Maximum Sum

Background

A problem that is simple to solve in one dimension is often much more difficult to solve in more than one dimension. Consider satisfying a boolean expression in conjunctive normal form in which each conjunct consists of exactly 3 disjuncts. This problem (3-SAT) is NP-complete. The problem 2-SAT is solved quite efficiently, however. In contrast, some problems belong to the same complexity class regardless of the dimensionality of the problem.

 

The Problem

Given a 2-dimensional array of positive and negative integers, find the sub-rectangle with the largest sum. The sum of a rectangle is the sum of all the elements in that rectangle. In this problem the sub-rectangle with the largest sum is referred to as the maximal sub-rectangle. A sub-rectangle is any contiguous sub-array of size or greater located within the whole array. As an example, the maximal sub-rectangle of the array:

is in the lower-left-hand corner:

and has the sum of 15.

 

Input and Output

The input consists of an array of integers. The input begins with a single positive integer N on a line by itself indicating the size of the square two dimensional array. This is followed by integers separated by white-space (newlines and spaces). These integers make up the array in row-major order (i.e., all numbers on the first row, left-to-right, then all numbers on the second row, left-to-right, etc.). N may be as large as 100. The numbers in the array will be in the range [-127, 127].

The output is the sum of the maximal sub-rectangle.

 

Sample Input

4

0 -2 -7  0 9  2 -6  2

-4  1 -4  1 -1

8  0 -2

 

Sample Output

15

 

using System;

using System.Collections.Generic;

using System.Linq;

using System.Text;

 

namespace MaxSum

{

    class Program

    {

        static void Main(string[] args)

        {

            //Array initialization

            int[,] matrix = new int[,] {

                                         {0, -2, -7, 0},

                                         {9, 2, -6, 2},

                                         {-4, 1, -4, 1},

                                         {-1, 8, 0, -2}

                                       };

            int rows = matrix.GetLength(0);

            int cols = matrix.GetLength(1);

            int flag = 0, sum = 0, row = 0, col = 0, rowSize = 0, colSize = 0;

            for(int i = 0; i < rows; i++)

            {

                for(int j = 0; j < cols; j++)

                {

                    //i and j will determine the varying window size here

                    for (int p = 0; p < rows; p++)

                    {

                        for (int q = 0; q < cols; q++)

                        {

                            //p and q are the matrix elements

                            if (p + i < rows && q + j < cols)

                            {

                                //Call the sum function here

                                if (flag == 0)

                                {

                                    sum = Sum(matrix, p, q, i, j);

                                    rowSize = i;

                                    colSize = j;

                                    row = p;

                                    col = q;

                                    flag = 1;

                                }

                                else

                                {

                                    if (sum < Sum(matrix, p, q, i, j))

                                    {

                                        sum = Sum(matrix, p, q, i, j);

                                        rowSize = i;

                                        colSize = j;

                                        row = p;

                                        col = q;

                                    }

                                }

                            }

                        }

                    }

                }

            }

            Console.WriteLine("Max Sum = " + sum);

            Console.WriteLine("Maximal Sub Matrix is:");

            for (int i = row; i <= row + rowSize; i++)

            {

                for (int j = col; j <= col + colSize; j++)

                {

                    Console.Write(matrix[i, j]);

                    Console.Write(" ");

                }

                Console.WriteLine();

            }

        }

        public static int Sum(int[,] matrix, int row, int col, int rowSize, int colSize)

        {

            int sum = 0;

            for (int i = row; i <= row + rowSize; i++)

            {

                for (int j = col; j <= col + colSize; j++)

                {

                    sum = sum + matrix[i, j];

                }

            }

            return sum;

        }

    }

}

 

Max Sum = 15
Maximal Sub Matrix is:
9 2
-4 1
-1 8
Press any key to continue . . .

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