Data Structure and Algorithm II - Lecture III PDF

# Determining the Run-Time of an Algorithm ## LECTURE III ### Lecturer Prof. A-B. Alhassan / Mr. S. Ibrahim ## Table of Contents - Introduction to Algorithm Analysis - Importance of Measuring Running Time - What is Running Time? - Measuring the Running Time of a Program - Growth of Functions -...

# Determining the Run-Time of an Algorithm ## LECTURE III ### Lecturer Prof. A-B. Alhassan / Mr. S. Ibrahim ## Table of Contents - Introduction to Algorithm Analysis - Importance of Measuring Running Time - What is Running Time? - Measuring the Running Time of a Program - Growth of Functions - Big-O Notation - Big-O Notation Examples - Big-Omega Notation - Big-Theta Notation - Further Explanation on Growth Rates - Comparing Different Growth Rates (Graph) - Best, Worst, and Average Cases - Example 1: Linear Search - Example 2: Binary Search - Example 3: Merge Sort - Example 4: Bubble Sort - Mathematical Proof: Big-O of a Quadratic Function - Mathematical Proof: Big-Omega of a Linear Function - Asymptotic Notations Recap - Common Time Complexities - Space Complexity vs. Time Complexity - Example: Optimizing Algorithms ## 1 Introduction to Algorithm Analysis ### What is Algorithm Analysis? Algorithm Analysis refers to the process of evaluating the performance of an algorithm, specifically in terms of its Time Complexity and Space Complexity, as a function of the size of its input. Goal: Evaluate how an algorithm's performance scales with input size. ### Time Complexity Time complexity assesses how the execution time of an algorithm changes as the size of the input grows. It provides a high-level understanding of the algorithm's efficiency by measuring the number of operations required. #### Common time complexities include: - O(1): Constant time (does not depend on input size) - O(log n): Logarithmic time (increases slowly as input grows) - O(n): Linear time (grows directly proportional to the input size) - O(n log n): Log-linear time (often seen in efficient sorting algorithms like mergesort) - O(n²): Quadratic time (commonly found in algorithms with nested loops, such as bubble sort) - O(2^n): Exponential time (typically seen in algorithms to solve problems through brute force, such as the traveling salesman problem) ## 2 Running Time ### What is Running Time? Running time refers to the amount of time an algorithm takes to complete its task, and it is typically expressed as a function of the size of its input (denoted as n). The running time is crucial for understanding how efficiently an algorithm will perform as the size of the input grows. #### Key Aspects of Running Time - Input Size (n): The running time depends on the size of the input. eg(sorting algorithm) - Operations Performed: The running time is directly influenced by the number and type of operations that the algorithm performs. eg (Simple Operations, Complex Operations) - Input Data: The structure or distribution of the input data can affect the running time ## 3 Measuring the Running Time of a Program ### Step-by-Step Flow When analyzing the efficiency of an algorithm, there are two primary approaches to measuring performance: empirical measurement and theoretical measurement. #### Empirical Measurement Involves actually running the algorithm with different inputs and observing how long it takes to execute. It provides real-world data about the algorithm's performance. #### Timing Functions Various programming languages offer functions to measure time directly. These functions allow you to record the execution time of a program or a function in order to empirically determine its performance. #### Example: - Python: time() - C++: clock() - JavaScript: console.time() - Java: System.nanoTime() ## 4 Empirical Measurement ### Steps for Empirical Measurement: - Prepare the Algorithm: Implement the algorithm to be tested. - Run with Different Inputs: Test the algorithm with various input sizes, such as small, medium, and large datasets. - Measure Time: Record the execution time for each test input size using timing functions. - Plot Results: (Optional) Plot the results (e.g., input size vs. time taken) to visually observe how the algorithm's performance scales with input size. ## 5 Advantages of Empirical Measurement #### Advantages - Real Data: It measures the actual time taken by an algorithm for different input sizes. - Practical Insight: It can take into account factors like CPU speed, memory access patterns, and other system-level influences that affect performance. - Useful for Benchmarking: It provides a concrete way to compare different algorithms in a specific context (e.g., which algorithm is faster for a particular input size). ## 6 Disadvantages of Empirical Measurement: #### Disadvantages - Environment Dependent: The results can vary depending on the system's hardware, operating system, or even the load on the system at the time of testing. - Limited Scope: It may not reveal the inherent scalability of an algorithm for very large input sizes (e.g., measuring time for n = 10^6 might not be feasible in a short amount of time). - Difficult to Generalize: The measurements are specific to the test case and may not reflect general performance across different problem domains. ## 7 Theoretical Measurement ### What is Theoretical measurement Theoretical measurement, on the other hand, involves analyzing the number of fundamental operations an algorithm performs, often without running it. This analysis gives an asymptotic understanding of the algorithm's time and space complexity, typically using Big-O notation. #### Fundamental Operations These are the basic operations that contribute to the algorithm's execution time, such as: - Assignments - Comparisons - Arithmetic operations - Memory accesses #### Asymptotic Analysis The goal is to estimate the growth rate of an algorithm's execution time as the input size increases. This is often expressed using Big-O, Big-Ω, and Big-O notations: - O(f(n)): Upper bound (worst-case time complexity). - Ω(f(n)): Lower bound (best-case time complexity). - θ(f(n)): Tight bound (exact time complexity). ## 8 Steps for Theoretical Measurement: ### Step by Step - Identify Fundamental Operations: Break down the algorithm into its basic operations (loops, recursion, etc.). - Count the Operations: Count how many times each operation is executed as a function of input size n. - Estimate Growth Rate: Determine how the number of operations increases as n grows (e.g., linear, quadratic, logarithmic). - Express in Big-O Notation: Express the time complexity of the algorithm using Big-O notation, which focuses on the dominant term (i.e., the one that grows the fastest as n increases). ## 9 Advantages of Theoretical Measurement: ### Benefits - General: Provides a broad understanding of how an algorithm behaves with respect to input size, independent of the specific machine or environment. - Scalable: It is useful for understanding how the algorithm will scale as the input size grows, especially for very large datasets. - Predictable: It offers a way to predict performance without actually running the algorithm. ## 10 Disadvantages of Theoretical Measurement: ### Limitations to Consider - Idealized: It assumes ideal conditions, and may not take into account practical considerations like constant factors, lower-order terms, or specific data characteristics. - Lacks Real-World Context: It does not provide empirical data about how the algorithm will actually perform in a specific system or with specific data. ## 11 Growth of a Function in Algorithm Analysis ### Growth of a Function T(n) The growth of a function refers to how the running time (or space usage) of an algorithm increases as the input size (denoted as n) grows. In algorithm analysis, we express the running time of an algorithm as a function of the input size, often denoted as T(n). where: n = Input Size T(n) = Running Time ## 12 Expressing Growth ### Example The running time function T(n) can often be expressed as a polynomial or a combination of terms, where the terms represent different operations performed by the algorithm. For example: T(n)=3n²+2n+1 This is a polynomial where: - 3n² represents a quadratic term, - 2n is a linear term, and - 1 is a constant. ## 13 Observing Growth as n Increases ### Tech Industry Usage To understand how the function T(n)=3n²+2n+1 grows, let's analyze each term: - 3n²: This term grows quadratically. As n increases, this term will dominate the growth of T(n) because the square of n increases faster than n or any constant term. - 2n: This term grows linearly with n. It increases at a slower rate than n² but still contributes to the overall running time. - Constant 1: This is a constant term and does not change as n increases. It becomes negligible for large n - For n=10: Subtitute n into the Equation (3n²+2n+1) T(10)=3(100)+20+1=321 - For n=100: Subtitute n into the Equation (3n²+2n+1) T(100)=3(100²)+2(100)+1=3(10000)+200+1=30201 ## 14 Asymptotic Analysis and Big-O Notation ### Asymptotic Analysis In algorithm analysis, we are primarily concerned with the asymptotic behavior of T(n) as n grows large. This helps us understand the algorithm's efficiency for large inputs. - The dominant term is the one that grows the fastest as n increases. In this case, the dominant term is 3n², because it grows much faster than 2n or the constant term 1. - The Big-O notation focuses on the dominant term, as lower-order terms and constant factors become irrelevant when n is large. - Thus, we can simplify T(n)=3n²+2n+1 to: T(n)=O(n²) ## 15 Big-O Notation ### Examples - Constant time: O(1) - Linear time: O(n) - Quadratic time: O(n²) - Logarithmic time: O(logn) - Linearithmic time: O(nlogn) ## 16 Time Complexity vs. Space Complexity ### Step-by-Step Proof To verify that a function T(n)=n³+2n² is Ω(n³), we need to show that there exist constants c and n₀ such that: T(n) ≥ c.n³ for all n ≥ n₀ #### Given the function: T(n)=n³+2n² #### We want to show that 7(n) ≥ C.n³ for all sufficiently large n, where C is a constant and n₀ is the smallest value of n where the inequality holds. #### Subtracting c.n³ from both sides of this Equation n³+2n² ≥ C.n³ n³+2n² - C.n³ ≥0 n³(1-c)+2n² ≥0 where C= 1 0+2n² ≥ 0 This is obviously true for all n≥0, since 2n² ≥ 0 ## 17 Time Complexity ### Achieving Success Time complexity refers to the amount of time an algorithm takes to complete as a function of the size of its input, typically denoted as n. It helps us understand how the execution time increases with larger input sizes. Goal: Measure how the running time increases with the size of the input. #### Common Notations: - O(1), - O(n), - O(n²), - O(logn), - O(nlogn), - (2n), etc. ### Why Time Complexity Matters Time complexity gives us insight into the scalability of an algorithm: - A fast algorithm (e.g., O(n)) will run efficiently even with large inputs. - A slow algorithm (e.g., O(n²) or (2n)) might become impractical for large inputs, as its running time increases rapidly with input size. ## Thank You! ### Questions Time