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This lecture covers a quick review of recursive and iterative quicksort algorithms, and an introduction to Matlab programming language.

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  1. Announcements • P3 due tomorrow • P4 handed out today -- Sorting and Matlab • Prelim leftovers -- pick them up after class • Grades will be done on a curve Lecture 14

  2. Today’s Topics • Quick review of recursive quicksort • Quick look at iterative quicksort • Matlab introduction Lecture 14

  3. Review of quicksort • Given an array b[0..k] • if b.length = 1, done • if b.length = 2, then swap if necessary, done. • partition the array around b[0], suppose value in b[0] ends up in position j • quicksort b[0..j-1] • quicksort b[j+1..k] • This version of quicksort is recursive (calls itself) Lecture 14

  4. h j k b <= x x >x Iterative quicksort (no recursion) • After partitioning the array, it looks like: • There are now two sections to sort, b[h..j-1] and b[j+1..k], and while one is being sorted, it must be remembered to sort the other. • Sorting b[h..j-1] will result in partitioning and the creation of two other sections to sort; these must also be “remembered”. Lecture 14

  5. Create a class Bounds // An instance represents the bound f and l of an // array section b[f..l] (for some array) publicclass Bounds { publicint f; public int l; // Constructor: instance with f=fp and l=lp public Bounds(int fp, int lp) {f= fp; l= lp;} } Lecture 14

  6. public void Quicksort(int [ ] b, int h, int k) { Bounds c [ ] = new Bounds [ k+1-h]; System.out.println(k); System.out.println(h); c[0]= new Bounds(h,k); int i= 1; // inv: b[h..k] is sorted iff all its subsegments // defined by elements of c[0..i-1] are sorted while (i > 0) { i = i -1; int f= c[i].f; int l= c[i].l; // Process segment b[f..l] Lecture 14

  7. if (l-f==1) {// b[f..l] has two elements if (b[f] > b[l]) { // Swap b[f] and b[l] int t= b[f]; b[f]= b[l]; b[l]= t; }} else if (l-f>1) { //b[f..l] has > 2 elements // Add bounds of b[f..j-1] and b[j+1..k] to c int j= partition(b,f,l); c[i]= new Bounds(f,j-1); i= i+1; c[i]= new Bounds(j+1,l); i= i+1;}}} Lecture 14

  8. Size of array c How big can array c get? Let b[h..k] have n values, and let b be already sorted. At each step, b[f..j-1] would be empty and b[j+1..k]would have all but one of the elements. After 3 loop iterations, we would have c[0] represents a segment of 0 elements c[1] represents a segment of 0 elements c[2] represents a segment of 0 elements c[3] represents a segment of n-3 elements In worst case array c needs almost n array elements! Lecture 14

  9. How to fix this. . . Put largest of the two segments b[f..j-1], b[j+1..k] on c first, then the smaller. Then, we can show that that if c[0] represents a segment of m elements, c looks like c[0] represents m elements c[1] represents < m/2 elements c[2] represents < m/4 elements c[3] represents < m/8 elements … c[i-1] represents m/ 2 i-1 elements c has at most 1+ log m elements So c has at most 1 + log n elements. Much better! Lecture 14

  10. Changes to algorithm • Changes to ensure that array c never gets bigger than log (l-f). If the array has 250 elements, array c need have no more than 50 elements. 1. Change allocation of c to Bounds c [ ] = new Bounds [ 50]; 2. Change implementation of “Add bounds …” to the following: Lecture 14

  11. Code Modifications // Add bounds of b[f..j-1] and b[j+1..k] to c // --put larger segment on first if (j-f > l-j) { c[i]= new Bounds (f,j-1); i= i+1; c[i+1]= new Bounds(j+1..k); i= i+1; } else { c[i]= new Bounds (j+1..k); i= i+1; c[i]= new Bounds(f,j-1); i= i+1; } } Lecture 14

  12. Recursive vs. Iterative • Which do you like better? • Which is easier to understand? • In P4, implement whichever you like • Choose better variable names that what you’ve seen in class • More on recursion later. . . Lecture 14

  13. Matlab Matlab Matlab • Yet another programming language • Think of it as a graphical calculator arrays and scalars • Pratap’s Getting Started with Matlab is on reserve in the Engineering library • Matlab is also available in Carpenter and Upson labs -- ask consultants how to access it Lecture 14

  14. Why Matlab? • The premier package for numerical computing, particularly arrays (matrices). Widely used in science/engineering. • Provides high-level interface to best-of-class numerical methods. Problem-solving without lower-level programming details. • Powerful graphics and visualization tools. • has variables, loops, conditionals, functions • but much array/matrix computation can be done directly without loops. Lecture 14

  15. The Matlab environment • Enter expressions or commands in the console window. Commands are executed immediately. An expression is evaluated and its value is immediately displayed. • Can define command scripts and new functions (future lecture). • Most important feature: help command. Type help to get a general list of available topics, or help topicfor information on topic. • Type more on in the console window to pause output after each full screen. Hit space to continue. • Anything following a % is ignored. Use it to include notes or comments in a session. • To leave Matlab, type quit Lecture 14

  16. Some expressions and operators • The usual basic arithmetic operations are provided (+, –, *, /, and ^). Everything is floating-point, although integer values are displayed without a fractional part. • 9/10 is 0.9 in Matlab • ^ is exponentiation (2 ^ 10) • Logical operations treat 1 as the value true and 0 as false. • Comparisons: <, <=, ==, ~=, >=, > • Logical Operators: &, |, ~ • Examples: 3 * 4 + 5 3 * 4 + 5 / 2 3 * (4 + 5) / 2 (3 < 2 ^ 2) & ~ (3 < 2) (3 < 2 ^ 2) | ~ (3 < 2) Lecture 14

  17. Variables • Variables are created when they are first assigned a value. x = 17 y = 3 * x • All variables are global (for now). • A variable exists from the time it is created until you quit Matlab. • Variable names are case-sensitive. Entering a = 17 A = 42 creates two separate variables. • Several variables containing useful constants are already defined pi 3.14159… Inf i, j sqrt(–1) NaN 0/0 Lecture 14

  18. Functions • Matlab provides a rich collections of standard functions. • Trigonometry: sin, cos, tan, cot, asin, acos, atan, atan2… • Exponential: exp, log, log10, sqrt • Complex: real, imag, abs, … • Rounding: floor, ceil, round, rem, sign • Specialized: bessel, gamma, erf, log2, rat, … Lecture 14

  19. Examples of function use x = 3; y = 4; d = sqrt(x ^ 2 + y ^ 2) sin(pi / 2) exp(1) sqrt(–1) Lecture 14

  20. Input and Output in Matlab • The value of a Matlab expression or statement is displayed immediately unless it is followed by a semicolon. z = x ^ 2 w = x ^ 3; • To change the precision of the output type format long format short Other formats are also available. (Enter help format for details.) • You can edit and reenter previous console input. Use the up- and down-arrow keys to access previous entries. Lecture 14

  21. Sample console window >> z = 1; >> x = 3; y = 4; >> d = sqrt(x^2 + y^2) d = 5 >> x = exp(1) x = 2.7183 >> y = sin(p1/2) ??? Undefined function or variable 'p1'. >> y = sin(pi/2) y = 1 Lecture 14

  22. Arrays in Matlab • All data in Matlab is actually an array — a 1- or 2-dimensional table of numbers. (We only consider 1-D arrays in this lecture.) • A single value, called a scalar, is simply an array of size 1. • To construct a 1-D array, list its elements surrounded by square brackets. y = [4 -5 10 0 5.2] x = [–5 sqrt(2) 17 2^3] >> a = [4, z, x, y] a = 4.0000 1.0000 2.7183 1.0000 Lecture 14

  23. More basics of Matlab arrays • Individual elements are accessed using a parenthesized subscript. x(3) • The first element of array x is x(1). • Can assign to elements of array as in Java. y(1) = 0 • The number of elements in array x is given by the built-in function length(x) Lecture 14

  24. Array functions • An array of evenly-spaced values can be generated by linspace(minVal, maxVal, nVals) Example: array of 100 values spaced from 0 to 2. v = linspace(0, 2*pi, 100); • There are many functions to compute facts about arrays. min(x) max(x) mean(x) sum(x) x(1) + … +x(length(x)) prod(x) x(1) * … * x(length(x)) Lecture 14

  25. Examples of previous >> a = linspace(0, 10, 5) a = 0 2.5000 5.0000 7.5000 10.0000 >> min(a) ans = 0 >> max (a) ans = 10 >> sum(a) ans = 25 >> prod(a) ans = 0 >> mean(a) ans = 5 Lecture 14

  26. Creating arrays • Two arrays can be combined with a comma and brackets: x = [1 2 3]; y = [4 5 6]; [x, y] (is [1 2 3 4 5 6]) z = [x, [x, y]]; • The colon can be used to generate a sequence of values. Forms: lowValue : highValue lowValue : step : highValue Lecture 14

  27. Examples of colon use 1 : 10 (yields 1 2 3 4 5 6 7 8 9 10) 1 : 2 : 10 (yields 1 3 5 7 9) 1 : 0.5 : 10 (yields 1 1.5 2 2.5 . . . 9 9.5 10) 10 : –1 : 1 (yields 10 9 8 7 . . . 1) 0 : 0.01 : 0.5 • A sequence of values is an array value. a = 0 : 2 : 16 b = [1 : 6] / 3 (yields b = 0.3333 0.6667 1.0000 1.3333 1.6667 2.0000 • A sequence of integers can also be used to select a segment of an array. a(3:6) Remember: arrays in Matlab start at 1 (in Java at 0) Lecture 14

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