1. Algorithms Java Methods A & AB Object-Oriented Programming and Data Structures Maria Litvin ● Gary Litvin Copyright © 2006 by Maria Litvin, Gary Litvin, and Skylight Publishing. All rights reserved. chapter  4

2. 4-2 Objectives: Understand general properties of algorithms Get familiar with pseudocode and flowcharts Learn about iterations and recursion Learn about working with lists Learn basic facts about OOP

3. 4-3 Define Algorithm... Hard to define formally A more or less compact, general, and abstract step-by-step recipe that describes how to perform a certain task or solve a certain problem. Examples:  Long division  Euclid’s Algorithm for finding the greatest common factor (circa 300 BC)  Binary Search (guess-the-number game)

4. 4-4 Tools for Describing Algorithms Pseudocode  A sequence of statements, more precise notation  Not a programming language, no formal syntax Flowcharts  Graphical representation of control flow

5. 4-5 Example: calculate 1 2 + 2 2 +... + n 2 Pseudocode Input: n sum  0 i  1 Repeat the following three steps while i  n: sq  i * i sum  sum + sq i  i + 1 Output: sum A  BA  B Set A to the value of B

6. 4-6 Example (cont’d) Flowchart n sum  0 i  1 i  n ? sq  i * i sum  sum + sq i  i + 1 sum No Yes Input / output Processing step Decision

7. 4-7 Another Example: 1. Start at pos0, facing dir0 2. If wall in front, turn 90º clockwise else go forward 3. If not back to initial position / direction proceed to Step 2 else stop dir  dir + 90º pos = pos0 and dir = dir0? Yes No Yes No pos  pos0 dir  dir0 Step forward Input: pos0, dir0 Stop Wall in front?

8. 4-8 Variables Algorithms usually work with variables A variable is a “named container” A variable is like a slate on which a value can be written and later erased and replaced with another value sum  sum + sq sum

9. 4-9 Properties of Algorithms Compactness: an algorithm can use iterations or recursion to repeat the same steps multiple times Generality: the same algorithm applies to any “size” of task or any input values Abstractness: an algorithm does not depend on a particular computer language or platform (although it may depend on the general computing model)

10. 4-10 Properties (cont’d) Input: n sum  0 i  1 Repeat the following three steps while i  n: sq  i * i sum  sum + sq i  i + 1 Output: sum Compact: the same length regardless of n, thanks to iterations  the algorithm repeats the same instructions many times, but with different values of the variables (The “running time” depends on n, of course) General: works for any n

11. 4-11 Properties (cont’d) function addSquares(n : integer) : integer; var i, sum : integer; begin sum := 0; for i := 1 to n do begin sum := sum + i * i end; addSquares := sum; end; Abstract: Pascal C/C++ Java public class MyMath { public static int addSquares(int n) { int sum = 0; for (int i = 1; i <= n; i++) sum += i * i; return sum; } int addSquares(int n) { int i, sum = 0; for (i = 1; i <= n; i++) sum += i * i; return sum; }

12. 4-12 Iterations Repeat the same sequence of instructions multiple times Start with initial values of variables Values of some of the variables change in each cycle Stop when the tested condition becomes false Supported by high-level programming languages

13. 4-13 Iterations: while Loop in Java For example: while ( ) {... // do something } while (i <= n) { sum += i * i; // add i * i to sum i++; // increment i by 1 }

14. 4-14 Iterations: for Loop in Java for ( ; ; ) {... // do something } for ( int i = 1; i <= n; i++) { sum += i * i; // add i * i to sum } For example: Increment i by 1

15. 4-15 Recursion A recursive solution describes a procedure for a particular task in terms of applying the same procedure to a similar but smaller task. Must have a base case when the task is so simple that no recursion is needed. Recursive calls must eventually converge to a base case.

16. 4-16 Recursion: an Example Procedure: Climb steps Base case: if no steps to climb  stop Recursive case: more steps to climb  1. Step up one step 2. Climb steps

17. 4-17 Recursive Methods A recursive method calls itself Must have a base case (can be implicit) Example: public class MyMath { public static int addSquares (int n) { if (n == 0) // if n is equal to 0 return 0; else return addSquares (n - 1) + n * n; } Base case Calls itself (with a smaller value of the parameter)

18. 4-18 Recursion: How Does it Work Implemented on a computer as a form of iterations, but hidden from the programmer Assisted by the system stack addSquares (4) addSquares (3) addSquares (2) addSquares (1) addSquares (0) 3 2 0 1 0 1 5 14 Base case 4 30

19. 4-19 Recursion (cont’d) Recursion is especially useful for dealing with nested structures or branching processes

20. 4-20 Recursion (cont’d) totalBytes (folder) { count  0 for each item X in folder { if X is a file count  count + the number of bytes in X else (if X is a folder) count  count + totalBytes(X) } return count } Base case (This is pseudocode, not Java!)

21. 4-21 Euclid’s Algorithm Finds the greatest common factor (GCF) of two positive integers Circa 300 BC

22. 4-22 Euclid’s Algorithm (cont’d) a, b a = b?a > b? a  a - bb  b - a a Yes No

23. 4-23 Euclid’s Algorithm (cont’d) With iterations With recursion public static int gcf (int a, int b) { while (a != b) // a not equal to b { if (a > b) a -= b; // subtract b from a else b -= a; // subtract a from b } return a; } public static int gcf (int a, int b) { if (a == b) // if a equals b return a; if (a > b) return gcf( a - b, b); else // if a < b return gcf(a, b - a); } Base case

24. 4-24 Working with Lists A list is a data structure in which the items are numbered We know how to get to the i-th item We know how to get from one item to the next quickly Amy 5 Ben 3 Cal 2 Dan 0 Eve 6 In Java, the elements are counted from 0 Fay 1 Guy 4

25. 4-25 List Traversal  Start at the first element  While more elements remain process the next element for (int i = 0; i < list.length; i++) System.out.println (list [ i ]); for (String word : list) System.out.println (word); Increment i by 1 Java’s “for each” loop (a.k.a. enhanced for loop)

26. 4-26 Sequential Search The number of comparisons is proportional to n, where n is the length of the list Amy 5 Ben 3 Cal 2 Dan 0 Eve 6 Fay 1 Guy 4 Amy? Amy!

27. 4-27 Binary Search “Divide and conquer” algorithm The elements of the list must be arranged in ascending (or descending) order The target value is always compared with the middle element of the remaining search range

28. 4-28 Binary Search (cont’d) Fay 5 Dan 3 Cal 2 Amy 0 Guy 6 Ben 1 Eve 4 Eve? Fay 5 Dan 3 Cal 2 Amy 0 Guy 6 Ben 1 Eve 4 Fay 5 Dan 3 Cal 2 Amy 0 Guy 6 Ben 1 Eve 4 Eve?Eve!

29. 4-29 Search: Sequential Binary The list can be in random order The number of comparisons is proportional to n For a list of 1,000,000 elements takes, on average, 500,000 comparisons The list must be sorted (e.g., in ascending order) The number of comparisons is proportional to log 2 n For a list of 1,000,000 elements takes, on average, only 20 comparisons

30. 4-30 Review: Why algorithms often use iterations? How is pseudocode different from Java code? Name three basic shapes used in flowcharts. Explain how variables are used in iterations. Which Java statements can be used to express iterations?

31. 4-31 Review (cont’d): What is called a base case in recursion? Suppose we define “word” as a sequence of letters. Turn this into a recursive definition. When does Binary Search apply? How many comparisons, on average, are needed to find one of the values in a list of 1000 elements using Sequential Search? Binary Search?