Beats-%97 ** with_explanations_and_comments ** easy_to_understand
Intuition\nThe general approach is straightforward:\n\nIterate through the 2D matrix. For each element, check if a 3x3 region can be formed.\nIf a valid region 
# Intuition\nThe general approach is straightforward:\n\nIterate through the 2D matrix. For each element, check if a 3x3 region can be formed.\nIf a valid region is found, calculate its sum and store this information, along with the number of regions the current node belongs to, in a separate data structure. For example, when a region is identified at <i,j>, we track its region sum divided by 9 for all elements within the region, from [i,j] to [i+2,j+2].\nIf no valid region is found, move on to the next element.\n\n# Approach\nBuilding on our previous discussion, we can improve the process by precomputing the sum of elements in a region and reusing it when the region is detected. This can be done using a method similar to prefix sum for 2D arrays.\n\nIn brief, we define an additional 2D matrix with dimensions (rows+1, cols+1), where rows and cols represent the number of rows and columns in the input matrix. The prefix sum is calculated as follows: presum[i][j] = presum[i-1][j] + presum[i][j-1] + image[i-1][j-1] - presum[i-1][j-1]. The reason for subtracting the overlapping part is that it gets counted twice in both presum[i][j-1] and presum[i-1][j], so we need to adjust for it.\n\nOnce the prefix sum matrix is computed, we can quickly calculate the sum of any region from (row1, col1) to (row2, col2) using the calculate_sum_from_to() method described in the solution.\n\n# Complexity\n- Time complexity:\n\npresum calculation -> O(mn)\nregion detection -> O(mn (9+9))\nres calculation -> -> O(mn)\n\n- Space complexity:\n- \npresum extra space -> O(mn)\nres extra space -> O(mn)\nregion sum and frequency space -> -> O(m*n)\n# Code\n```cpp []\nclass Solution {\npublic:\n    vector<vector<int>> pre_sum;\n    int rows;\n    int cols;\n    vector<vector<int>> resultGrid(vector<vector<int>>& image, int t) {\n        rows = image.size();\n        cols = image[0].size();\n\n        vector<vector<pair<double, int> >> v(rows, vector<pair<double, int>>(cols));\n        vector<vector<int>> res(rows, vector<int>(cols));\n\n        pre_sum = vector<vector<int>> (rows+1, vector<int> (cols+1,0));\n\n        calculate_pre_sum(image);\n       \n       for(int i=0; i< rows-2; ++i) {\n            for (int j=0; j<cols-2; ++j) {\n                int temp_sum =0;\n                bool region = true;\n\n                for(int pp = i; pp<i+3; ++pp) {\n                    int qq =j;\n                    for(; qq<j+3; ++qq) {\n                        /* \n                        To detect a region, we compare each node with its right and bottom neighbors. For example, if we are detecting a region with the top-left corner at (i=2, j=4), when checking the neighbors of (i=3, j=5), we only compare it with (i=4, j=5) and (i=3, j=6). The left and top neighbors have already been checked in previous comparisons.\n                        */\n                        if ( (pp <i+2) && (abs(image[pp][qq] - image[pp+1][qq]) > t ) ||\n                             (qq <j+2) && (abs(image[pp][qq] - image[pp][qq+1]) >t)\n                            ) { \n                                region = false;\n                                break; \n                        }\n\n                    }\n                    if (!region) {\n                        \n                        break;\n                    }\n                }\n\n                if(region) {\n                    /* If a region is detected, update the sum and frequency for all the nodes within that region. I initially considered optimizing this by only tracking the top-left corner indices of the region, but then realized that the number of regions a node can belong to may vary, so this approach wouldn\'t work consistently.\n                    */\n                    temp_sum = calculate_sum_from_to(i, j, i+2, j+2);\n                    for (int k = i; k < i + 3 && k < rows; k++) {\n                            for (int l = j; l < j + 3 && l < cols; l++) {\n                                v[k][l].first += temp_sum / 9;\n                                v[k][l].second++;\n                            }\n                    }\n                }\n            }\n        }\n\n        for (int i = 0; i < rows; i++) {\n            for (int j = 0; j < cols; j++) {\n                res[i][j] = v[i][j].second == 0 ? image[i][j] : v[i][j].first / v[i][j].second;\n                }\n        }\n        return res;\n    }\n\n    void calculate_pre_sum(vector<vector<int>>& image) {\n        for(int i=1; i<=rows; ++i) {\n            for(int j=1; j<=cols ; ++j) {\n                pre_sum[i][j] = image [i-1][j-1] //note pre_sum size is larger\n                                + pre_sum[i-1][j] +  pre_sum[i][j-1] - pre_sum[i-1][j-1];\n\n            }\n        }\n    }\n\n    int calculate_sum_from_to (int row1, int col1, int row2, int col2) {\n        return pre_sum[row2+1][col2+1] - pre_sum[row2+1][col1] - pre_sum[row1][col2+1] + pre_sum[row1][col1];\n    }\n\n\n    void print(vector<vector<int>> &m) {\n        for(auto e: m) {\n            for (auto ee: e) {\n                cout<<ee<<" ";\n            }\n            cout<<endl;\n        }\n    }\n};\n```
['Matrix', 'Prefix Sum', 'C++']