Dynamic programming/matrix chain multiplication (#10562)

* updating DIRECTORY.md

* spell changes

* updating DIRECTORY.md

* real world applications

* updating DIRECTORY.md

* Update matrix_chain_multiplication.py

Add a non-dp solution with benchmarks.

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update matrix_chain_multiplication.py

* Update matrix_chain_multiplication.py

* Update matrix_chain_multiplication.py

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: Pooja Sharma <poojasharma@MyBigMac.local>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

DIRECTORY.md

@@ -182,6 +182,7 @@
     * [Permutations](data_structures/arrays/permutations.py)
     * [Prefix Sum](data_structures/arrays/prefix_sum.py)
     * [Product Sum](data_structures/arrays/product_sum.py)
+    * [Sparse Table](data_structures/arrays/sparse_table.py)
   * Binary Tree
     * [Avl Tree](data_structures/binary_tree/avl_tree.py)
     * [Basic Binary Tree](data_structures/binary_tree/basic_binary_tree.py)
@@ -340,6 +341,7 @@
   * [Longest Increasing Subsequence O(Nlogn)](dynamic_programming/longest_increasing_subsequence_o(nlogn).py)
   * [Longest Palindromic Subsequence](dynamic_programming/longest_palindromic_subsequence.py)
   * [Longest Sub Array](dynamic_programming/longest_sub_array.py)
+  * [Matrix Chain Multiplication](dynamic_programming/matrix_chain_multiplication.py)
   * [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
   * [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
   * [Max Product Subarray](dynamic_programming/max_product_subarray.py)
@@ -370,6 +372,7 @@
   * [Builtin Voltage](electronics/builtin_voltage.py)
   * [Carrier Concentration](electronics/carrier_concentration.py)
   * [Charging Capacitor](electronics/charging_capacitor.py)
+  * [Charging Inductor](electronics/charging_inductor.py)
   * [Circular Convolution](electronics/circular_convolution.py)
   * [Coulombs Law](electronics/coulombs_law.py)
   * [Electric Conductivity](electronics/electric_conductivity.py)
@@ -524,6 +527,7 @@
   * [Simplex](linear_programming/simplex.py)
 
 ## Machine Learning
+  * [Apriori Algorithm](machine_learning/apriori_algorithm.py)
   * [Astar](machine_learning/astar.py)
   * [Data Transformations](machine_learning/data_transformations.py)
   * [Decision Tree](machine_learning/decision_tree.py)
@@ -554,7 +558,6 @@
   * [Word Frequency Functions](machine_learning/word_frequency_functions.py)
   * [Xgboost Classifier](machine_learning/xgboost_classifier.py)
   * [Xgboost Regressor](machine_learning/xgboost_regressor.py)
-  * [Apriori Algorithm](machine_learning/apriori_algorithm.py)
 
 ## Maths
   * [Abs](maths/abs.py)

dynamic_programming/matrix_chain_multiplication.py

@@ -0,0 +1,143 @@
+"""
+Find the minimum number of multiplications needed to multiply chain of matrices.
+Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/
+
+The algorithm has interesting real-world applications. Example:
+1. Image transformations in Computer Graphics as images are composed of matrix.
+2. Solve complex polynomial equations in the field of algebra using least processing
+   power.
+3. Calculate overall impact of macroeconomic decisions as economic equations involve a
+   number of variables.
+4. Self-driving car navigation can be made more accurate as matrix multiplication can
+   accurately determine position and orientation of obstacles in short time.
+
+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
+the ith matrix is arr[i-1]*arr[i].
+So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of dimensions
+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 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
+"""
+from collections.abc import Iterator
+from contextlib import contextmanager
+from functools import cache
+from sys import maxsize
+
+
+def matrix_chain_multiply(arr: list[int]) -> int:
+    """
+    Find the minimum number of multiplcations required to multiply the chain of matrices
+
+    Args:
+        arr: The input array of integers.
+
+    Returns:
+        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
+        >>> matrix_chain_multiply(list(range(1, 100)))
+        323398
+
+        # >>> matrix_chain_multiply(list(range(1, 251)))
+        # 2626798
+    """
+    if len(arr) < 2:
+        return 0
+    # initialising 2D dp matrix
+    n = len(arr)
+    dp = [[maxsize 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]
+
+
+def matrix_chain_order(dims: list[int]) -> int:
+    """
+    Source: https://en.wikipedia.org/wiki/Matrix_chain_multiplication
+    The dynamic programming solution is faster than cached the recursive solution and
+    can handle larger inputs.
+    >>> matrix_chain_order([1, 2, 3, 4, 3])
+    30
+    >>> matrix_chain_order([10])
+    0
+    >>> matrix_chain_order([10, 20])
+    0
+    >>> matrix_chain_order([19, 2, 19])
+    722
+    >>> matrix_chain_order(list(range(1, 100)))
+    323398
+
+    # >>> matrix_chain_order(list(range(1, 251)))  # Max before RecursionError is raised
+    # 2626798
+    """
+
+    @cache
+    def a(i: int, j: int) -> int:
+        return min(
+            (a(i, k) + dims[i] * dims[k] * dims[j] + a(k, j) for k in range(i + 1, j)),
+            default=0,
+        )
+
+    return a(0, len(dims) - 1)
+
+
+@contextmanager
+def elapsed_time(msg: str) -> Iterator:
+    # print(f"Starting: {msg}")
+    from time import perf_counter_ns
+
+    start = perf_counter_ns()
+    yield
+    print(f"Finished: {msg} in {(perf_counter_ns() - start) / 10 ** 9} seconds.")
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    with elapsed_time("matrix_chain_order"):
+        print(f"{matrix_chain_order(list(range(1, 251))) = }")
+    with elapsed_time("matrix_chain_multiply"):
+        print(f"{matrix_chain_multiply(list(range(1, 251))) = }")
+    with elapsed_time("matrix_chain_order"):
+        print(f"{matrix_chain_order(list(range(1, 251))) = }")
+    with elapsed_time("matrix_chain_multiply"):
+        print(f"{matrix_chain_multiply(list(range(1, 251))) = }")