Object-Oriented Design CSC 212

Announcements Ask more questions!  Your fellow students have the same questions (remember, I grade the daily quizzes)  A different explanation can often help clarify the matter Homework #1 on web  Due before class on Thursday  No user I/O required I explicitly state when this is required

Recursion A recursive function using old results to define new values  E.g.,  Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, … Value of new term is sum of two preceding ones

Recursion Recursive definitions have two parts:  Base Case – solved non-recursively (often with constant definitions) n! = 1, if n = 1 First two terms of Fibonacci sequence defined as 1  Recursive Case – solved using the function being defined n! = n * (n-1)!, n > 1

Recursion Continued Using recursion can simplify code: public static int factorial(int num) { if (n == 1) return 1; else return n * factorial(num – 1); } But, recursion can also bring problems  What does factorial(-2) return? Recursion can also be very slow

Another Recursion Example Can have multiple base cases Recursion can also be done multiple times public int fibonacci(int n) { if (n < 0) { return 0; } else if (n < 2) { return 1; } else { return fibonacci(n – 1) + fibonacci(n-2); }

Mutual Recursion Recursion can also occur across methods: public static int factEven(int n) { return n * factOdd(n – 1); } public static int factOdd(int n) { if (n == 1) return 1; else return n * factEven(n – 1); } We will see more complex methods of recursion later

Aside: Logarithms If B * K = N, then log B N = K  B is the base of the logarithm Unless stated, in CSC logarithm base is 2 So log N really means log 2 N  log N = K if and only if 2 K = N log 16 is 4 log 1,024 = 10 log 1,000,000,000 ≈ 30 Logarithmic functions grow very slowly

Logarithm Examples Number of bits required to store a binary number is logarithmic  8 bits stores 256 values log 256 = 8  Maximum value of Java word = 2,147,483,648 log 2,147,483,648 = 31

Logarithm Examples Inventor of chess asked Emperor to be paid like this:  1 grain of rice on the first square, 2 on next… So each square has twice the grain as previous  Function grows exponentially i.e. 2 n, the inverse of a logarithm  Emperors like clever games, but not always the game designers Chess inventor was beheaded

Analysis Techniques Running time is important when coding  Obviously true for real-time systems  But also holds for most systems But not always possible to compare times of all algorithms  Lots of ways to solve a single problem  Many different implementations possible for each solution

Analysis Techniques Want way of examining algorithm that ignores affect of compiler, hardware, etc. Consider algorithm across different inputs, including (especially) worst possible case How to do this analysis without dealing with implementation issues?

The Pseudo-Code Answer Analysis is only for human eyes  Do bother with details needed to make code compile  Instead use "pseudo-code" Pseudo-code isn't real  Name used when writing algorithm in a computer language-like manner

The Pseudo-Code Answer Pseudo-code includes all important code  E.g., Loops, assignments, method calls, etc.  Helps better analyze algorithm Pseudo-code isn't formal – only used to understand algorithm  Ignore unimportant punctuation, formalisms  Write pseudo-code so people can understand and analyze it

Pseudo-code Example What is this function computing? int exampleFunction(int n, n > 0) returnVariable  1 while (n > 0) returnVariable = returnVariable * n n  n – 1 return returnVariable

Algorithm Analysis When comparing algorithms, do not want to measure exact times  Do not want to do all the coding and testing  Instead want back-of-the-envelope measures  Provide quick and easy evaluation and comparison  Implementation often affects execution times, anyway!

Big-Oh Notation Big-Oh computes code complexity  Provides worst-case analysis of performance  Execution time related to code complexity  Enables comparison between algorithms Can use pseudo-code description of algorithm Do not need to implement all approaches Avoids comparing details not related to algorithms  E.g., Compiler, CPU, Users typing speed

Algorithmic Analysis

Algorithm Analysis Approximate time to run a program with n inputs on 1GHz machine: n = 10n = 50n = 100n = 1000n = 10 6 O(n log n)35 ns200 ns700 ns10000 ns20 ms O(n 2 )100 ns2500 ns10000 ns1 ms17 min O(n 5 )0.1 ms0.3 s10.8 s11.6 days3x10 13 years O(2 n )1000 ns13 days4 x years Too long! O(n!)4 ms Too long!

Big-Oh Notation Want correct results for any data set  Only consider details affecting large data sets  Ignore multipliers: O(5n) = O(2n) = O(n) Constant multipliers affected by implementation Coding tricks can often reduce these factors, anyway  Use only dominating term: O(n 5 + n 2 ) = O(n 5 ) Does extra 17 minutes matter after 3x10 13 years?

Analysis of Algorithms Individual statements  E.g., method calls, assignments, arithmetic…  O(1) Complexity Also called “constant time”  Also holds for sequence of statements Provided statements (including loops) execute constant number of times for all input sizes Remember: only want rough estimate – we ignore constant multipliers

Analysis of Algorithms Simple Loops for (int i = 0; i < n; i++) { S }  for statement executed n times  If S is simple sequence of statements (e.g., complexity of O(1)), total complexity is n*O(1) = O(n)

Analysis of Algorithms, cont. Slightly more complicated loops for (int i = 0; i < n; i += 2) { S }  i takes values 0, 2, 4,... until it is larger than n  for loop executes n/2 times  If S executes in O(1) time, loop complexity is n/2 * O(1) = O(n/2) = O(½n) = O(n)

Analysis of Algorithms, cont. Nested Loops for (int i=0; i<n; i++) { for (int j = 0; j < n; j++) { S } }  If S executes in constant time (e.g., O(1)) complexity of j loop is n * O(1) = O(n)  i loop's complexity = n * j loop's complexity = n * O(n) = O(n 2 )

Analysis of Algorithms, cont. Complex Nested Loops for (int m = 0; m < n; m++) { for (int i = 0; i < m; i++) { S } }  Outer loop executes n times  Inner loop executes m times  Assume S executes in constant time

Analysis of Algorithms, cont. Total number of executions is: = n n n -1 = (1 + n-1) + (2 + n-2) ( (n-1)/2 +(n+1)/2) = (n) + (n) + (n) (n) = n * n/2 = 0.5 * n 2 = O(n 2 )

Analysis of Algorithms, cont. Complex Nested Loops  Big-Oh notation matches previous nested loop  Minor improvement in inner loop didn't change big picture  But execution time may be half as much Big-Oh cannot be used to measure small improvements

Analysis of Algorithms, cont. Loops with ‘jumps’ for (int i = 0; i < n; i *= 2) { S }  i equals 1, 2, 4,... until it exceeds n  for loop executes 1 + log 2 n times  If S executes in O(1) time, loop complexity is: (log 2 n + 1) * O(1) = O(log n + 1) = O(log n)

Quick Analysis Tricks Analyzing nested loops  Complexity is product of loops’ complexity.  What is complexity of the following code? for (int i = 0; i < n; i++) for (int j = 0; j < i; j++) for (int k = 0; k < n; k++) for (int m = 0; m < k; m += 2) for (int q = 0; q < j; q++) { S }

Quick Analysis Tricks Analyzing consecutive loops  Complexity will be longest loop's complexity  What is complexity of 5 consecutive loops: 4 loop from 1 - n; 1 loops from 1 - n 2 ?

Experimental Verification Occasionally, may want to verify you have determined the correct complexity To do this verification  Implement the algorithm in a real programming language  Pick initial number of inputs (i.e. n)  Measure time needed to run program

Experimental Verification Verifying Big-Oh analysis  Increase n by a factor of 10 and run it again if logarithmic (O(log n)), takes 3x longer if linear (O(n)), takes 10x longer if O(n log n), takes 13x longer if quadratic (O(n 2 )), takes 100x longer if exponential (O(2 n )), takes over 1000x longer

Limitations of Big-Oh Analysis Constants can make a difference  For n < 8, 3n is larger than n log n  Ignores differences in time needed to access data in main memory versus on a disk Disks can take thousands of times longer to read  Worst case rarely happens Big-Oh notation often overestimates total time

Daily Quiz #1 Finish writing the findMin and findMax methods without using any loops: public void printMinAndMax(int[] a) { if (a.length < 1) return; System.out.println(“Min entry value: ” + Integer.toString(findMin(a,0))); System.out.println(“Max entry value: ” + Integer.toString(findMax(a,0))); } public int findMin(int[] a, int n) { … } public int findMax(int[] a, int n) { … }  Hint: Use n to determine when to stop the recursion

Daily Quiz #2 GIVEN: public abstract class Person {... } public interface Worker {... } public class Student extends Person {... } public class Employee extends Person implements Worker {... } public class StudentEmp extends Student implements Worker {... } WHICH OF THESE ARE ILLEGAL? OK? NEED CASTING? Person p1 = new Person(); Person p2 = new Student(); Person p3 = new Employee(); Person p4 = new StudentEmp(); Worker w2 = p2; Worker w3 = p3; Worker w4 = p4; Student s2 = p2; Student s3 = p3; Student s4 = p4; Employee e3 = p3; Employee e4 = p4; StudentEmp se3 = p3; StudentEmp se4 = p4;

Daily Quiz #2 Not using these slides, write each of the following:  A method executing in O(1) time  A method executing in O(n) time  A method executing in O(n log n) time  A method executing in O(n 2 ) time  A method executing in O(n 4 log n) time