Analyzing an Algorithm Computing the Order of Magnitude Big O Notation

Measuring the Performance of an Algorithm Order of magnitude describes the growth/running time of an algorithmic process. Example: As the size of the array gets larger, what is the increase in processing time? What happens when we double the data? The running time of an algorithm is the number of primitive operations executed. Big-O notation is used to describe the worst-case scenario of an algorithm.

Computing the runtime of an algorithm It's hard to compute the exact runtime of an algorithm. Runtime depends on: The speed of the processor Additional tasks being performed and executed in the background. Instead of talking about the runtime directly, we use big O notation to show how quickly the runtime grows.

Measuring how quickly runtime grows It is only possible to use units of time, such as seconds, when the runtime is measured directly. Algorithms are measured by how quickly the runtime grows, not speed time. Order of magnitude uses the size of the data. Data can become arbitrarily large. An algorithm may seem inefficient when the data size is small but becomes more efficient when the data size increases. Big O notation examines the worst case scenario.

Example: What is the runtime of this algorithm? int sum = 0; int k = 0; while (k < n){ sum += a[k]; k++; } Task 1: Determine the number of operations processed by this algorithm.

Counting the operations in an algorithm int sum = 0; int k = 0; while (k < n){ sum += a[k]; k++; } Initialization: 2 operations Loop Statement: n + 1 iterations of two operations 2 + 2 * (n + 1)

Count only basic operations in an algorithm int sum = 0; int k = 0; while (k < n){ sum += a[k]; k++; } Initialization should be counted as a single basic operation. Loop Statement: Each loop iteration represents a single basic operation 1 + n

Simplify the further O(n) int sum = 0; int k = 0; while (k < n){ Rule 1: For larger data sizes, constants become insignificant. Rule 2: Only the most dominant term of the expression is useful. int sum = 0; int k = 0; while (k < n){ sum += a[k]; k++; } 1 + n Final Answer: O(n)

Practice : Examine the efficiency expression of an algorithm Practice : Examine the efficiency expression of an algorithm. Determine the Big O notation: Ans : 1000n O(n) 8n log2 n + 14n O(n log2 n) n3 – n2 – 3 ? 4n2 + 7n + 2 ? 3 log2 n + 1 ?

Answers to Practice Problems : Examine the efficiency expression of an algorithm. Determine the Big O notation: 1000n O(n) 8n log2 n + 14n O(n log2 n) n3 – n2 – 3 O (n3) 4n2 + 7n + 2 O(n2) 3 log2 n + 1 O(log2 n)

Searching Efficiency for a Linear Search Describe the worst case scenario when asked to perform a linear search. How many comparisons are required? How many comparisons are required for performing a linear search in an average situation? What is the order of magnitude of efficiency for both the worst and the average case for the linear search?

Searching Efficiency for a Binary Search Describe the worst case scenario when asked to perform a binary search. How many comparisons are required? How many comparisons are required for performing a binary search in an average situation? What is the order of magnitude of efficiency for both the worst and the average case for the binary search?