Lecture 18 Oct 24 Cell arrays (Ch 4), structures recursion (Ch 12)

1. Lecture 18 Oct 24 • Cell arrays (Ch 4), structures • recursion (Ch 12)

2. Cell arrays • suppose we want to represent a collection of sets such as: • {1, 2, 4}, {3, 4, 6}, {7, 8} • Each set can be represented by vector: • [1, 2, 4], [3, 4, 6], [7, 8] • >> A = [[ 1, 2, 4], [3, 4, 6], [7, 8]] • A becomes [1, 2, 4, 3, 4, 6, 7, 8]

3. Cell arrays

4. Cell array – examples

5. Cell – operations

6. More about cell arrays and struct arrays Cell Arrays • Creating Cell Arrays • Accessing Cell Arrays • Using Cell Arrays • Processing Cell Arrays MATLAB Structures • Constructing and Accessing One Structure • Constructor Functions Structure Arrays • Constructing Structure Arrays • Accessing Structure Elements • Manipulating Structures

7. Concept: Collecting Dissimilar Objects Heterogeneous collectionspermit objects of different data types to be grouped in a collection. • They allow data abstraction to apply to a much broader range of content. • However, the fact that the contents of these collections may be of any data type severely restricts the operations that can be performed on the collections as a whole. • Whereas a significant number of arithmetic and logical operations can be performed on whole number arrays, algorithms that process heterogeneous collections must deal with the data contents one item at a time.

8. Cell Arrays • Cell arrays, as the name suggests, have the general form of arrays and can be indexed numerically as arrays. • However, each element of a cell array should be considered as a container in which one data object of any class can be stored. • They can be treated as arrays of containers for the purpose of concatenation and slicing. • However, if you wish to access or modify the contents of the containers, the cells must be accessed individually.

9. Creating Cell Arrays • By assigning values individually to a variable indexed with braces: >> A{1} = 42 A = [42] • By assigning containers individually to a variable indexed with brackets: >> B[1] = {[4 6]}; B = [1x2 double] • By concatenating cell contents using braces {. . .}: >> C = {3, [1,2,3], 'abcde'} C = [3] [1x3 double] 'abcde' • By concatenating cell containers: >> D = [A B C {'xyz'}] D = [42] [1x2 double] [3] [1x3 double] 'abcde' 'xyz'

10. Accessing Cell Arrays Continuing the previous examples, we have the following: >> E = D(2) % parentheses - a container E = [4 6] However, braces are used to access the contents of the containers as follows: >> D{2} % braces - the contents ans = 4 6 If the right-hand side of an assignment statement results in multiple cell arrays, the assignment must be to the same number of variables.

11. Using Cell Arrays • Containing lists of possible values for switch/case statements • Substituting for parameter lists in function calls • For example, suppose you have a function largest(a, b, c)that consumes three variables and produces the largest of the three values provided. It can be used in the following styles: A = 4; B = 6; C = 5; N = largest(A, B, C) params = { 4, 6, 5 }; N = largest(params{1:3})

12. Processing Cell Arrays • Checking the class of the element can be achieved in one of two ways: • The function class(item) returns a string specifying the item type that can be used in a switch statement. • Individual test functions can be used in an if... elseif construct; • examples of the individual test functions are isa(item, 'class'), • iscell(...), ischar(...), islogical(...), isnumeric(...), and • isstruct(...)

13. MATLAB Structures • Structures allow items in the collection to be indexed by field name. • The data contained in a structure is referenced by field name, e.g., item1. • The rules for making a field name are the same as those for a variable. • Fields of a structure, like the elements of a cell array, are heterogeneous—they can contain any MATLAB object.

14. Constructing and Accessing a Structure • To set the value of items in a structure A, the syntax is as follows: >> A.item1 = 'abcde' A = item1: 'abcde' >> A.item2 = 42 A = item1: 'abcde' item2: 42 • Fields in a structure are accessed in the same way—by using the dotted notation. >> A.item2 = A.item2 ./ 2 A = item1: 'abcde' item2: 21

15. Manipulating Field Names • To determine the names of the fields in a structure, the built-in function fieldnames(...) returns a cell array containing the field names as strings. >> names = fieldnames(A) names = 'item1' 'item2’ • Fields can also be accessed “indirectly” by setting a variable to the name of the field, and then using parentheses to indicate that the variable contents should be used as the field name: >> fn = names{1}; >> A.(fn) = [A.(fn) 'fg'] A = item1: 'abcdefg' item2: 21

16. More about Field Names • You can remove a field from a structure using the built-in function rmfield(...). • Be careful. rmfield(...) returns a new structure with the requested field removed. It does not remove that field from your original structure. • If you want the field removed from the original, you must assign the result from rmfield(...) to replace the original structure: >> A = rmfield(A, 'item1') A = item2: 21

17. Why Constructor Functions? Use constructor functions, as opposed to “manually” entering data into structures, for the following reasons: • Manual entry can result in strange behavior due to typographical errors or having fields in the wrong order • The resulting code is generally more compact and easier to understand • When constructing collections of structures, it enforces consistency across the collections

18. Built-in Constructor Function struct(…) >> struct('first','Fred', ... 'last','Jones', ... 'phone','(123) 555-1212', ... 'birth', struct( 'day', 31, ... 'month', 'February', ... 'year', 1965 )) ans = first: 'Fred' last: 'Jones' phone: '(123) 555-1212' birth: [1x1 struct]

19. Custom Constructor Functions • A typical custom constructor function function ans = makeCD(gn, ar, ti, yr, st, pr) % integrate CD data into a structure ans.genre = gn ; ans.artist = ar ; ans.title = ti; ans.year = yr; ans.stars = st; ans.price = pr; • Usage: >> CD = makeCD('Blues', 'Charles, Ray’, 'Genius Loves Company', 2004, 4.5, 15.35 ) CD = genre: 'Blues' artist: 'Charles, Ray' title: 'Genius Loves Company' year: 2004 stars: 4.5000 price: 15.3500

20. Building Structure Arrays Manually >> entry(1).first = 'Fred'; >> entry(1).last = 'Jones'; >> entry(1).age = 37; >> entry(1).phone = ' (123) 555-1212'; >> entry(2).first = 'Sally’; >> entry(2).last = 'Smith’; >> entry(2).age = 29; >> entry(2).phone = '(000) 555-1212' entry = 1x2 structure array with fields: first last age phone

21. Building Structure Arrays with struct(…) genres = {'Blues', 'Classical', 'Country' }; artists = {'Clapton, Eric', 'Bocelli, Andrea', … 'Twain, Shania' }; years = { 2004, 2004, 2004 }; stars = { 2, 4.6, 3.9 }; prices = { 18.95, 14.89, 13.49 }; cds = struct( ‘genre’, genres, … 'artist', artists, … 'year', years, … 'stars', stars, … 'price', prices);

22. Building Structure Arrays with makeCD(…) cds(1) = makeCD('Blues', 'Clapton, Eric', ... 'Sessions For Robert J', 2004, 2, 18.95 ) cds(2) = makeCD('Classical', ... 'Bocelli, Andrea', 'Andrea', 2004, 4.6, 14.89 ) cds(3) = makeCD( 'Country', 'Twain, Shania', ... 'Greatest Hits', 2004, 3.9, 13.49 ) cds(4) = makeCD( 'Latin', 'Trevi, Gloria', ... 'Como Nace El Universo', 2004, 5, 12.15 ) cds(5) = makeCD( 'Rock/Pop', 'Ludacris', ... 'The Red Light District', 2004, 4, 13.49 ) cds(6) = makeCD( 'R & B', '2Pac', ... 'Loyal To The Game', 2004, 3.9, 13.49 ) cds(7) = makeCD( 'Rap', 'Eminem', ... 'Encore', 2004, 3.5, 15.75 ) cds(8) = makeCD( 'Heavy Metal', 'Rammstein', ... 'Reise, Reise', 2004, 4.2, 12.65 )

23. Recursive functions Before we conclude this chapter, we will discuss recursive functions, those that can call themselves. We have examples of functions that call other functions, e.g. insertionsort calling insert etc. If f(n) is defined in terms of f(n – 1), as for example, in the case of factorial, why not let f call itself? n! = n x (n – 1)! Or in matlab: fact(n) = n .* fact(n – 1)

24. Rules for recursive functions • there should be exit from recursion. (i.e., there should be some conditional branch path in which there is no recursive call). Such cases are called the base cases. • 2. recursive calls should make towards base case (usually by calling itself with smaller input values). • 3. Recursion may be more time or memory consuming so should be careful with their use.

25. Example 1: Write a recursive function to compute n! • function out = fact(n) • if n <= 1 out = 1; • else out = n .* fact(n-1); • end;

26. Example 2: Write a recursive function in Matlab to perform binary search. Assume array A is sorted in ascending order. Search(A, low, high, x) will return the largest t such that A(t) <= x. Pre-condition: A(low) <= x <= A(high) Thus low <= t <= high. Initially, low = 1, high = size(A)

27. Recursive binary search program function out = search(A, low, high, x) % returns the largest t s.t. A(t) <= x where low <= t <= high % A is assumed to be sorted if high - low == 1 if A(high) == x out = high; else out = low; end; else mid = floor((low + high)/2); if A(mid) == x out = mid; elseif A(mid) < x out = search(A, mid + 1, high, x); else out = search(A, low, mid - 1, x); end; end;

28. Binary search program function out = binary_search(A, x) if A(1) < x out = 0; else out = search(A, 1, length(A), x) end

29. Project: Generating all subsets of a given set Given a set, like [1, 3, 4], the subsets are: [ ] [1] [3] [4] [1 3] [1 4] [3 4] [1 3 4] We want to write a program to generate all the subsets of a given collection

30. Idea behind algorithm – recursion This is a problem for which non-recursive solutions are significantly harder than recursive solutions. Idea: input array is a of length n. Recursively find all subsets of a(2:n) Then add a(1) to each of the subsets. Combine the two collections.

32. Since we need to represent a collection of sets, we have two choices: • use of cell arrays • use of two-dimensional arrays • The latter is not suitable for this problem since the sizes of the subsets are not the same • We use recursion to generate the solution. • We need a function insert that inserts a given number into all the sets of a given collection.

33. Example showing how insert works

34. Code for insert function out = insert(i, lst) % inserts i into each membet of lst for j = 1:length(lst) out{j}= [i, lst{j}]; end;

35. Code for subsets function L = subsets(lst) % generates all subsets of lst if length(lst) == 0 L = {[]}; elseif length(lst) == 1 L = {[lst(1)],[]}; else L1 = subsets(lst(2:end)); L2 = insert(lst(1), L1); L = [L1, L2]; end;

36. Printing the contents of a cell array function setprint(cellset) % prints every member of the cell in one line % assume cellset is a collection of sets of integers for k=1:length(cellset) aprint(cellset{k}); fprintf('\n'); end; function aprint(r) for j = 1:length(r) fprintf('%d', r(j)); fprintf(' '); end; fprintf('\n')

37. A recursive image tiling problem Given an input image as in the left-side, we want to produce the image on the right-side.

38. Solution to the recursive image tiling problem Observation 1: 3rd quadrant has the input image, but shrunk into quarter. We need to write a function shrink that shrinks the image to quarter size. Quadrant 1 of the output is the shrunk version of the whole output. Equivalently, if F is the function that outputs the image on the right when given as input on the left, then F applied to the image on the 3rd quadrant of the output is what goes into the 1st quadrant of the output. Quadrants 2 and 4 are now easy to determine. They can be obtained by copying a part of the image in quadrant 1.