Question

Suppose a metropolitan area is divided into four school restricts: \(1,2,3,4\). The Seate Beard af Etupcation beeppos tack of the number of student transfers fram ane district ted ancether and the shudent transiers within a district. This information is nesorded each year in a \(4 x\) table ss shown here. The entry in row 1 , cohimn \(3 \mid 314\), for exsmple, stiows the number of student tronefers trom district 1 to district 3 for the yeary the entry in now 1 , wien w.column \(1(243)\) shows the number of student transfers within district 1 . \begin{tabular}{l|llll} & 1 & 2 & 3 & 4 \\ \hline 1 & 243 & 187 & 314 & 244 \\ 2 & 215 & 420 & 345 & 172 \\ 3 & 197 & 352 & 385 & 261 \\ 4 & 340 & 135 & 217 & 344 \end{tabular} Suppose there are \(n\) school districts, and the Board of Education maintains an \(n \times n\) table. a. Write a pseudocode algorithm to print the table, that is, to print each of the entries in the table. Write an expression for the number of print statements the algorithm executes. b. Write a pseudocode algorithm to print \(n\) copies of the table, one to give to each of the \(n\) school district supervisors. Write an expression for the number of print statements the algorithm executes. c. What is the order of magnitude of the work done by the algorithm in part \(b\) if the unit of work is printing a table element?

Short answer

a. The table prints with n^2 statements. b. Printing n tables results in n^3 statements. c. The order of magnitude is O(n^3).

Step-by-step solution

Understand the Table Structure

The table provided is a 4x4 matrix, where each entry represents the number of student transfers between or within districts. The entry in row i, column j, denotes transfers from district i to district j. The diagonal entries represent transfers within the same district, while off-diagonal entries indicate transfers between different districts.

Write Pseudocode for Printing the Table

To print the entire table, an algorithm needs to iterate over all rows and columns. A simple pseudocode would look like this: 1. FOR each row i from 1 to n 2. FOR each column j from 1 to n 3. PRINT table[i][j] This pseudocode uses two nested loops, meaning it will execute a print statement for each table entry.

Analyze Number of Print Statements in Table Printing

The algorithm from Step 2 includes two loops, each iterating n times, where n is the number of districts. Therefore, the total number of print statements executed is given by the expression: \( n \cdot n = n^2 \).

Pseudocode for Printing n Copies of the Table

To provide each district with a copy of the table, you need to print the entire table n times. The modified pseudocode will be: 1. FOR each district k from 1 to n 2. FOR each row i from 1 to n 3. FOR each column j from 1 to n 4. PRINT table[i][j] With this approach, the loop responsible for iterating districts ensures the table prints n times.

Analyze Number of Print Statements in Printing Multiple Copies

The pseudocode in Step 4 involves three nested loops: the outer loop (for districts) running n times, and the two inner loops each running n times, resulting in \( n^3 \) print statements for printing n copies of the table.

Determine Order of Magnitude

The number of print operations in the algorithm from Step 5 is \( n^3 \), indicating the algorithm's order of magnitude is O(n^3). Generally, O(n^3) exemplifies cubic complexity, which increases significantly with n.

Key concepts

Algorithm Complexity

Algorithm complexity helps us understand how efficient an algorithm is based on its operations. It is important to evaluate the efficiency, particularly in terms of time and space as the input size grows. Complexity is often expressed using Big O notation, which gives a high-level understanding of the algorithm's performance in the worst-case scenario.
For instance, in the given exercise that involves matrix and student transfers, understanding the complexity can reveal how feasible it is to execute operations as the number of districts increases. We observe that when printing the student transfer matrix for each district (as seen in part b), the number of print statements grows cubically, resulting in an algorithm complexity of \(O(n^3)\). This denotes that the amount of work increases rapidly with the number of districts \(n\).

• \(O(1)\) implies constant time complexity—rarely grows.
• \(O(n)\) indicates linear complexity—grows proportionally to \(n\).
• \(O(n^2)\) is quadratic complexity—common in nested loops.
• \(O(n^3)\) is cubic and grows even faster.

When working with larger datasets, a cubic complexity like \(O(n^3)\) can become prohibitive due to the significant increase in computational resources.

Nested Loops

Nested loops occur when a loop is placed inside another loop. They are often utilized when working with data structures like matrices, where each loop iterates over different dimensions, such as rows and columns. In our exercise, we have a two-level nested loop necessary to traverse a 2D matrix representing student transfers.
Each step of the outer loop triggers the full execution of the inner loop, effectively looping through all elements of a specific row each time. If there are three nested loops, as in part b, it means the code will perform the innermost operations for each combination of the outer and middle loop conditions.

• Outer loop: Corresponds to iterating over districts.
• Middle loop: Traverses rows within a district.
• Inner loop: Goes over columns in each row.

Understanding how nested loops work reassures us of the high computational cost involved, specifically as these add up to compound the complexity.

Matrix Printing

Matrix printing involves iterating over each element of a two-dimensional dataset and displaying it, row by row. In the context of the exercise, a \(4 \times 4\) table is an example where each element represents student transfer data. Visualizing this data requires understanding both rows and columns.
When printing a matrix:

• Ensure you cover each row sequentially before moving to the next row.
• Within each row, traverse through each column element.
• Nested loops facilitate addressing each element with specific index combinations.

For example, the pseudocode provided simply uses two loops for iterating through all rows and columns, but when a third loop is added, the table is printed multiple times, once for each district. Effectively implementing matrix printing logic helps represent data clearly by maintaining structure and order.

Student Transfers

Student transfers refer to the movement of students from one district to another, either inwardly within the same district or outwardly among different districts. This exercise contextualizes student transfers with precise data, well-outlined in a matrix form.
Each entry in the matrix denotes transfers with diagonal elements capturing students who remain within the same district, and off-diagonal elements showing students who move between districts.

• Diagonal Entries: Secure internal transfers within the district.
• Off-Diagonal Entries: Reflect inter-district student movements.

This data is crucial for educational policy planning and resource allocation. An efficient algorithm to print this data helps enhance communication and reporting between various school districts.