mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-27 15:01:08 +00:00
Consolidate duplicate implementations of max subarray (#8849)
* Remove max subarray sum duplicate implementations
* updating DIRECTORY.md
* Rename max_sum_contiguous_subsequence.py
* Fix typo in dynamic_programming/max_subarray_sum.py
* Remove duplicate divide and conquer max subarray
* updating DIRECTORY.md
---------
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
This commit is contained in:
Tianyi Zheng
GitHub
parent
c9ee6ed188
commit
a0eec90466
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@ -293,7 +293,7 @@
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* [Inversions](divide_and_conquer/inversions.py)
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* [Kth Order Statistic](divide_and_conquer/kth_order_statistic.py)
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* [Max Difference Pair](divide_and_conquer/max_difference_pair.py)
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* [Max Subarray Sum](divide_and_conquer/max_subarray_sum.py)
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* [Max Subarray](divide_and_conquer/max_subarray.py)
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* [Mergesort](divide_and_conquer/mergesort.py)
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* [Peak](divide_and_conquer/peak.py)
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* [Power](divide_and_conquer/power.py)
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@ -324,8 +324,7 @@
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* [Matrix Chain Order](dynamic_programming/matrix_chain_order.py)
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* [Max Non Adjacent Sum](dynamic_programming/max_non_adjacent_sum.py)
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* [Max Product Subarray](dynamic_programming/max_product_subarray.py)
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* [Max Sub Array](dynamic_programming/max_sub_array.py)
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* [Max Sum Contiguous Subsequence](dynamic_programming/max_sum_contiguous_subsequence.py)
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* [Max Subarray Sum](dynamic_programming/max_subarray_sum.py)
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* [Min Distance Up Bottom](dynamic_programming/min_distance_up_bottom.py)
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* [Minimum Coin Change](dynamic_programming/minimum_coin_change.py)
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* [Minimum Cost Path](dynamic_programming/minimum_cost_path.py)
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@ -591,12 +590,10 @@
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* [Is Square Free](maths/is_square_free.py)
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* [Jaccard Similarity](maths/jaccard_similarity.py)
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* [Juggler Sequence](maths/juggler_sequence.py)
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* [Kadanes](maths/kadanes.py)
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* [Karatsuba](maths/karatsuba.py)
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* [Krishnamurthy Number](maths/krishnamurthy_number.py)
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* [Kth Lexicographic Permutation](maths/kth_lexicographic_permutation.py)
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* [Largest Of Very Large Numbers](maths/largest_of_very_large_numbers.py)
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* [Largest Subarray Sum](maths/largest_subarray_sum.py)
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* [Least Common Multiple](maths/least_common_multiple.py)
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* [Line Length](maths/line_length.py)
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* [Liouville Lambda](maths/liouville_lambda.py)
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@ -733,7 +730,6 @@
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* [Linear Congruential Generator](other/linear_congruential_generator.py)
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* [Lru Cache](other/lru_cache.py)
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* [Magicdiamondpattern](other/magicdiamondpattern.py)
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* [Maximum Subarray](other/maximum_subarray.py)
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* [Maximum Subsequence](other/maximum_subsequence.py)
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* [Nested Brackets](other/nested_brackets.py)
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* [Number Container System](other/number_container_system.py)
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112
divide_and_conquer/max_subarray.py
Normal file
112
divide_and_conquer/max_subarray.py
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@ -0,0 +1,112 @@
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"""
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The maximum subarray problem is the task of finding the continuous subarray that has the
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maximum sum within a given array of numbers. For example, given the array
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[-2, 1, -3, 4, -1, 2, 1, -5, 4], the contiguous subarray with the maximum sum is
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[4, -1, 2, 1], which has a sum of 6.
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This divide-and-conquer algorithm finds the maximum subarray in O(n log n) time.
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"""
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from __future__ import annotations
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import time
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from collections.abc import Sequence
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from random import randint
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from matplotlib import pyplot as plt
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def max_subarray(
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arr: Sequence[float], low: int, high: int
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) -> tuple[int | None, int | None, float]:
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"""
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Solves the maximum subarray problem using divide and conquer.
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:param arr: the given array of numbers
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:param low: the start index
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:param high: the end index
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:return: the start index of the maximum subarray, the end index of the
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maximum subarray, and the maximum subarray sum
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>>> nums = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
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>>> max_subarray(nums, 0, len(nums) - 1)
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(3, 6, 6)
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>>> nums = [2, 8, 9]
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>>> max_subarray(nums, 0, len(nums) - 1)
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(0, 2, 19)
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>>> nums = [0, 0]
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>>> max_subarray(nums, 0, len(nums) - 1)
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(0, 0, 0)
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>>> nums = [-1.0, 0.0, 1.0]
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>>> max_subarray(nums, 0, len(nums) - 1)
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(2, 2, 1.0)
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>>> nums = [-2, -3, -1, -4, -6]
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>>> max_subarray(nums, 0, len(nums) - 1)
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(2, 2, -1)
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>>> max_subarray([], 0, 0)
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(None, None, 0)
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"""
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if not arr:
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return None, None, 0
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if low == high:
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return low, high, arr[low]
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mid = (low + high) // 2
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left_low, left_high, left_sum = max_subarray(arr, low, mid)
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right_low, right_high, right_sum = max_subarray(arr, mid + 1, high)
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cross_left, cross_right, cross_sum = max_cross_sum(arr, low, mid, high)
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if left_sum >= right_sum and left_sum >= cross_sum:
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return left_low, left_high, left_sum
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elif right_sum >= left_sum and right_sum >= cross_sum:
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return right_low, right_high, right_sum
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return cross_left, cross_right, cross_sum
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def max_cross_sum(
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arr: Sequence[float], low: int, mid: int, high: int
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) -> tuple[int, int, float]:
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left_sum, max_left = float("-inf"), -1
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right_sum, max_right = float("-inf"), -1
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summ: int | float = 0
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for i in range(mid, low - 1, -1):
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summ += arr[i]
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if summ > left_sum:
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left_sum = summ
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max_left = i
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summ = 0
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for i in range(mid + 1, high + 1):
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summ += arr[i]
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if summ > right_sum:
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right_sum = summ
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max_right = i
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return max_left, max_right, (left_sum + right_sum)
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def time_max_subarray(input_size: int) -> float:
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arr = [randint(1, input_size) for _ in range(input_size)]
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start = time.time()
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max_subarray(arr, 0, input_size - 1)
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end = time.time()
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return end - start
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def plot_runtimes() -> None:
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input_sizes = [10, 100, 1000, 10000, 50000, 100000, 200000, 300000, 400000, 500000]
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runtimes = [time_max_subarray(input_size) for input_size in input_sizes]
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print("No of Inputs\t\tTime Taken")
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for input_size, runtime in zip(input_sizes, runtimes):
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print(input_size, "\t\t", runtime)
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plt.plot(input_sizes, runtimes)
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plt.xlabel("Number of Inputs")
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plt.ylabel("Time taken in seconds")
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plt.show()
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if __name__ == "__main__":
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"""
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A random simulation of this algorithm.
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"""
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from doctest import testmod
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testmod()
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@ -1,78 +0,0 @@
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"""
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Given a array of length n, max_subarray_sum() finds
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the maximum of sum of contiguous sub-array using divide and conquer method.
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Time complexity : O(n log n)
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Ref : INTRODUCTION TO ALGORITHMS THIRD EDITION
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(section : 4, sub-section : 4.1, page : 70)
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"""
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def max_sum_from_start(array):
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"""This function finds the maximum contiguous sum of array from 0 index
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Parameters :
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array (list[int]) : given array
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Returns :
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max_sum (int) : maximum contiguous sum of array from 0 index
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"""
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array_sum = 0
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max_sum = float("-inf")
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for num in array:
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array_sum += num
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if array_sum > max_sum:
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max_sum = array_sum
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return max_sum
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def max_cross_array_sum(array, left, mid, right):
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"""This function finds the maximum contiguous sum of left and right arrays
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Parameters :
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array, left, mid, right (list[int], int, int, int)
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Returns :
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(int) : maximum of sum of contiguous sum of left and right arrays
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"""
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max_sum_of_left = max_sum_from_start(array[left : mid + 1][::-1])
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max_sum_of_right = max_sum_from_start(array[mid + 1 : right + 1])
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return max_sum_of_left + max_sum_of_right
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def max_subarray_sum(array, left, right):
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"""Maximum contiguous sub-array sum, using divide and conquer method
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Parameters :
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array, left, right (list[int], int, int) :
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given array, current left index and current right index
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Returns :
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int : maximum of sum of contiguous sub-array
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"""
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# base case: array has only one element
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if left == right:
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return array[right]
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# Recursion
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mid = (left + right) // 2
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left_half_sum = max_subarray_sum(array, left, mid)
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right_half_sum = max_subarray_sum(array, mid + 1, right)
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cross_sum = max_cross_array_sum(array, left, mid, right)
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return max(left_half_sum, right_half_sum, cross_sum)
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if __name__ == "__main__":
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array = [-2, -5, 6, -2, -3, 1, 5, -6]
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array_length = len(array)
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print(
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"Maximum sum of contiguous subarray:",
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max_subarray_sum(array, 0, array_length - 1),
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)
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@ -1,93 +0,0 @@
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"""
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author : Mayank Kumar Jha (mk9440)
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"""
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from __future__ import annotations
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def find_max_sub_array(a, low, high):
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if low == high:
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return low, high, a[low]
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else:
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mid = (low + high) // 2
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left_low, left_high, left_sum = find_max_sub_array(a, low, mid)
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right_low, right_high, right_sum = find_max_sub_array(a, mid + 1, high)
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cross_left, cross_right, cross_sum = find_max_cross_sum(a, low, mid, high)
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if left_sum >= right_sum and left_sum >= cross_sum:
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return left_low, left_high, left_sum
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elif right_sum >= left_sum and right_sum >= cross_sum:
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return right_low, right_high, right_sum
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else:
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return cross_left, cross_right, cross_sum
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def find_max_cross_sum(a, low, mid, high):
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left_sum, max_left = -999999999, -1
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right_sum, max_right = -999999999, -1
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summ = 0
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for i in range(mid, low - 1, -1):
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summ += a[i]
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if summ > left_sum:
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left_sum = summ
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max_left = i
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summ = 0
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for i in range(mid + 1, high + 1):
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summ += a[i]
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if summ > right_sum:
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right_sum = summ
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max_right = i
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return max_left, max_right, (left_sum + right_sum)
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def max_sub_array(nums: list[int]) -> int:
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"""
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Finds the contiguous subarray which has the largest sum and return its sum.
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>>> max_sub_array([-2, 1, -3, 4, -1, 2, 1, -5, 4])
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6
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An empty (sub)array has sum 0.
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>>> max_sub_array([])
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0
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If all elements are negative, the largest subarray would be the empty array,
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having the sum 0.
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>>> max_sub_array([-1, -2, -3])
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0
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>>> max_sub_array([5, -2, -3])
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5
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>>> max_sub_array([31, -41, 59, 26, -53, 58, 97, -93, -23, 84])
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187
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"""
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best = 0
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current = 0
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for i in nums:
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current += i
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current = max(current, 0)
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best = max(best, current)
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return best
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if __name__ == "__main__":
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"""
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A random simulation of this algorithm.
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"""
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import time
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from random import randint
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from matplotlib import pyplot as plt
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inputs = [10, 100, 1000, 10000, 50000, 100000, 200000, 300000, 400000, 500000]
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tim = []
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for i in inputs:
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li = [randint(1, i) for j in range(i)]
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strt = time.time()
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(find_max_sub_array(li, 0, len(li) - 1))
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end = time.time()
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tim.append(end - strt)
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print("No of Inputs Time Taken")
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for i in range(len(inputs)):
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print(inputs[i], "\t\t", tim[i])
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plt.plot(inputs, tim)
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plt.xlabel("Number of Inputs")
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plt.ylabel("Time taken in seconds ")
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plt.show()
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60
dynamic_programming/max_subarray_sum.py
Normal file
60
dynamic_programming/max_subarray_sum.py
Normal file
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@ -0,0 +1,60 @@
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"""
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The maximum subarray sum problem is the task of finding the maximum sum that can be
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obtained from a contiguous subarray within a given array of numbers. For example, given
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the array [-2, 1, -3, 4, -1, 2, 1, -5, 4], the contiguous subarray with the maximum sum
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is [4, -1, 2, 1], so the maximum subarray sum is 6.
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Kadane's algorithm is a simple dynamic programming algorithm that solves the maximum
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subarray sum problem in O(n) time and O(1) space.
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Reference: https://en.wikipedia.org/wiki/Maximum_subarray_problem
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"""
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from collections.abc import Sequence
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def max_subarray_sum(
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arr: Sequence[float], allow_empty_subarrays: bool = False
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) -> float:
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"""
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Solves the maximum subarray sum problem using Kadane's algorithm.
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:param arr: the given array of numbers
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:param allow_empty_subarrays: if True, then the algorithm considers empty subarrays
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>>> max_subarray_sum([2, 8, 9])
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19
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>>> max_subarray_sum([0, 0])
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0
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>>> max_subarray_sum([-1.0, 0.0, 1.0])
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1.0
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>>> max_subarray_sum([1, 2, 3, 4, -2])
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10
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>>> max_subarray_sum([-2, 1, -3, 4, -1, 2, 1, -5, 4])
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6
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>>> max_subarray_sum([2, 3, -9, 8, -2])
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8
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>>> max_subarray_sum([-2, -3, -1, -4, -6])
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-1
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>>> max_subarray_sum([-2, -3, -1, -4, -6], allow_empty_subarrays=True)
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0
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>>> max_subarray_sum([])
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0
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"""
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if not arr:
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return 0
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max_sum = 0 if allow_empty_subarrays else float("-inf")
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curr_sum = 0.0
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for num in arr:
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curr_sum = max(0 if allow_empty_subarrays else num, curr_sum + num)
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max_sum = max(max_sum, curr_sum)
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return max_sum
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if __name__ == "__main__":
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from doctest import testmod
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testmod()
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nums = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
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print(f"{max_subarray_sum(nums) = }")
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@ -1,20 +0,0 @@
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def max_subarray_sum(nums: list) -> int:
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"""
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>>> max_subarray_sum([6 , 9, -1, 3, -7, -5, 10])
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17
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"""
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if not nums:
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return 0
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n = len(nums)
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res, s, s_pre = nums[0], nums[0], nums[0]
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for i in range(1, n):
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s = max(nums[i], s_pre + nums[i])
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s_pre = s
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res = max(res, s)
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return res
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if __name__ == "__main__":
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nums = [6, 9, -1, 3, -7, -5, 10]
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print(max_subarray_sum(nums))
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@ -1,63 +0,0 @@
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"""
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Kadane's algorithm to get maximum subarray sum
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https://medium.com/@rsinghal757/kadanes-algorithm-dynamic-programming-how-and-why-does-it-work-3fd8849ed73d
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https://en.wikipedia.org/wiki/Maximum_subarray_problem
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"""
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test_data: tuple = ([-2, -8, -9], [2, 8, 9], [-1, 0, 1], [0, 0], [])
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def negative_exist(arr: list) -> int:
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"""
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>>> negative_exist([-2,-8,-9])
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-2
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>>> [negative_exist(arr) for arr in test_data]
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[-2, 0, 0, 0, 0]
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"""
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arr = arr or [0]
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max_number = arr[0]
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for i in arr:
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if i >= 0:
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return 0
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elif max_number <= i:
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max_number = i
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return max_number
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||||
|
||||
|
||||
def kadanes(arr: list) -> int:
|
||||
"""
|
||||
If negative_exist() returns 0 than this function will execute
|
||||
else it will return the value return by negative_exist function
|
||||
|
||||
For example: arr = [2, 3, -9, 8, -2]
|
||||
Initially we set value of max_sum to 0 and max_till_element to 0 than when
|
||||
max_sum is less than max_till particular element it will assign that value to
|
||||
max_sum and when value of max_till_sum is less than 0 it will assign 0 to i
|
||||
and after that whole process, return the max_sum
|
||||
So the output for above arr is 8
|
||||
|
||||
>>> kadanes([2, 3, -9, 8, -2])
|
||||
8
|
||||
>>> [kadanes(arr) for arr in test_data]
|
||||
[-2, 19, 1, 0, 0]
|
||||
"""
|
||||
max_sum = negative_exist(arr)
|
||||
if max_sum < 0:
|
||||
return max_sum
|
||||
|
||||
max_sum = 0
|
||||
max_till_element = 0
|
||||
|
||||
for i in arr:
|
||||
max_till_element += i
|
||||
max_sum = max(max_sum, max_till_element)
|
||||
max_till_element = max(max_till_element, 0)
|
||||
return max_sum
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
try:
|
||||
print("Enter integer values sepatated by spaces")
|
||||
arr = [int(x) for x in input().split()]
|
||||
print(f"Maximum subarray sum of {arr} is {kadanes(arr)}")
|
||||
except ValueError:
|
||||
print("Please enter integer values.")
|
||||
|
|
@ -1,21 +0,0 @@
|
|||
from sys import maxsize
|
||||
|
||||
|
||||
def max_sub_array_sum(a: list, size: int = 0):
|
||||
"""
|
||||
>>> max_sub_array_sum([-13, -3, -25, -20, -3, -16, -23, -12, -5, -22, -15, -4, -7])
|
||||
-3
|
||||
"""
|
||||
size = size or len(a)
|
||||
max_so_far = -maxsize - 1
|
||||
max_ending_here = 0
|
||||
for i in range(0, size):
|
||||
max_ending_here = max_ending_here + a[i]
|
||||
max_so_far = max(max_so_far, max_ending_here)
|
||||
max_ending_here = max(max_ending_here, 0)
|
||||
return max_so_far
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
a = [-13, -3, -25, -20, 1, -16, -23, -12, -5, -22, -15, -4, -7]
|
||||
print(("Maximum contiguous sum is", max_sub_array_sum(a, len(a))))
|
||||
|
|
@ -1,32 +0,0 @@
|
|||
from collections.abc import Sequence
|
||||
|
||||
|
||||
def max_subarray_sum(nums: Sequence[int]) -> int:
|
||||
"""Return the maximum possible sum amongst all non - empty subarrays.
|
||||
|
||||
Raises:
|
||||
ValueError: when nums is empty.
|
||||
|
||||
>>> max_subarray_sum([1,2,3,4,-2])
|
||||
10
|
||||
>>> max_subarray_sum([-2,1,-3,4,-1,2,1,-5,4])
|
||||
6
|
||||
"""
|
||||
if not nums:
|
||||
raise ValueError("Input sequence should not be empty")
|
||||
|
||||
curr_max = ans = nums[0]
|
||||
nums_len = len(nums)
|
||||
|
||||
for i in range(1, nums_len):
|
||||
num = nums[i]
|
||||
curr_max = max(curr_max + num, num)
|
||||
ans = max(curr_max, ans)
|
||||
|
||||
return ans
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
n = int(input("Enter number of elements : ").strip())
|
||||
array = list(map(int, input("\nEnter the numbers : ").strip().split()))[:n]
|
||||
print(max_subarray_sum(array))
|
||||
Loading…
Reference in New Issue
Block a user