2 d kadane

2D Kadane

To find out the maximum sum sub-matrix in a given matrix

Objective

Given a matrix of numbers, we need to find a sub-matrix the sum of whose elements is the maximum of sums of sub-matrices of all possible sizes

Example

9 8 5

6 -1 2

-3 -4 -1

Possible Sub-matrices

1X1

8

6 2

-3 -4

-1

-1

9 5

1X2

9 8

6 -1

-4 -1-3 -4

-1 2

8 5

2X1

5

2

9

6

8

-1

-1

-4

2

-1

6

-3

1X3

-3 -4 -1

6 -1 2

9 8 5

3X1

9

6

-3

8

-1

-4

5

2

-1

2X2

6 -1

-3 -4

-1 2

-4 -1

8 5

-1 2

9 8

6 -1

2X3

6 -1 2

-3 -4 -1

9 8 5

6 -1 2

3X2

9 8

6 -1

-3 -4

8 5

-1 2

-4 -1

3X3

9 8 5

6 -1 2

-3 -4 -1

Of all these sub-matrices

Maximum sum is of the sub-matrix

Its sum is 29

9 8 5

6 -1 2

2D Kadane Intuition

Let us take the same sub-matrix

We will work through it step by step

This is the given matrix

9 8 5

6 -1 2

-3 -4 -1

STEP 1

For each row calculate the sum of elements along the columns from row 1 to row i

For row 1 to 1

The sum is:

9 8 5

For row 1 to 2

Add the elements of the 2 rows

9 8 5

6 -1 2

9+6=15 8-1=7 5+2=7

For row 1 to 3

Add the elements of the three rows

9 8 5

6 -1 2

-3 -4 -1

9+6-3=12 8-1-4=3 5+2-1=6

We have the following 3 rows

9 8 5

12 3 6

15 7 7

STEP 2

For each pair of rows i & j calculate the sum of elements along the columns from row i to row j

Sum of elements from row i to j = (sum from row 1 to j) – (sum from 1 to i-1)

Row 1 to 1 Row 2 to 2

Row 1 to 2 Row 2 to 3

Row 1 to 3 Row 3 to 3

9 8 5

12 3 6

15 7 7

6 -1 2

-3 -4 -1

3 -5 1

STEP 3

To each of the pairwise row sums found in the previous step apply 1D Kadane algorithm to find maximum contiguous sum in each pair-wise row sum. The maximum of all such contiguous sums found is the answer.

Row 1 to 1 – max_sum = 22 for col 1 to 3 Row 2 to 2 – max_sum = 7 for col 1 to 3

Row 1 to 2 – max_sum = 29 for col 1 to 3 Row 2 to 3 – max_sum = 3 for col 1 to 1

Row 1 to 3 – max_sum = 21 for col 1 to 3 Row 3 to 3 – max_sum = -1 for col 3 to 3

9 8 5

12 3 6

15 7 7

6 -1 2

-3 -4 -1

3 -5 1

Maximum sum is for all 3 columns rows 1 and 2

i.e. the max sum sub-matrix is

Maximum sum = 29

9 8 5

6 -1 2

Algorithm

Consider a matrix of dimensions M X N1. For each row i calculate the sum of elements

along the columns from row 1 to row i2. For each pair of rows i and j, calculate the sum

of elements from along the columns from row ito row j. The sums calculated in step 1 are helpful. We get C(M,2) rows

3. For each of the pair-wise row sum calculated in step 2, calculate the maximum sum using 1D Kadane algo.

4. The maximum over all the sum found in step 3 is the required answer.

Complexity

For step 1 of algo: O(MN)

For step 2 of algo: O(NM2)

For step 3 of algo: O(NM2)

Thus overall complexity O(NM2)

Considering N and M to be of the same order complexity is O(N3)