1.4 Frequency Count Method

Understanding Algorithm Complexity

Introduction to Algorithm Complexity

• The discussion begins with the concept of algorithm complexity, specifically focusing on frequency count methods to analyze algorithms.

• An example is provided using an array of size five, illustrating how to find the sum of all elements in that array.

Time Complexity Analysis

• The time taken by an algorithm can be assessed by assigning one unit of time for each statement executed, considering repetitions.

• In C language code, a loop executes based on conditions checked multiple times; for instance, checking if i < n occurs n + 1 times when n = 5.

• The execution count leads to a total time function expressed as 2n + 2, simplified to n + 1.

Understanding Loop Execution

• It is noted that the inner statements within loops execute for n times while the loop itself runs for n + 1.

• This results in a time function denoted as f(n) = 2n + 1, indicating linear complexity or order of O(n).

Space Complexity Considerations

• Space complexity is also discussed, where variables used contribute to space requirements. For this case, it totals n + 3, leading to an order of space complexity as well.

Matrix Operations and Their Complexities

Summing Two Matrices

• The next topic covers finding the sum of two square matrices (e.g., dimensions 3 times 3).

• The algorithm's time complexity is analyzed; it involves nested loops executing for each element in both matrices.

Time Complexity Calculation

• The resulting time function from summing two matrices is identified as O(n^2), with the highest degree being squared.

Space Complexity in Matrix Operations

• Variables involved include matrix elements and indices. Thus, space complexity also reflects quadratic growth: O(n^2).

Multiplication of Matrices

Introduction to Matrix Multiplication

• A new algorithm focuses on multiplying two matrices which involves three nested loops.

Detailed Time Analysis

Time Complexity and Space Complexity Analysis

Time Function Derivation

• The time function f(n) is expressed as a polynomial, specifically C + Q2 + U + 3n^2 , indicating the complexity of the algorithm.

• The degree of the polynomial is determined to be O(n^3) , suggesting that the algorithm's time complexity grows cubically with respect to input size.

Space Complexity Discussion

• The space complexity is analyzed in terms of matrix dimensions, leading to an expression involving n^2 .