|
| 1 | +""" |
| 2 | +The maximum subarray problem is the task of finding the continuous subarray that has the |
| 3 | +maximum sum within a given array of numbers. For example, given the array |
| 4 | +[-2, 1, -3, 4, -1, 2, 1, -5, 4], the contiguous subarray with the maximum sum is |
| 5 | +[4, -1, 2, 1], which has a sum of 6. |
| 6 | +
|
| 7 | +This divide-and-conquer algorithm finds the maximum subarray in O(n log n) time. |
| 8 | +""" |
| 9 | +from __future__ import annotations |
| 10 | + |
| 11 | +import time |
| 12 | +from collections.abc import Sequence |
| 13 | +from random import randint |
| 14 | + |
| 15 | +from matplotlib import pyplot as plt |
| 16 | + |
| 17 | + |
| 18 | +def max_subarray( |
| 19 | + arr: Sequence[float], low: int, high: int |
| 20 | +) -> tuple[int | None, int | None, float]: |
| 21 | + """ |
| 22 | + Solves the maximum subarray problem using divide and conquer. |
| 23 | + :param arr: the given array of numbers |
| 24 | + :param low: the start index |
| 25 | + :param high: the end index |
| 26 | + :return: the start index of the maximum subarray, the end index of the |
| 27 | + maximum subarray, and the maximum subarray sum |
| 28 | +
|
| 29 | + >>> nums = [-2, 1, -3, 4, -1, 2, 1, -5, 4] |
| 30 | + >>> max_subarray(nums, 0, len(nums) - 1) |
| 31 | + (3, 6, 6) |
| 32 | + >>> nums = [2, 8, 9] |
| 33 | + >>> max_subarray(nums, 0, len(nums) - 1) |
| 34 | + (0, 2, 19) |
| 35 | + >>> nums = [0, 0] |
| 36 | + >>> max_subarray(nums, 0, len(nums) - 1) |
| 37 | + (0, 0, 0) |
| 38 | + >>> nums = [-1.0, 0.0, 1.0] |
| 39 | + >>> max_subarray(nums, 0, len(nums) - 1) |
| 40 | + (2, 2, 1.0) |
| 41 | + >>> nums = [-2, -3, -1, -4, -6] |
| 42 | + >>> max_subarray(nums, 0, len(nums) - 1) |
| 43 | + (2, 2, -1) |
| 44 | + >>> max_subarray([], 0, 0) |
| 45 | + (None, None, 0) |
| 46 | + """ |
| 47 | + if not arr: |
| 48 | + return None, None, 0 |
| 49 | + if low == high: |
| 50 | + return low, high, arr[low] |
| 51 | + |
| 52 | + mid = (low + high) // 2 |
| 53 | + left_low, left_high, left_sum = max_subarray(arr, low, mid) |
| 54 | + right_low, right_high, right_sum = max_subarray(arr, mid + 1, high) |
| 55 | + cross_left, cross_right, cross_sum = max_cross_sum(arr, low, mid, high) |
| 56 | + if left_sum >= right_sum and left_sum >= cross_sum: |
| 57 | + return left_low, left_high, left_sum |
| 58 | + elif right_sum >= left_sum and right_sum >= cross_sum: |
| 59 | + return right_low, right_high, right_sum |
| 60 | + return cross_left, cross_right, cross_sum |
| 61 | + |
| 62 | + |
| 63 | +def max_cross_sum( |
| 64 | + arr: Sequence[float], low: int, mid: int, high: int |
| 65 | +) -> tuple[int, int, float]: |
| 66 | + left_sum, max_left = float("-inf"), -1 |
| 67 | + right_sum, max_right = float("-inf"), -1 |
| 68 | + |
| 69 | + summ: int | float = 0 |
| 70 | + for i in range(mid, low - 1, -1): |
| 71 | + summ += arr[i] |
| 72 | + if summ > left_sum: |
| 73 | + left_sum = summ |
| 74 | + max_left = i |
| 75 | + |
| 76 | + summ = 0 |
| 77 | + for i in range(mid + 1, high + 1): |
| 78 | + summ += arr[i] |
| 79 | + if summ > right_sum: |
| 80 | + right_sum = summ |
| 81 | + max_right = i |
| 82 | + |
| 83 | + return max_left, max_right, (left_sum + right_sum) |
| 84 | + |
| 85 | + |
| 86 | +def time_max_subarray(input_size: int) -> float: |
| 87 | + arr = [randint(1, input_size) for _ in range(input_size)] |
| 88 | + start = time.time() |
| 89 | + max_subarray(arr, 0, input_size - 1) |
| 90 | + end = time.time() |
| 91 | + return end - start |
| 92 | + |
| 93 | + |
| 94 | +def plot_runtimes() -> None: |
| 95 | + input_sizes = [10, 100, 1000, 10000, 50000, 100000, 200000, 300000, 400000, 500000] |
| 96 | + runtimes = [time_max_subarray(input_size) for input_size in input_sizes] |
| 97 | + print("No of Inputs\t\tTime Taken") |
| 98 | + for input_size, runtime in zip(input_sizes, runtimes): |
| 99 | + print(input_size, "\t\t", runtime) |
| 100 | + plt.plot(input_sizes, runtimes) |
| 101 | + plt.xlabel("Number of Inputs") |
| 102 | + plt.ylabel("Time taken in seconds") |
| 103 | + plt.show() |
| 104 | + |
| 105 | + |
| 106 | +if __name__ == "__main__": |
| 107 | + """ |
| 108 | + A random simulation of this algorithm. |
| 109 | + """ |
| 110 | + from doctest import testmod |
| 111 | + |
| 112 | + testmod() |
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