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"""
MSP(max sub-array problem)
wiki(https://en.wikipedia.org/wiki/Maximum_subarray_problem).
1-D:
kadene's algorithm time complexity: O(n)
2-D:
~=O(n^3)
"""
import numpy as np
from itertools import combinations, accumulate
def (A):
"""
1D array MSP.
Parameters
----------
A: list or np.ndarray
Returns
-------
max_so_far: max sum of subarraies.
>>> A = np.array([1, 2, 3, 0, -1, -2, 3, 4, 1, -2, 0])
>>> max_so_far = msp_1D(A)
>>> max_so_far == 11
True
"""
max_ending_here = max_so_far = 0
for x in A:
max_ending_here = max(0, max_ending_here + x)
max_so_far = max(max_so_far, max_ending_here)
return max_so_far
def msp_1D_with_index(A):
"""
solve 1D max sub-array problem.
Parameters
----------
A: list or np.ndarray
Returns
-------
start_ind: int
end_ind: int
max_so_far: max sum of subarraies.
That is to say:
>>> A = np.array([1, 2, 3, 0, -1, -2, 3, 4, 1, -2, 0])
>>> start_ind, end_ind, max_so_far = msp_1D_with_index(A)
>>> start_ind, end_ind, max_so_far
(0, 8, 11)
>>> sum(A[start_ind:end_ind+1]) == max_so_far
True
"""
start_ind, end_ind = 0, 0
max_ending_here, max_so_far = 0, 0
for idx, x in enumerate(A):
max_ending_here1 = max_ending_here + x
if max_ending_here1 > 0:
max_ending_here = max_ending_here1
else:
start_ind = idx + 1
max_ending_here = 0
if max_so_far < max_ending_here:
end_ind = idx
max_so_far = max_ending_here
return start_ind, end_ind, max_so_far
def msp_2D(A):
"""
2D array MSP.
Parameters
----------
A: np.ndarray
Returns
-------
start_ind: int
end_ind: int
max_so_far: max sum of subarraies.
Steps
-----
1. caculate acc matrix(accumulate of A).
2. for all indexes i, j pairs(i<j),
calculating sum of block for a k column(using acc matrix).
do 1-d MSP(get a max sum block under i, j)
3. return the max.
>>> A = np.array([
[0, -2, -7, 0],
[9, 2, -6, 2],
[-4, 1, -4, 1],
[-1, 8, 0, -2]])
>>> m = msp_2D(A)
>>> m == 15
True
"""
A = np.array(A)
acc = np.array([*accumulate(A)])
m = 0
for i, j in combinations(range(A.shape[0]), 2):
m_tmp = msp_1D([
acc[j, k] - acc[i - 1, k] if i != 0 else acc[j, k]
for k in range(A.shape[1])
])
if m_tmp > m:
m = m_tmp
return m
def msp_2D_with_index(A):
"""
2D array MSP.
Parameters
----------
A: np.ndarray
Returns
-------
start_ind: tuple
max sum block left-top position.
end_ind: tuple
max sum block right-bottom position.
max_so_far: max sum of subarraies.
That is to say:
>>> A = np.array([
[0, -2, -7, 0],
[9, 2, -6, 2],
[-4, 1, -4, 1],
[-1, 8, 0, -2]])
>>> start_ind, end_ind, m = msp_2D_with_index(A)
>>> A[start_ind[0]:end_ind[0]+1, start_ind[1]:end_ind[1]+1].sum() == m
True
"""
A = np.array(A)
acc = np.array([*accumulate(A)])
start_ind, end_ind, m = None, None, 0
for i, j in combinations(range(A.shape[0]), 2):
sind, eind, m_tmp = msp_1D_with_index([
acc[j, k] - acc[i - 1, k] if i != 0 else acc[j, k]
for k in range(A.shape[1])
])
if m_tmp > m:
m = m_tmp
start_ind = (i, sind)
end_ind = (j, eind)
return start_ind, end_ind, m
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