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Problem solution · Python

Max Sum of Sub Matrix No Larger Than K

Max Sum of Sub Matrix No Larger Than K: a Python solution using binary search. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Binary search
Source
Kamyu LeetCode Solutions
Length
104 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Binary search

For Max Sum of Sub Matrix No Larger Than K, the implementation exploits a monotonic condition to discard half of the remaining search space after every check.

  1. Identify the ordered answer range or sorted search domain.
  2. Write a predicate whose truth changes only once.
  3. Move the appropriate boundary after each midpoint check and return the final feasible position.

Code notes

  • 104 lines of Python from the credited upstream file max-sum-of-sub-matrix-no-larger-than-k.py.
  • The implementation visibly relies on sequence storage.
  • No explicit loop blocks detected.

Complexity

Multiply the logarithmic number of midpoint checks by the cost of one predicate evaluation.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeMax Sum of Sub Matrix No Larger Than K · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(min(m, n)^2 * max(m, n) * log(max(m, n)))# Space: O(max(m, n)) from bisect import bisect_left, insort class Solution(object):    def maxSumSubmatrix(self, matrix, k):        """        :type matrix: List[List[int]]        :type k: int        :rtype: int        """        if not matrix:            return 0         m = min(len(matrix), len(matrix[0]))        n = max(len(matrix), len(matrix[0]))        result = float("-inf")         for i in xrange(m):            sums = [0] * n            for j in xrange(i, m):                for l in xrange(n):                    sums[l] += matrix[j][l] if m == len(matrix) else matrix[l][j]                 # Find the max subarray no more than K.                accu_sum_set, accu_sum = [0], 0                for sum in sums:                    accu_sum += sum                    it = bisect_left(accu_sum_set, accu_sum - k)  # Time: O(logn)                    if it != len(accu_sum_set):                        result = max(result, accu_sum - accu_sum_set[it])                    insort(accu_sum_set, accu_sum)  # Time: O(n)         return result  # Time:  O(min(m, n)^2 * max(m, n) * log(max(m, n))) ~ O(min(m, n)^2 * max(m, n)^2)# Space: O(max(m, n))class Solution_TLE(object):    def maxSumSubmatrix(self, matrix, k):        """        :type matrix: List[List[int]]        :type k: int        :rtype: int        """        class BST(object):  # not avl, rbtree            def __init__(self, val):                self.val = val                self.left = None                self.right = None             def insert(self, val):  # Time: O(h) = O(logn) ~ O(n)                curr = self                while curr:                    if curr.val >= val:                        if curr.left:                            curr = curr.left                        else:                            curr.left = BST(val)                            return                    else:                        if curr.right:                            curr = curr.right                        else:                            curr.right = BST(val)                            return             def lower_bound(self, val):  # Time: O(h) = O(logn) ~ O(n)                result, curr = None, self                while curr:                    if curr.val >= val:                        result, curr = curr, curr.left                    else:                        curr = curr.right                return result          if not matrix:            return 0         m = min(len(matrix), len(matrix[0]))        n = max(len(matrix), len(matrix[0]))        result = float("-inf")         for i in xrange(m):            sums = [0] * n            for j in xrange(i, m):                for l in xrange(n):                    sums[l] += matrix[j][l] if m == len(matrix) else matrix[l][j]                 # Find the max subarray no more than K.                accu_sum_set = BST(0)                accu_sum = 0                for sum in sums:                    accu_sum += sum                    node = accu_sum_set.lower_bound(accu_sum - k)                    if node:                        result = max(result, accu_sum - node.val)                    accu_sum_set.insert(accu_sum)         return result

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