The 1D version of this is pretty textbook so I got there fast.
Use a 2D prefix sum array where each cell stores the sum of the submatrix from (0,0) to (i,j). Preprocess the matrix in O(m*n) time, then answer each query in O(1) using inclusion-exclusion. Discuss trade-offs like memory usage and potential optimizations for sparse matrices.
Pro tip: Mention that the prefix sum array can be stored as an integer array, and for very large matrices, consider using a 1D array with index arithmetic to improve cache performance. Also, clarify that the matrix is immutable, so no updates are needed, making prefix sums ideal.
Confirm the matrix dimensions, query frequency, and whether the matrix is truly immutable. Ask about memory constraints and expected query patterns.
Explain that a 2D prefix sum array allows O(1) queries after O(m*n) preprocessing. Describe how to build it using dynamic programming: prefix[i][j] = matrix[i][j] + prefix[i-1][j] + prefix[i][j-1] - prefix[i-1][j-1].
Show how to compute the sum for a rectangle using inclusion-exclusion: sum = prefix[r2][c2] - prefix[r1-1][c2] - prefix[r2][c1-1] + prefix[r1-1][c1-1], handling boundaries with padding.
State time complexity: O(m*n) preprocessing, O(1) per query. Space complexity: O(m*n) extra space. Discuss alternatives like sparse representations if the matrix has many zeros.
Address edge cases like empty rectangles, single cells, and full matrix queries. Mention potential optimizations such as using a 1D array for cache efficiency or bit manipulation for binary matrices.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.