|
| 1 | +""" |
| 2 | +Find the minimum number of multiplications needed to multiply chain of matrices. |
| 3 | +Reference: https://www.geeksforgeeks.org/matrix-chain-multiplication-dp-8/ |
| 4 | +
|
| 5 | +The algorithm has interesting real-world applications. Example: |
| 6 | +1. Image transformations in Computer Graphics as images are composed of matrix. |
| 7 | +2. Solve complex polynomial equations in the field of algebra using least processing |
| 8 | + power. |
| 9 | +3. Calculate overall impact of macroeconomic decisions as economic equations involve a |
| 10 | + number of variables. |
| 11 | +4. Self-driving car navigation can be made more accurate as matrix multiplication can |
| 12 | + accurately determine position and orientation of obstacles in short time. |
| 13 | +
|
| 14 | +Python doctests can be run with the following command: |
| 15 | +python -m doctest -v matrix_chain_multiply.py |
| 16 | +
|
| 17 | +Given a sequence arr[] that represents chain of 2D matrices such that the dimension of |
| 18 | +the ith matrix is arr[i-1]*arr[i]. |
| 19 | +So suppose arr = [40, 20, 30, 10, 30] means we have 4 matrices of dimensions |
| 20 | +40*20, 20*30, 30*10 and 10*30. |
| 21 | +
|
| 22 | +matrix_chain_multiply() returns an integer denoting minimum number of multiplications to |
| 23 | +multiply the chain. |
| 24 | +
|
| 25 | +We do not need to perform actual multiplication here. |
| 26 | +We only need to decide the order in which to perform the multiplication. |
| 27 | +
|
| 28 | +Hints: |
| 29 | +1. Number of multiplications (ie cost) to multiply 2 matrices |
| 30 | +of size m*p and p*n is m*p*n. |
| 31 | +2. Cost of matrix multiplication is associative ie (M1*M2)*M3 != M1*(M2*M3) |
| 32 | +3. Matrix multiplication is not commutative. So, M1*M2 does not mean M2*M1 can be done. |
| 33 | +4. To determine the required order, we can try different combinations. |
| 34 | +So, this problem has overlapping sub-problems and can be solved using recursion. |
| 35 | +We use Dynamic Programming for optimal time complexity. |
| 36 | +
|
| 37 | +Example input: |
| 38 | +arr = [40, 20, 30, 10, 30] |
| 39 | +output: 26000 |
| 40 | +""" |
| 41 | +from collections.abc import Iterator |
| 42 | +from contextlib import contextmanager |
| 43 | +from functools import cache |
| 44 | +from sys import maxsize |
| 45 | + |
| 46 | + |
| 47 | +def matrix_chain_multiply(arr: list[int]) -> int: |
| 48 | + """ |
| 49 | + Find the minimum number of multiplcations required to multiply the chain of matrices |
| 50 | +
|
| 51 | + Args: |
| 52 | + arr: The input array of integers. |
| 53 | +
|
| 54 | + Returns: |
| 55 | + Minimum number of multiplications needed to multiply the chain |
| 56 | +
|
| 57 | + Examples: |
| 58 | + >>> matrix_chain_multiply([1, 2, 3, 4, 3]) |
| 59 | + 30 |
| 60 | + >>> matrix_chain_multiply([10]) |
| 61 | + 0 |
| 62 | + >>> matrix_chain_multiply([10, 20]) |
| 63 | + 0 |
| 64 | + >>> matrix_chain_multiply([19, 2, 19]) |
| 65 | + 722 |
| 66 | + >>> matrix_chain_multiply(list(range(1, 100))) |
| 67 | + 323398 |
| 68 | +
|
| 69 | + # >>> matrix_chain_multiply(list(range(1, 251))) |
| 70 | + # 2626798 |
| 71 | + """ |
| 72 | + if len(arr) < 2: |
| 73 | + return 0 |
| 74 | + # initialising 2D dp matrix |
| 75 | + n = len(arr) |
| 76 | + dp = [[maxsize for j in range(n)] for i in range(n)] |
| 77 | + # we want minimum cost of multiplication of matrices |
| 78 | + # of dimension (i*k) and (k*j). This cost is arr[i-1]*arr[k]*arr[j]. |
| 79 | + for i in range(n - 1, 0, -1): |
| 80 | + for j in range(i, n): |
| 81 | + if i == j: |
| 82 | + dp[i][j] = 0 |
| 83 | + continue |
| 84 | + for k in range(i, j): |
| 85 | + dp[i][j] = min( |
| 86 | + dp[i][j], dp[i][k] + dp[k + 1][j] + arr[i - 1] * arr[k] * arr[j] |
| 87 | + ) |
| 88 | + |
| 89 | + return dp[1][n - 1] |
| 90 | + |
| 91 | + |
| 92 | +def matrix_chain_order(dims: list[int]) -> int: |
| 93 | + """ |
| 94 | + Source: https://en.wikipedia.org/wiki/Matrix_chain_multiplication |
| 95 | + The dynamic programming solution is faster than cached the recursive solution and |
| 96 | + can handle larger inputs. |
| 97 | + >>> matrix_chain_order([1, 2, 3, 4, 3]) |
| 98 | + 30 |
| 99 | + >>> matrix_chain_order([10]) |
| 100 | + 0 |
| 101 | + >>> matrix_chain_order([10, 20]) |
| 102 | + 0 |
| 103 | + >>> matrix_chain_order([19, 2, 19]) |
| 104 | + 722 |
| 105 | + >>> matrix_chain_order(list(range(1, 100))) |
| 106 | + 323398 |
| 107 | +
|
| 108 | + # >>> matrix_chain_order(list(range(1, 251))) # Max before RecursionError is raised |
| 109 | + # 2626798 |
| 110 | + """ |
| 111 | + |
| 112 | + @cache |
| 113 | + def a(i: int, j: int) -> int: |
| 114 | + return min( |
| 115 | + (a(i, k) + dims[i] * dims[k] * dims[j] + a(k, j) for k in range(i + 1, j)), |
| 116 | + default=0, |
| 117 | + ) |
| 118 | + |
| 119 | + return a(0, len(dims) - 1) |
| 120 | + |
| 121 | + |
| 122 | +@contextmanager |
| 123 | +def elapsed_time(msg: str) -> Iterator: |
| 124 | + # print(f"Starting: {msg}") |
| 125 | + from time import perf_counter_ns |
| 126 | + |
| 127 | + start = perf_counter_ns() |
| 128 | + yield |
| 129 | + print(f"Finished: {msg} in {(perf_counter_ns() - start) / 10 ** 9} seconds.") |
| 130 | + |
| 131 | + |
| 132 | +if __name__ == "__main__": |
| 133 | + import doctest |
| 134 | + |
| 135 | + doctest.testmod() |
| 136 | + with elapsed_time("matrix_chain_order"): |
| 137 | + print(f"{matrix_chain_order(list(range(1, 251))) = }") |
| 138 | + with elapsed_time("matrix_chain_multiply"): |
| 139 | + print(f"{matrix_chain_multiply(list(range(1, 251))) = }") |
| 140 | + with elapsed_time("matrix_chain_order"): |
| 141 | + print(f"{matrix_chain_order(list(range(1, 251))) = }") |
| 142 | + with elapsed_time("matrix_chain_multiply"): |
| 143 | + print(f"{matrix_chain_multiply(list(range(1, 251))) = }") |
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