at least k elements that are less than nums[i] to the left of i and also at least k elements that are less than nums[i] to the right of i. The task is to calculate the total number of such "k-big" indices.

Intuition

The key to solving this problem efficiently lies in being able to quickly determine how many elements are smaller than a given element to its left and right. A naive approach might try to compare each element to all others, resulting in an impractical time complexity.

To optimize this, we can use a Binary Indexed Tree (BIT), also known as a Fenwick tree. This data structure allows us to efficiently compute prefix sums and update values, which is exactly what we need to keep track of the count of elements smaller than the current one.

The solution approach involves two Binary Indexed Trees:

• tree1 tracks the number of smaller elements to the left of each index.
• tree2 tracks the number of smaller elements to the right of each index.

We initially build tree2 with the counts of all elements. As we traverse the elements in nums, we update tree2 to remove the count of the current element, essentially keeping the count of elements to the right. We then check if the current index satisfies the "k-big" condition using both tree1 and tree2. If so, we increment our answer. Simultaneously, we update tree1 to reflect this element for the counts of the subsequent indices.

This way, as we iterate over the array, we accumulate the "k-big" indices by efficiently querying and updating the prefix sums using the Binary Indexed Trees.

Learn more about Segment Tree, Binary Search, Divide and Conquer and Merge Sort patterns.

Solution Approach

The solution applies a Binary Indexed Tree to manage the computation of the counts of elements smaller than a certain value to the left and right of a given index efficiently.

Here are the steps of the implementation:

Binary Indexed Tree (BIT) Structure:

• A class BinaryIndexedTree is defined to encapsulate the BIT operations such as update and query. Each instance of this class maintains an internal array c, where n+1 is the size of the given nums array.

Update Operation:

• In the update method of BIT, we add a value delta to a specific index x and then update all subsequent elements in the array c that are responsible for that index. This is done by iterating through the array using the bitwise operation x += x & -x.

Query Operation:

• The query method calculates the prefix sum up to a given index x. It iterates through the array c while decrementing the index using the bit manipulation x -= x & -x until x is zero. The result is the sum of elements for the range [1, x].

Solution Class and Method:

• In the Solution class, the method kBigIndices implements the main logic.
• We initialize two instances of the BinaryIndexedTree: tree1 for counting elements to the left and tree2 for counting elements to the right.

Building tree2:

• The full tree2 is built with the input array by incrementing the count for each value found in nums.

Traverse the Array:

• As we iterate over the elements in nums, for each element v:
  • We first decrement the count in tree2 for the current element since we want to exclude it from the right count.
  • Then we check if the element v is "k-big" using both tree1 and tree2:
    • tree1.query(v - 1) checks for at least k elements less than v to the left.
    • tree2.query(v - 1) checks for at least k elements less than v to the right.
  • If both conditions are true, we increment our answer (ans).
  • We update tree1 to include the current element in the left count for future iterations by using tree1.update(v, 1).

Following these steps ensures an efficient computation of the "k-big" indices, with time complexity that allows the problem to be solved in practical scenarios, even for larger input arrays.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have the following integer array nums and integer k:

nums = [3, 5, 1, 2, 4]
k = 2

With this input, we want to find the number of indices where the "k-big" property is satisfied, meaning there are at least 2 elements smaller to both the left and right of that index. Here's how the solution works step by step:

Step 1: Initialize two Binary Indexed Trees (BITs)

• We will have tree1 to track counts to the left and tree2 to track counts to the right.

Step 2: Building tree2

• Before we start checking each element, build tree2 with the count of all elements in nums. After populating tree2, it would look something like this (assuming 1-based indexing for the tree and also assuming nums elements as their indices):

tree2 based on values: [0, 1, 1, 1, 1, 1]

Step 3: Traverse the Array and Perform Operations

• Now, we iterate through each element in nums.

For index 0 (nums[0] = 3):

• Update tree2 by decrementing the count of the value 3.
• Check if there are at least k (2) elements less than 3 to the left using tree1.query(2) (which would return 0 right now).
• Check if there are at least k (2) elements less than 3 to the right using tree2.query(2) (which returns 2, since the values 1 and 2 are less than 3).
• Condition for "k-big" is not satisfied because we don't have enough elements on the left.
• Update tree1 with the current value (3) by using tree1.update(3, 1).

For index 1 (nums[1] = 5):

• Update tree2 by decrementing the count for the value 5.
• tree1.query(4) checks elements less than 5 to the left, returns 1 (only 3 is less than 5).
• tree2.query(4) checks elements less than 5 to the right, returns 2 (since 1 and 2 are less than 5).
• "k-big" condition is not satisfied for the same reason as above.
• Update tree1 with the current value (5) by using tree1.update(5, 1).

For index 2 (nums[2] = 1):

• Same procedures as above, but there will be no updates since it's clear there are no elements less than 1 to the left or right.

For index 3 (nums[3] = 2):

• Update tree2, decrementing the count of value 2.
• Use tree1.query(1) to check the left side which returns 1, not meeting the "k-big".
• Use tree2.query(1) to check the right side which returns 1 as well.
• "k-big" condition is not met.
• Update tree1 with the current value (2).

For index 4 (nums[4] = 4):

• Update tree2, decrementing the count of value 4.
• Use tree1.query(3) to find if there are at least 2 numbers that are < 4 on the left; it returns 2 (since 3 and 2 are less than 4).
• Use tree2.query(3) to find if there are at least 2 numbers that are < 4 on the right; it returns 0 (since we are at the end, no elements to the right).
• "k-big" condition is not met.

After we have finished iterating through the elements in nums, we come to realize that none of the indices satisfy the "k-big" property for k = 2. Therefore, our answer would be 0.

This example walked through each element and showed how we check the "k-big" property using the two Binary Indexed Trees. In a larger input array, the efficiency of the BIT operations for update and query becomes crucial, making the solution scalable.

Solution Implementation

Time and Space Complexity

The provided code uses two Binary Indexed Trees (BITs) to store and update the frequency of elements in nums. The method update will be called n times for each tree, and each call takes O(log n) time as it iterates through the tree updating nodes. The query method is also called n times for each tree within the for loop, and it likewise takes O(log n) time to aggregate values from the tree.

Given that we have two BITs and each operation (update and query) is O(log n), the overall time complexity is O(n log n), where n is the number of elements in nums.

For space complexity, each tree has an array of size n + 1 to store cumulative frequencies, which results in O(n) space complexity. Since there are two such arrays, the total space used by the data structures is O(2n) which simplifies to O(n).

The code will trigger these operations:

• update(x, delta) is called 2n times, resulting in O(2n log n) operations.
• query(x) is called 2n times, also resulting in O(2n log n) operations.

When combined and simplified, the computational complexity remains O(n log n) for time and O(n) for space. The reference answer aligns with this analysis.

Learn more about how to find time and space complexity quickly using problem constraints.