A very Brief and precise explanation with the Implementation of the Kadane's algorithm on 1-D and 2-D Array.

1. Maximum Sum Subarray
Presented by Shaheen Kousar

2. Topics
• Introduction
– Maximum sum subarray problem
– WHY MSS?
• Approaches
• Kadane’s Algorithm
– One-Dimensional Array
– Two-Dimensional Array

3. Maximum Sum Problem
• Given array of numbers, find the subarray
(contiguous elements) with the largest
aggregate
-3 7 -12 1 6 -3 5 -2

4. WHY MSS?
• Maximum Sum Subarray problem is widely
used in applications such as pattern
recognition, image processing and data
mining.

5. Approaches
– Brute-Force Algorithm
– Divide And Conquer
– Kadane’s Algorithm

6. One-Dimensional Array
1 -3 2 -5 7 6 -1 -4 11 -23

7. • Let’s Find the Max subarray ending with A [4]=7
7-5
-5
-52
-5
7
-3 2
7
7-3 2
7
1
7
2
4
1
2
1 -3 2 -5 7 6 -1 -4 11 -23

8. Let’s Find the Max subarray ending with A [6]=-1
7-5
-5
-52
-5
7
-3 2
7
7-3 2
7
1
-1
-1
-1
-1
6
6
6
6
6
6
-1
-1
-1
-1
5
12
7
9
6
7

9. Two-Dimensional Array
Index 0 1 2 3 4
0 2 1 -3 -4 5
1 0 6 3 4 1
2 2 -2 -1 4 -5
3 -3 3 1 0 3

10. Let’s find the Rectangle having Maximum sum
in the 2D Matrix
Index 0 1 2 3 4
0 2 1 -3 -4 5
1 0 6 3 4 1
2 2 -2 -1 4 -5
3 -3 3 1 0 3
2
0
2
-3
L = 0 , R = 0, Current Sum = 4 , Max Sum= 4,
Max-left = 0, Max-right = 0,Max-up= 0, Max-down = 2
0
1
2
3

11. Let’s find the Rectangle having Maximum sum
in the 2D Matrix
Index 0 1 2 3 4
0 2 1 -3 -4 5
1 0 6 3 4 1
2 2 -2 -1 4 -5
3 -3 3 1 0 3
3
6
0
0
Max-Sum = 4
L = 0 , R = 1, Current Sum = 9, Max Sum= 9,
Max-left = 0, Max-right = 1,Max-up= 0, Max-down = 1
0
1
2
3
2
0
2
-3
0
1
2
3

12. Let’s find the Rectangle having Maximum sum
in the 2D Matrix
Index 0 1 2 3 4
0 2 1 -3 -4 5
1 0 6 3 4 1
2 2 -2 -1 4 -5
3 -3 3 1 0 3
0
9
-1
1
Max-Sum = 9
• Current sum is not greater than max-sum here!
L = 0 , R = 2, Current Sum = 9, Max Sum= 9,
Max-left = 0, Max-right = 1,Max-up= 0, Max-down = 1
0
1
2
3
3
6
0
0
0
1
2
3

13. Let’s find the Rectangle having Maximum sum
in the 2D Matrix
Index 0 1 2 3 4
0 2 1 -3 -4 5
1 0 6 3 4 1
2 2 -2 -1 4 -5
3 -3 3 1 0 3
-4
13
3
1
Max-Sum = 9
L = 0 , R = 3, Current Sum = 17, Max Sum= 17,
Max-left = 0, Max-right = 1,Max-up= 1, Max-down = 3
0
1
2
3
0
9
-1
1
0
1
2
3

14. Let’s find the Rectangle having Maximum sum
in the 2D Matrix
Index 0 1 2 3 4
0 2 1 -3 -4 5
1 0 6 3 4 1
2 2 -2 -1 4 -5
3 -3 3 1 0 3
-6
13
1
4
Max-Sum = 17
L = 1 , R = 3, Current Sum = 18, Max Sum= 18,
Max-left = 1, Max-right = 3,Max-up= 1, Max-down = 3
0
1
2
3
-2
9
-3
4
0
1
2
3

15. • Iterate for each value of Outer loop for
L=1,2,3,4 and for each outer loop perform
inner loop for R=1,2,3,4.

16. Let’s find the Rectangle having Maximum sum
in the 2D Matrix
Index 0 1 2 3 4
0 2 1 -3 -4 5
1 0 6 3 4 1
2 2 -2 -1 4 -5
3 -3 3 1 0 3
-6
13
1
4
Max-Sum = 18
Max-left = 1, Max-right = 3,Max-up= 1, Max-down = 3
0
1
2
3

17. Time Complexity
Time Complexity for Kadane’s Algorithm is O(N^2).
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