Algorithm Design: Optimization using recursion.
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The document discusses algorithm design, focusing on optimization through recursion to enhance efficiency. It outlines the characteristics of efficient algorithms, compares different design techniques including loops and recursion, and highlights the importance of functions and recursive design strategies. It concludes by mentioning applications of recursion in computer science and mathematics, such as fractals.
Algorithm Design: Optimization using recursion.
- 1. ALGORITHM DESIGN: OPTIMIZATION USING RECURSION COMPUTER SCIENCE HL NANJING FOREIGN LANGUAGE FERRY KEMPERMAN APRIK 2019
- 2. ALGORITHM DESIGN: THE THINKING PROCESS • Algorithm design is designing a solution for a complex problem that is …… • Non-ambiguous • Executable • Stepwise based • Flow controlled • Optimized for efficiency! • What does this mean?
- 3. OPTIMIZED ALGORITHM DESIGN → EFFICIENCY • An algorithm is considered designed well if • 1. The algorithm works according to specs • 2. The algorithm works as efficient as possible. • Optimization is the process needed to design an efficient algorithm.
- 4. WHAT IS AN EFFICIENT ALGORITHM EXACTLY? • An efficient algorithm need to meet these three requirements: • 1. A minimal number of instructions used to execute the algorithm • 2. The minimal amount of RAM used during execution of the algorithm • 3. The fastest retrieval of data and instructions from RAM to the CPU possible. • An optimized algorithm = an algorithm that is designed in such a way that it meets these requirements. • To put it simply: a non-optimized algorithm works correctly, but is slow and resources heavy (CPU+RAM!).
- 5. TECHNIQUES TO DESIGN EFFICIENT ALGORITHMS • So how to design an efficient algorithm? • You create a flowchart or pseudocode as a design. • How do you know this design is efficient and optimized? • Well, how many instructions did you use? Can it be less? • How does your algorithm store and retrieve data? Variables and their datatypes? Can you make smarter choices here? • Is the flow of your algorithm smartly designed? Do you make use of repetitions (loops) and selection (if/case statements) in such a way that the number of executed statements is minimized
- 6. LET’S COMPARE TWO DESIGNS……… Design 1: Using FOR loop • DECLARE Factorial : INTEGER • DECLARE Terms : INTEGER • Factorial 0 • Terms 0 • INPUT “Specify N to calculate N!” Terms • FOR Counter 1 TO Terms • Factorial Factorial * Counter • ENDFOR • OUTPUT “The factorial “ + Terms + “! = “ + Factorial Design 2: Using WHILE loop • DECLARE Factorial : INTEGER • DECLARE Terms : INTEGER • Factorial 0 • Terms 0 • Counter 0 • INPUT “Specify N to calculate N!” Terms • WHILE Counter <= Terms • Factorial Factorial * Terms • Counter Counter + 1 • ENDWHILE • OUTPUT “The factorial “ + Terms + “! = “ + Factorial
- 7. OR AN EVEN MORE EFFICIENT DESIGN…..USING RECURSION! • FUNCTION CalculateFactorial (N: INTEGER) RETURNS INTEGER • IF N = 1 • THEN RETURN 1 • ELSE RETURN N * CalculateFactorial(N-1) • ENDIF • ENDFUNCTION • DECLARE Terms : INTEGER • DECLARE Factorial : INTEGER • INPUT “Give me a number N to calculate N!” Terms • Factorial CalculateFactorial(Terms) • OUTPUT “The factorial of “ + Terms + “! = “ + Factorial
- 8. LET’S BREAK IT DOWN • FUNCTION CalculateFactorial (N: INTEGER) RETURNS INTEGER • IF N = 1 • THEN RETURN 1 • ELSE RETURN N * CalculateFactorial(N-1) • ENDIF • ENDFUNCTION • On the right I have defined a function. • A function is a top-down design concept that I am going to explain in SL later on. • A function is technically speaking a reusable code and it follows the mathematical definition of a function: • Input → Function (process) → Output • A program calls a function by its name and the code inside the function is then executed. • A function is comprised of four parts: • Name → A unique identifier call the function • Body → The indented code inside the function that will be executed on function call • Input parameter(s) → What values does the function need to process? • Return value → What is the output value of the function? • IMPORTANT: Parameters and return values have FIXED datatypes.
- 9. • FUNCTION CalculateFactorial (N: INTEGER) RETURNS INTEGER • IF N = 1 • THEN RETURN 1 • ELSE RETURN N * CalculateFactorial(N-1) • ENDIF • ENDFUNCTION Datatype of return value Function name Input parameter with datatype integer: it expects an integer! Entire first line: function header Function body
- 10. HOW TO USE THE FUNCTION? THE FUNCTION CALL • A function is simply used by referencing his name with the correct input. • This is called a function call. • Let’s take look at my earlier code: • DECLARE Terms : INTEGER • DECLARE Factorial : INTEGER • INPUT “Give me a number N to calculate N!” Terms • Factorial CalculateFactorial(Terms) • OUTPUT “The factorial of “ + Terms + “! = “ + Factorial The function is called here by this statement! Let’s have a look on how this works….
- 11. Main program • DECLARE Terms : INTEGER • DECLARE Factorial : INTEGER • INPUT “Give me a number N to calculate N!” Terms • Factorial CalculateFactorial(Terms) • OUTPUT “The factorial of “ + Terms + “! = “ + Factorial Function definition • FUNCTION CalculateFactorial (N: INTEGER) RETURNS INTEGER • IF N = 1 • THEN RETURN 1 • ELSE RETURN N * CalculateFactorial(N-1) • ENDIF • ENDFUNCTION Execute the main. User enters 3 to calculate 3!, variable Terms will contain 3. Factorial CalculateFactorial(3) // the function call How to evaluate CalculateFactorial(3)? Execute the body of the function with input parameter 3, so N=3 inside the function! IF N=1, nope, so ELSE part executed….RETURN N * CalculateFactorial(N-1), but N=3 here. So…..RETURN 3 * CalculateFactorial(3-1)……but the function is called again, by itself!!!! This is called recursion! Let’s continue……how to evaluate CalculateFactorial(3-1)? Well, do the same. CalculateFactorial(2)….. N=1, nope, so ELSE part….RETURN N+CalculateFactorial(N-1), with N=2 this time. RETURN 2*CalculateFactorial(1)…. And again…….CalculateFactorial(1), so N=1…then RETURN 1…….Which make 3*2*1 = 6, which is 3!, stored in Factorial variable.
- 13. RECURSION, AN DESIGN STRATEGY • Recursion is a design technique in which you design a function that calls itself. • Every time a function calls itself we call this an incarnation. • Recursive design is not easy! You need to recognize patterns that are recurring to implement recursion. • Recursion can optimize your algorithm’s efficiency significantly. • No need for repetitive code. • Recursion is not looping! What is the difference? • Incarnation depth for recursion is sometimes limited depending on the platform.
- 14. APPLICATIONS OF RECURSIVE DESIGN IN CS AND MATH: FRACTALS • FRACTALS: Patterns that repeat • themselves on a smaller scale each • time the function calls itself…. • A very famous one in math is • The Mandelbrot Set in the picture. • Koch curves, binary trees etc. • Even….CS Art!!!