Maximum Sum Interval

Presented By:
1. Al-Amin Islam Hridoy
ID: 2016-1-60-023
2. Md. Golam Rasul
ID: 2016-1-60-080
3. Nishat Jahan Nishi
ID: 2016-1-60-015

Overview
 Introduction
• What is Maximum Sum Interval?
 Approaches
• Brute Force Algorithm
• Divide and Conquer Algorithm
 Complexity Analysis
 Applications
 Conclusion

Introduction
 A part of an array is an interval which can be one element or two
elements or all of the elements.
{3, -2, 5, -1}
{3, -2, 5} , {-2, 5, -1}
{3, -2} , {-2, 5} , {5, -1}
{3} , {-2} , {5} , {-1}
3 -2 5 -1
3 + (-2) + 5 = 6
Maximum Sum
Interval

History
Ulf Grenander (23 July 1923 – 12 May 2016) was a Swedish statistician and professor
of applied mathematics at Brown University. He first posed this problem in 1977.
Grenander was looking to find the maximum sum of an m * n rectangular region of real
numbers. He designed an algorithm that ran in O(n6) time. His algorithm was too slow
and the problem was too complex to be solved, so he simplified it to one dimension. This
made the problem easier to comprehend and as a result, he was able to solve the original
problem with a faster running time of O(n3).

Approaches
1. Brute Force Algorithm
A[] =
Sum =
3 -2 5 -1
INT_MIN
3 = 3
3 > INT_MIN

Approaches
A[] =
Sum =
3 -2 5 -1
3
-2
3
5
= 3
= -2
= 5
5 > 3

Approaches
A[] =
Sum =
3 -2 5 -1
5
-2
3
5
-1
= 3
= -2
= 5
= -1

Approaches
A[] =
Sum =
3 -2 5 -1
5
-23
-2 5
= 1
= 3
5 -1 = 4

Approaches
A[] =
Sum =
3 -2 5 -1
5
-23 5 = 6
6 > 5

Approaches
A[] =
Sum =
3 -2 5 -1
6
-23
-2
5 = 6
= 25 -1

Approaches
A[] =
Sum =
3 -2 5 -1
6
-23 5 = 5-1

Approaches
A[] =
Sum =
3 -2 5 -1
6 Maximum Sum
Interval

Approaches
max = INT_MIN;
For i = 0 to N-1
For j = i to N-1 {
sum = 0;
For x = i to j
sum = sum + A[x];
If sum > max then
max = sum;
}
Return (max);
𝑶(𝒏 𝟑
)

Approaches
2. Divide and Conquer Algorithm
Divide and Conquer is an algorithm design paradigm based on multi-branched recursion. A divide and
conquer algorithm works by recursively breaking down a problem into two or more sub-problems of
the same or related type, until these become simple enough to be solved directly. The solutions to
the sub-problems are then combined to give a solution to the original problem.

Approaches
3 -2 5 -1
3 -2 5 -1
3 -2 5 -1

Approaches
3 -2 5 -1
3 -2 5 -1
3 -2 5 -1
3 -2 5 -1

Max Left sum =
Max Right sum =
Max Cross sum =
Approaches
3 -2 5 -1
3 -2 5 -1
3 -2 5 -1
3 -2 5 -11
3
-2
1
3

Max Left sum =
Max Right sum =
Max Cross sum =
Approaches
3 -2 5 -1
3 -2 5 -1
3 -2 5 -1
3 -2 5 -11
3
4
5
-1
4
5

Max Left sum =
Max Right sum =
Max Cross sum =
Approaches
3 -2 5 -1
3 -2 5 -1
3 -2 5 -1
3 -2 5 -11
3
4
56
3
5
6
6
Max Cross sum Calculation:
Left sum = max {(-2) , (-2+3)} = 1
Right sum = max {(5) , (5-1)} = 5
Cross sum = Left sum + Right sum = 6

Approaches
3 -2 5 -1
3 -2 5 -1
3 -2 5 -1
3 -2 5 -11
3
4
56
6
Maximum Sum
Interval 6

MSI (A , i , j)
If i == j
Return ( A[i] );
Else
Find MSI ( A , i ,
𝑖+𝑗
2
);
Find MSI ( A ,
𝑖+𝑗
2
+1 , j );
Find MSI that contains both A [
𝑖+𝑗
2
] and A[
𝑖+𝑗
2
+1];
Return maximum of the three sequences found;
Approaches
𝑶(𝒏𝒍𝒐𝒈𝒏)

Complexity Analysis
𝑂(𝑛3
) 𝑂(𝑛2
)
𝑂(𝑛𝑙𝑜𝑔𝑛)
𝑂(𝑛)
Complexity Analysis of the Approaches
Kadane’s
Algorithm

Applications
Genomic sequence analysis
Protein domain sequence analysis

Applications
Computer Vision

Applications
Computer Vision
Data Mining

Conclusion
‘Maximum Sum Interval’ is a very important problem to be solved
to implement very important tasks in our life. The innovative of
this project is to solving this problem using ‘Divide and Conquer’
algorithm. Although ‘Dynamic Programming’ can be a faster
approach to solve it, ‘Divide and Conquer’ is not that a slow
approach. In compare to the ‘Brute Force’ approach we can say,
this is really a very fast approach to solve this problem.
In summary, we learned about ‘Maximum Sum Interval’ problem
and learned how to solve it in different approaches with different
complexities. We also achieved a much better understanding
‘Divide and Conquer’ algorithm than before.

References
• www.wikipedia.org
• www.geeksforgeeks.org
• www.stackoverflow.com
• www.shutterstock.com
• Google Images (public domain)

Maximum Sum Interval

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