Add Range Maximum Query

dynamic_programming/range_maximum_query.py

@@ -0,0 +1,97 @@
+"""
+Range Maximum Query Problem:
+
+    Given an array of integers and q queries, for each query, find the maximum value in the range [l, r).
+    The array is 0-indexed and the queries are 0-indexed.
+
+    More details:
+        https://www.geeksforgeeks.org/range-maximum-query-using-sparse-table/
+
+"""
+
+
+class RangeMaximumQuery:
+    def __init__(self, array: list) -> None:
+        """
+        Initialize RangeMaximumQuery with given array.
+
+        Parameters:
+            array: list[int]
+
+        Example:
+            >>> rmq = RangeMaximumQuery([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
+        """
+        self.array = array
+        self.size = len(array)
+        self._preprocess()
+
+    def _preprocess(self) -> None:
+        """
+        Preprocess the DP array for RMQ in O(n log n).
+        ------------------------------
+        >>> rmq = RangeMaximumQuery([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
+        >>> rmq.log[7]
+        2
+        >>> rmq.log[8]
+        3
+        >>> rmq.log[9]
+        3
+        >>> rmq.dp[0][0]
+        1
+        >>> rmq.dp[0][9]
+        10
+        >>> rmq.dp[1][0]
+        2
+        >>> rmq.dp[1][9]
+        10
+        >>> rmq.dp[1][8]
+        10
+        """
+        # Fill logarithm array for O(1) usage
+        self.log = [0] * (self.size + 1)
+        for i in range(2, self.size + 1):
+            self.log[i] = self.log[i >> 1] + 1
+        self.size_log = self.log[self.size] + 1
+
+        # Create empty DP 2D array with size log(n) x n
+        self.dp = [[0] * self.size for _ in range(self.size_log)]
+        for i in range(self.size):
+            self.dp[0][i] = self.array[i]
+        for p in range(1, self.size_log):
+            for i in range(self.size):
+                if i + (1 << p) <= self.size:
+                    self.dp[p][i] = max(self.dp[p - 1][i], self.dp[p - 1][i + (1 << (p - 1))])
+                else:
+                    self.dp[p][i] = self.dp[p - 1][i]
+
+    def query(self, left: int, right: int) -> int:
+        """
+        Finds the maximum value in the range [left, right) in O(1).
+
+        Parameters:
+            left: int, the left index of the range (inclusive)
+            right: int, the right index of the range (exclusive)
+
+        Returns:
+            int, the maximum value in the range [left, right)
+
+        Example:
+            >>> array = [1, 3, 5, 7, 9, 11]
+            >>> queries = [(0, 1), (1, 4), (0, 5), (2, 3), (0, 6)]
+            >>> rmq = RangeMaximumQuery(array)
+            >>> for l, r in queries:
+            ...     print(rmq.query(l, r))
+            1
+            7
+            9
+            5
+            11
+        """
+        p = self.log[right - left]  # Find the logarithm of the size of range
+        return max(self.dp[p][left], self.dp[p][right - (1 << p)])
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()