Lec 14 Oct 22

• more examples of recursive programs

• more about cell arrays

• structures in Matlab

Example showing how insert works

Code for insert

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

Code for subsets

function L = subsets(lst)% generates all subsets of lstif 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;

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 integersfor 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')

Generating all permutations of a given set

Example: set [ 1 3 4]

Permutations:

[1 3 4][1 4 3][3 1 4][3 4 1][4 1 3][4 3 1]

We will discuss the algorithm and implementation details in lecture. The problem is part of HW # 4.

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

Concept: Collecting Dissimilar Objects

Heterogeneous collections permit 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.

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.

Creating Cell Arrays• By assigning values individually to a variable indexed

with braces:>> A{1} = 42A = [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'

Accessing Cell Arrays

Continuing the previous examples, we have the following:

>> E = D(2) % parentheses - a containerE = [4 6]

However, braces are used to access the contents of the containers as follows:

>> D{2} % braces - the contentsans = 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. .

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})

Processing Cell Arrays

• The template for processing cell arrays is:<initialize result>

for <index specification>

<extract an element>

<check the element accordingly>

<process the element accordingly>

end

<finalize result>

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(...)

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.

Constructing and Accessing One Structure

• To set the value of items in a structure A, the syntax is as follows:

>> A.item1 = 'abcde'A = item1: 'abcde'>> A.item2 = 42A =item1: 'abcde'item2: 42

• Fields in a structure are accessed in the same way—by using the dotted notation.

>> A.item2 = A.item2 ./ 2A =item1: 'abcde'item2: 21

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

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

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

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]

Custom Constructor Functions

• A typical custom constructor functionfunction ans = makeCD(gn, ar, ti, yr, st, pr)% integrate CD data into a structureans.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: 2004stars: 4.5000price: 15.3500

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:firstlastagephone

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);

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 )

Summary

■ Cell arrays are vectors of containers; their elements can be manipulated either as vectors of containers, or individually by inserting or extracting the contents of the container using braces in place of parentheses.■ The elements of a structure are accessed by name rather than by indexing, using the dot operator to specify the field name to be used.■ Structures can be collected into structure arrays whose elements are structures all with the same field names. These elements can then be indexed and manipulated in the same manner as the cells in a cell array.

Recursive backtracking – project

The next problem introduces a general technique of problem solving that can be used in a wide range of settings. We will use it to solve the following problem: a Latin square of order n is a n by n square matrix in which each row and each column contains exactly one occurrence of numbers 1 to n. (In mid-term, you wrote a program to test if a given matrix is a Latin square.)

Our problem is more complicated – given a partially filled Latin square, we want to complete it into a Latin square, if possible.