Lecture 12
•Recursion – more examples
• Chapter 8 problems and solutions
• Cell arrays, Chapter 10
Oct 15
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)
Rules for recursive functions
1. 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.
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;
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)
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;
Binary search program
function out = binary_search(A, x)
if A(1) < x out = 0;
else
out = search(A, 1, length(A), x)
end
Merge Sorting – recursive algorithm
Write a program in Matlab to merge two
arrays. Merging involves combining two sorted
arrays into a single sorted array.
>> A = merge([2 8 11 20], [4 5 7 13 19 24])
A =
2 4 5 7 8 11 13 19 20 24
An example involving 1-d array
Given two segments of sorted arrays A and B, output
the result of merging them as a single sorted array.
B
A
merge(A, 2, 3, B, 9, 13) should return
0 1
2
3
4
5
6
12 2 3 4 4 5
merge(A, 2, 2, B, 9,10) should return the array 1 2 4
merge(A, 3, 4, B, 9, 8) should return 5 7 since the second
array is empty. (high index < low index means the array is
empty.)
What are the base cases?
What are the base cases?
• one array segment is empty.
In this case, what is the output?
What are the base cases?
• one array segment is empty.
In this case, what is the output?
• The other array segment, so we just have to
copy the segment to the output.
What are the base cases?
• one array segment is empty.
In this case, what is the output?
• The other array segment, so we just have to
copy the segment to the output.
• what if both segments are not empty? We
need to make recursive call.
• Before we proceed, we need to make a
change to the prototype of the function merge.
Why?
• We need to add two more parameters - the
name of the array and the starting index in the
output array at which we want to write.
Merge function
function out = merge(A, B)
if length(A)==0
out = B;
elseif length(B) == 0
out = A;
else
if A(1) <= B(1)
temp =A(1);
out = [temp, merge(A(2:end),B)];
else
temp = B(1);
out = [temp, merge(A,B(2:end))];
end;
end;
Merge-sorting
To sort an array, recursively sort the two halves, then
apply merge.
function out=mergesort(A)
if length(A)==1
out = A;
else
s = length(A);
mid = floor(s/2);
out1 = mergesort(A(1:mid));
out2 =
mergesort(A(mid+1:end));
out = merge(out1, out2);
end
Exercise 8.1.
>> repeat([2 3; 3 1; 4 2]
ans =
[222344]
Exercise 8.1.
>> repeat([2 3; 3 1; 4 2]
ans =
[222344]
function out = repeat1(m)
out = [];
[r, c] = size(m);
for i = 1 : r %go through each row
for j = 1 : m(i, 2)
%Repeat the number of times specified in the second column
out(end + 1) = m(i, 1);
end
end
Exercise 8.4. Write a function myfind, that mimics the
behavior of the built-in function find. That is, it takes
as input a boolean vector of 1’s and 0’s.
Exercise 8.4. Write a function myfind, that mimics the
behavior of the built-in function find. That is, it takes
as input a boolean vector of 1’s and 0’s.
function out = myfind(A, x)
out = [];
for j = 1:length(A)
if A(j) == x
out = [out, j];
end;
end;
Exercise 8.8
Write a function intoBits to take an integer number
as input and output a string of 0’s and 1’s representing
the number in base 2.
Exercise 8.11
Write functions minAll, minCol and minRow that find
the overall minimum, column minima and row minima
of a matrix A.
Exercise 8.11
Write functions minAll, minCol and minRow that find
the overall minimum, column minima and row minima
of a matrix A.
min(min(matrix)) %Minimum element of the whole matrix
min(matrix); %Minimum element of each column
min(transpose(matrix)); %Minimum element of each row
Lec 13
Oct 17