new algorithm matrix_chain_multiplication

dynamic_programming/matrix_chain_multiplication.py

@@ -0,0 +1,78 @@
+"""
+Find the minimum number of multiplications needed to multiply chain of matrices.
+Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/
+
+Python doctests can be run with the following command:
+python -m doctest -v matrix_chain_multiply.py
+
+Given a sequence arr[] that represents chain of 2D matrices such that
+the dimension of ith matrix is arr[i-1]*arr[i].
+So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of
+dimesions 40*20, 20*30, 30*10 and 10*30.
+
+matrix_chain_multiply() returns an integer denoting
+minimum number of multiplications to multiply the chain.
+
+We do not need to perform actual multiplication here.
+We only need to decide the order in which to perform the multiplication.
+
+Hints:
+1. Number of multiplications (ie cost) to multiply 2 matrices
+of size m*p and p*n is m*p*n.
+2. Cost of matrix multiplication is neither associative ie (M1*M2)*M3 != M1*(M2*M3)
+3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done.
+4. To determine the required order, we can try different combinations.
+So, this problem has overlapping sub-problems and can be solved using recursion.
+We use Dynamic Programming for optimal time complexity.
+
+Example input :
+arr = [40, 20, 30, 10, 30]
+output : 26000
+"""
+
+
+def matrix_chain_multiply(arr: list[int]) -> int:
+    """
+    Find the minimum number of multiplcations to multiply
+    chain of matrices.
+
+    Args:
+        arr : The input array of integers.
+
+    Returns:
+        int: Minimum number of multiplications needed to multiply the chain
+
+    Examples:
+        >>> matrix_chain_multiply([1,2,3,4,3])
+        30
+        >>> matrix_chain_multiply([10])
+        0
+        >>> matrix_chain_multiply([10, 20])
+        0
+        >>> matrix_chain_multiply([19, 2, 19])
+        722
+    """
+    # first edge case
+    if len(arr) < 2:
+        return 0
+    # initialising 2D dp matrix
+    n = len(arr)
+    dp = [[float("inf") for j in range(n)] for i in range(n)]
+    # we want minimum cost of multiplication of matrices
+    # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j].
+    for i in range(n - 1, 0, -1):
+        for j in range(i, n):
+            if i == j:
+                dp[i][j] = 0
+                continue
+            for k in range(i, j):
+                dp[i][j] = min(
+                    dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j]
+                )
+    return dp[1][n - 1]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()