Q12Data Structures

Two-dimensional arrays, despite their conceptual matrix-like grid structure, must ultimately be stored linearly within a computer's one-dimensional memory space. This linear flattening is typically achieved via two standard memory mapping algorithms: Row-Major Order (storing data row by sequential row) and Column-Major Order (storing data column by sequential column). Calculating the exact physical memory address of a specific element requires applying defined mathematical formulas utilizing the array's declared bounds, the base memory address, and the byte size of a single element.

Extracting Matrix Parameters

The problem explicitly defines a 2D array structurally bounded as $A[L1\dots U1,L2\dots U2]=A[-15\dots 10,15\dots 40]$. From this declaration, we systematically extract the critical operational parameters: - Row Lower Bound ($L1$) = -15 - Row Upper Bound ($U1$) = 10 - Column Lower Bound ($L2$) = 15 - Column Upper Bound ($U2$) = 40 - Base Starting Memory Address ($Base$) = 100 - Element Size in Bytes ($W$) = 4 bytes - Target Element Row Index ($i$) = 15 - Target Element Column Index ($j$) = 25 We first calculate the absolute dimensions of the matrix. The total number of rows $M=U1-L1+1=10-(-15)+1=26$. The total number of columns $N=U2-L2+1=40-15+1=26$.

Calculation via Row-Major Order

The universal formula for Address Calculation in Row-Major Order is mathematically defined as: $Address=Base+W\times [(i-L1)\times N+(j-L2)]$.

Calculation via Column-Major Order

Conversely, the universal formula for Address Calculation in Column-Major Order is defined as: $Address=Base+W\times [(j-L2)\times M+(i-L1)]$.

Through meticulous substitution and execution, the specific memory address is 3260 under Row-Major constraints and 1260 under Column-Major constraints.