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* Create sparse_table.py * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Descriptive names for variables * Fix ruff check error * Update sparse_table.py * Add comments, change variable names * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Fix typo * Update sparse_table.py --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com> Co-authored-by: Christian Clauss <cclauss@me.com>
95 lines
3.3 KiB
Python
95 lines
3.3 KiB
Python
"""
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Sparse table is a data structure that allows answering range queries on
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a static number list, i.e. the elements do not change throughout all the queries.
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The implementation below will solve the problem of Range Minimum Query:
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Finding the minimum value of a subset [L..R] of a static number list.
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Overall time complexity: O(nlogn)
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Overall space complexity: O(nlogn)
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Wikipedia link: https://en.wikipedia.org/wiki/Range_minimum_query
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"""
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from math import log2
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def build_sparse_table(number_list: list[int]) -> list[list[int]]:
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"""
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Precompute range minimum queries with power of two length and store the precomputed
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values in a table.
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>>> build_sparse_table([8, 1, 0, 3, 4, 9, 3])
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[[8, 1, 0, 3, 4, 9, 3], [1, 0, 0, 3, 4, 3, 0], [0, 0, 0, 3, 0, 0, 0]]
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>>> build_sparse_table([3, 1, 9])
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[[3, 1, 9], [1, 1, 0]]
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>>> build_sparse_table([])
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Traceback (most recent call last):
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...
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ValueError: empty number list not allowed
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"""
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if not number_list:
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raise ValueError("empty number list not allowed")
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length = len(number_list)
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# Initialise sparse_table -- sparse_table[j][i] represents the minimum value of the
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# subset of length (2 ** j) of number_list, starting from index i.
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# smallest power of 2 subset length that fully covers number_list
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row = int(log2(length)) + 1
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sparse_table = [[0 for i in range(length)] for j in range(row)]
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# minimum of subset of length 1 is that value itself
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for i, value in enumerate(number_list):
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sparse_table[0][i] = value
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j = 1
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# compute the minimum value for all intervals with size (2 ** j)
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while (1 << j) <= length:
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i = 0
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# while subset starting from i still have at least (2 ** j) elements
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while (i + (1 << j) - 1) < length:
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# split range [i, i + 2 ** j] and find minimum of 2 halves
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sparse_table[j][i] = min(
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sparse_table[j - 1][i + (1 << (j - 1))], sparse_table[j - 1][i]
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)
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i += 1
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j += 1
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return sparse_table
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def query(sparse_table: list[list[int]], left_bound: int, right_bound: int) -> int:
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"""
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>>> query(build_sparse_table([8, 1, 0, 3, 4, 9, 3]), 0, 4)
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0
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>>> query(build_sparse_table([8, 1, 0, 3, 4, 9, 3]), 4, 6)
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3
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>>> query(build_sparse_table([3, 1, 9]), 2, 2)
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9
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>>> query(build_sparse_table([3, 1, 9]), 0, 1)
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1
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>>> query(build_sparse_table([8, 1, 0, 3, 4, 9, 3]), 0, 11)
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Traceback (most recent call last):
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...
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IndexError: list index out of range
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>>> query(build_sparse_table([]), 0, 0)
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Traceback (most recent call last):
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...
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ValueError: empty number list not allowed
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"""
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if left_bound < 0 or right_bound >= len(sparse_table[0]):
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raise IndexError("list index out of range")
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# highest subset length of power of 2 that is within range [left_bound, right_bound]
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j = int(log2(right_bound - left_bound + 1))
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# minimum of 2 overlapping smaller subsets:
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# [left_bound, left_bound + 2 ** j - 1] and [right_bound - 2 ** j + 1, right_bound]
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return min(sparse_table[j][right_bound - (1 << j) + 1], sparse_table[j][left_bound])
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if __name__ == "__main__":
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from doctest import testmod
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testmod()
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print(f"{query(build_sparse_table([3, 1, 9]), 2, 2) = }")
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