Added data_structures/arrays/sparse_table.py (#10437)

* Create sparse_table.py

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* Update sparse_table.py

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* Update sparse_table.py

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Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <[email protected]>

Author: 3 people

Diff for: data_structures/arrays/sparse_table.py

@@ -0,0 +1,94 @@
+"""
+    Sparse table is a data structure that allows answering range queries on
+    a static number list, i.e. the elements do not change throughout all the queries.
+
+    The implementation below will solve the problem of Range Minimum Query:
+    Finding the minimum value of a subset [L..R] of a static number list.
+
+    Overall time complexity: O(nlogn)
+    Overall space complexity: O(nlogn)
+
+    Wikipedia link: https://en.wikipedia.org/wiki/Range_minimum_query
+"""
+from math import log2
+
+
+def build_sparse_table(number_list: list[int]) -> list[list[int]]:
+    """
+    Precompute range minimum queries with power of two length and store the precomputed
+    values in a table.
+
+    >>> build_sparse_table([8, 1, 0, 3, 4, 9, 3])
+    [[8, 1, 0, 3, 4, 9, 3], [1, 0, 0, 3, 4, 3, 0], [0, 0, 0, 3, 0, 0, 0]]
+    >>> build_sparse_table([3, 1, 9])
+    [[3, 1, 9], [1, 1, 0]]
+    >>> build_sparse_table([])
+    Traceback (most recent call last):
+    ...
+    ValueError: empty number list not allowed
+    """
+    if not number_list:
+        raise ValueError("empty number list not allowed")
+
+    length = len(number_list)
+    # Initialise sparse_table -- sparse_table[j][i] represents the minimum value of the
+    # subset of length (2 ** j) of number_list, starting from index i.
+
+    # smallest power of 2 subset length that fully covers number_list
+    row = int(log2(length)) + 1
+    sparse_table = [[0 for i in range(length)] for j in range(row)]
+
+    # minimum of subset of length 1 is that value itself
+    for i, value in enumerate(number_list):
+        sparse_table[0][i] = value
+    j = 1
+
+    # compute the minimum value for all intervals with size (2 ** j)
+    while (1 << j) <= length:
+        i = 0
+        # while subset starting from i still have at least (2 ** j) elements
+        while (i + (1 << j) - 1) < length:
+            # split range [i, i + 2 ** j] and find minimum of 2 halves
+            sparse_table[j][i] = min(
+                sparse_table[j - 1][i + (1 << (j - 1))], sparse_table[j - 1][i]
+            )
+            i += 1
+        j += 1
+    return sparse_table
+
+
+def query(sparse_table: list[list[int]], left_bound: int, right_bound: int) -> int:
+    """
+    >>> query(build_sparse_table([8, 1, 0, 3, 4, 9, 3]), 0, 4)
+    0
+    >>> query(build_sparse_table([8, 1, 0, 3, 4, 9, 3]), 4, 6)
+    3
+    >>> query(build_sparse_table([3, 1, 9]), 2, 2)
+    9
+    >>> query(build_sparse_table([3, 1, 9]), 0, 1)
+    1
+    >>> query(build_sparse_table([8, 1, 0, 3, 4, 9, 3]), 0, 11)
+    Traceback (most recent call last):
+    ...
+    IndexError: list index out of range
+    >>> query(build_sparse_table([]), 0, 0)
+    Traceback (most recent call last):
+    ...
+    ValueError: empty number list not allowed
+    """
+    if left_bound < 0 or right_bound >= len(sparse_table[0]):
+        raise IndexError("list index out of range")
+
+    # highest subset length of power of 2 that is within range [left_bound, right_bound]
+    j = int(log2(right_bound - left_bound + 1))
+
+    # minimum of 2 overlapping smaller subsets:
+    # [left_bound, left_bound + 2 ** j - 1] and [right_bound - 2 ** j + 1, right_bound]
+    return min(sparse_table[j][right_bound - (1 << j) + 1], sparse_table[j][left_bound])
+
+
+if __name__ == "__main__":
+    from doctest import testmod
+
+    testmod()
+    print(f"{query(build_sparse_table([3, 1, 9]), 2, 2) = }")