Lec 15
Oct 27
• more examples of recursive programs
• more about cell arrays
• structures in Matlab
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 = 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
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 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
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
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 )
Binary search tree – a recursive structure
Recursively defined structures can be useful in some
applications.
Define a binary search tree as a struct with three fields:
key – a real number
left, right – binary search trees.
base case: empty tree (represented by empty cell)
Ex: { 12, { 8, {}, {} }, { 7, { 9, {}, {}}, { 4, {}, {} } } }
14
Binary Search Trees (BST)
•
A data structure for efficient searching, inser-tion
and deletion (dictionary operations)
•
Binary search tree property
•
•
•
For every node x:
All the keys in its left
subtree are smaller than
the key value in x
All the keys in its right
subtree are larger than the
key value in x
Binary Search Trees
Example:
Tree height = 4
A binary search tree
Not a binary search tree
Key requirement of a BST: all the keys in a BST are
distinct, no duplication
Binary Search Trees
The same set of keys may have different BSTs
Searching BST
Example: Suppose T is the tree being searched:
• If we are searching for 15, then we are done.
• If we are searching for a key < 15, then we should
search in the left subtree.
• If we are searching for a key > 15, then we should
search in the right subtree.
Search
•
Find X: return true (false) if x is in tree T.
function out = search(T, x)
if isempty(T)
out = false; return;
elseif abs(T.key – x) < 0.0001
out = true; return;
elseif T.key > x
out = search(T.left, x);
else
out = search(T.right, x);
end;
Inorder Traversal of BST
•
Inorder traversal of BST prints out all the keys in sorted
order
treePrint(T.left);
Print(T.key);
treePrint(T.right);
Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20
findMin/ findMax
 Goal: return the node containing the smallest (largest)
key in the tree
 Algorithm: Start at the root and go left (right) as long as
there is a left (right) child. The stopping point is the
smallest (largest) element
Insertion
To insert(X):
 Proceed down the tree as you would for search.
 If x is found, do nothing (or update some secondary
record)
 Otherwise, insert X at the last spot on the path traversed
X = 13
Another example of insertion
Example: insert(11). Show the path taken and the position
at which 11 is inserted.
Note: There is a unique place where a new key can be
inserted.
Binary Search Trees / Slide 25
Insertion
X = 13
function out = insert(A, x)
if isempty(A)
out.key = x;
out.left = {};
out.right = {};
elseif A.key < x
A.right = insert(A.right, x);
out = A;
else
A.left = insert(A.left, x);
out = A;
end;
25
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.
Test case – a simple example
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.