Add: Matrix Prefix Sum

dynamic_programming/distinct_sequences.py

@@ -0,0 +1,74 @@
+"""
+author: Aayush Soni
+Given two strings s and t, return the number of distinct subsequences
+of s which equals t.
+Input: s = "babgbag", t = "bag"
+Output: 5
+Explanation: As shown below, there are 5 ways you can generate "bag" from s.
+>>> "babgbag"
+>>> "babgbag"
+>>> "babgbag"
+>>> "babgbag"
+>>> "babgbag"
+Leetcode link: https://leetcode.com/problems/distinct-subsequences/description/
+"""
+
+def count_distinct_subsequences(s: str, t: str) -> int:
+ """
+ This function calculates the number of distinct
+ subsequences of s that equal t.
+
+ :param s: The input string s.
+ :param t: The target string t.
+ :return: The number of distinct subsequences of s that equal t.
+
+ >>> countDistinctSubsequences("babgbag", "bag")
+ 5
+ >>> countDistinctSubsequences("rabbbit", "rabbit")
+ 3
+ """
+
+ m = len(t) # Length of t
+ n = len(s) # Length of s
+
+ # If t is longer than s, it can't be a subsequence.
+ if m > n:
+ return 0
+
+ # Create a matrix to store counts of subsequences.
+ # dp[i][j] stores the count of occurrences of t(1..i) in s(1..j).
+ dp = [[0 for _ in range(n + 1)] for __ in range(m + 1)]
+
+ # Initialize the first column with all 0s.
+ # An empty string can't have another string as a subsequence.
+ for i in range(1, m + 1):
+ dp[i][0] = 0
+
+ # Initialize the first row with all 1s.
+ # An empty string is a subsequence of all strings.
+ for j in range(n + 1):
+ dp[0][j] = 1
+
+ # Fill dp[][] in a bottom-up manner
+ for i in range(1, m + 1):
+ for j in range(1, n + 1):
+ # If the last characters don't match,
+ # the value is the same as the value without the last character in s.
+ if t[i - 1] != s[j - 1]:
+ dp[i][j] = dp[i][j - 1]
+
+ # If the last characters match,
+ # the value is obtained by considering two cases:
+ # a) All subsequences without the last character in s.
+ # b) All subsequences without the last characters in both s and t.
+ else:
+ dp[i][j] = dp[i][j - 1] + dp[i - 1][j - 1]
+
+ return dp[m][n]
+
+
+if __name__ == "__main__":
+ s = "babgbag"
+ t = "bag"
+ result = count_distinct_subsequences(s, t)
+ print(f"The number of distinct subsequences of '{t}' in '{s}' is {result}.")

matrix/matrix_prefix_sum.py

@@ -0,0 +1,86 @@
+"""
+Python Program to find prefix sum of a 2D array
+"""
+
+
+def calculate_prefix_sum(matrix: list[list[int]]) -> list[list[int]]:
+ """
+ Calculate the prefix sum of a 2D matrix.
+ Prefix Sum Formula:
+ prefix_sum[i][j] = prefix_sum[i - 1][j] + prefix_sum[i][j - 1]
+ - prefix_sum[i - 1][j - 1] + matrix[i][j]
+
+ :param matrix: A 2D matrix.
+ :return: A matrix containing the prefix sums.
+
+ >>> calculate_prefix_sum([[1, 1, 1], [1, 1, 1], [1, 1, 1]])
+ [[1, 2, 3], [2, 4, 6], [3, 6, 9]]
+
+ >>> calculate_prefix_sum([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+ [[1, 3, 6], [5, 12, 21], [12, 27, 45]]
+ """
+ rows = len(matrix)
+ cols = len(matrix[0])
+
+ # Initialize the prefix sum matrix with zeros, with the
+ # same dimensions as the original matrix.
+ prefix_sum = [[0 for _ in range(cols)] for _ in range(rows)]
+
+ # Calculate the prefix sum for the top-left cell.
+ prefix_sum[0][0] = matrix[0][0]
+
+ # Calculate cumulative sums for the first row.
+ for i in range(1, cols):
+ prefix_sum[0][i] = prefix_sum[0][i - 1] + matrix[0][i]
+
+ # Calculate cumulative sums for the first column.
+ for i in range(1, rows):
+ prefix_sum[i][0] = prefix_sum[i - 1][0] + matrix[i][0]
+
+ # Update the values in the cells using the general formula.
+ for i in range(1, rows):
+ for j in range(1, cols):
+ # The value in each cell is the sum of:
+ # - The cell above it
+ # - The cell to the left of it
+ # - Subtracting the overlapping cell
+ # - Adding the value from the original matrix
+ prefix_sum[i][j] = (
+ prefix_sum[i - 1][j]
+ + prefix_sum[i][j - 1]
+ - prefix_sum[i - 1][j - 1]
+ + matrix[i][j]
+ )
+
+ return prefix_sum
+
+
+def display_matrix(matrix: list[list[int]]) -> None:
+ """
+ Display a 2D matrix.
+
+ :param matrix: A 2D matrix.
+
+ Display a 2D matrix by printing each row's elements separated by spaces.
+
+ >>> display_matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+ 1 2 3
+ 4 5 6
+ 7 8 9
+
+ >>> display_matrix([[10, 20, 30], [40, 50, 60]])
+ 10 20 30
+ 40 50 60
+ """
+ for row in matrix:
+ # Join the elements of each row with spaces and print the result.
+ print(" ".join(map(str, row)))
+
+
+if __name__ == "__main__":
+ matrix = [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]
+ # Calculate the prefix sum of the 2D matrix
+ prefix_sum_matrix = calculate_prefix_sum(matrix)
+
+ # Display the prefix sum matrix
+ display_matrix(prefix_sum_matrix)