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()