Added dynamic_programming/range_sum_query.py (TheAlgorithms#12592)

* Create prefix_sum.py

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* Rename prefix_sum.py to range_sum_query.py

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

Author: 3 people

Diff for: dynamic_programming/range_sum_query.py

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+"""
+Author: Sanjay Muthu <https://github.com/XenoBytesX>
+
+This is an implementation of the Dynamic Programming solution to the Range Sum Query.
+
+The problem statement is:
+    Given an array and q queries,
+    each query stating you to find the sum of elements from l to r (inclusive)
+
+Example:
+    arr = [1, 4, 6, 2, 61, 12]
+    queries = 3
+    l_1 = 2, r_1 = 5
+    l_2 = 1, r_2 = 5
+    l_3 = 3, r_3 = 4
+
+    as input will return
+
+    [81, 85, 63]
+
+    as output
+
+0-indexing:
+NOTE: 0-indexing means the indexing of the array starts from 0
+Example: a = [1, 2, 3, 4, 5, 6]
+         Here, the 0th index of a is 1,
+               the 1st index of a is 2,
+               and so forth
+
+Time Complexity: O(N + Q)
+* O(N) pre-calculation time to calculate the prefix sum array
+* and O(1) time per each query = O(1 * Q) = O(Q) time
+
+Space Complexity: O(N)
+* O(N) to store the prefix sum
+
+Algorithm:
+So, first we calculate the prefix sum (dp) of the array.
+The prefix sum of the index i is the sum of all elements indexed
+from 0 to i (inclusive).
+The prefix sum of the index i is the prefix sum of index (i - 1) + the current element.
+So, the state of the dp is dp[i] = dp[i - 1] + a[i].
+
+After we calculate the prefix sum,
+for each query [l, r]
+the answer is dp[r] - dp[l - 1] (we need to be careful because l might be 0).
+For example take this array:
+    [4, 2, 1, 6, 3]
+The prefix sum calculated for this array would be:
+    [4, 4 + 2, 4 + 2 + 1, 4 + 2 + 1 + 6, 4 + 2 + 1 + 6 + 3]
+    ==> [4, 6, 7, 13, 16]
+If the query was l = 3, r = 4,
+the answer would be 6 + 3 = 9 but this would require O(r - l + 1) time ≈ O(N) time
+
+If we use prefix sums we can find it in O(1) by using the formula
+prefix[r] - prefix[l - 1].
+This formula works because prefix[r] is the sum of elements from [0, r]
+and prefix[l - 1] is the sum of elements from [0, l - 1],
+so if we do prefix[r] - prefix[l - 1] it will be
+[0, r] - [0, l - 1] = [0, l - 1] + [l, r] - [0, l - 1] = [l, r]
+"""
+
+
+def prefix_sum(array: list[int], queries: list[tuple[int, int]]) -> list[int]:
+    """
+    >>> prefix_sum([1, 4, 6, 2, 61, 12], [(2, 5), (1, 5), (3, 4)])
+    [81, 85, 63]
+    >>> prefix_sum([4, 2, 1, 6, 3], [(3, 4), (1, 3), (0, 2)])
+    [9, 9, 7]
+    """
+    # The prefix sum array
+    dp = [0] * len(array)
+    dp[0] = array[0]
+    for i in range(1, len(array)):
+        dp[i] = dp[i - 1] + array[i]
+
+    # See Algorithm section (Line 44)
+    result = []
+    for query in queries:
+        left, right = query
+        res = dp[right]
+        if left > 0:
+            res -= dp[left - 1]
+        result.append(res)
+
+    return result
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()