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This approach uses a Binary Indexed Tree (BIT) data structure to efficiently count the number of elements less than and greater than the current element during the insertion process.
We maintain a BIT to store the frequency of each element from the instructions array. For each element, we calculate the cost as the minimum of the count of elements that are less than it and the count of elements that are greater than it that have been inserted so far. We use BIT to efficiently calculate prefix sums, which helps in calculating these counts.
Time Complexity: O(n log n), where n is the number of instructions. The operations of update and query on the BIT are O(log n).
Space Complexity: O(n), to store the BIT.
1class CreateSortedArray:
2 def __init__(self):
3 self.BIT = [0] * 100001
4 self.MOD = 10**9
The Python class CreateSortedArray uses a method-based approach to define the BIT functionality. The class maintains an array for BIT and uses methods to update and query counts. This helps to compute the desired cost efficiently.
This approach employs a Segment Tree to perform range query operations to efficiently count the number of elements less than and greater than the current element during insertion. The Segment Tree helps maintain the frequency of elements within a range and can provide range sum in logarithmic time, allowing for quick calculations of the insertion cost.
Time Complexity: O(n log n), where n is the number of instructions. The Segment Tree operations take O(log n).
Space Complexity: O(n), to store the Segment Tree.
The C implementation uses a Segment Tree to efficiently count elements in a range for insertion cost calculation. The update function alters the Segment Tree to reflect changes in element frequency, while query retrieves summations of elements over a range.