Ch4-recursion.ppt

Recursion 1
Recursion
© 2013 Goodrich, Tamassia, Goldwasser

Recursion 2
The Recursion Pattern
 Recursion: when a method calls itself
 Classic example--the factorial function:
 n! = 1· 2· 3· ··· · (n-1)· n
 Recursive definition:
 As a Python method:
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Recursion 3
Content of a Recursive Method
 Base case(s)
 Values of the input variables for which we perform
no recursive calls are called base cases (there
should be at least one base case).
 Every possible chain of recursive calls must
eventually reach a base case.
 Recursive calls
 Calls to the current method.
 Each recursive call should be defined so that it
makes progress towards a base case.
© 2013 Goodrich, Tamassia,
Goldwasser

Visualizing Recursion
 Recursion trace
 A box for each
recursive call
 An arrow from each
caller to callee
 An arrow from each
callee to caller
showing return value
 Example
Recursion 4
recursiveFactorial (4)
return 1
call
call
call
call
return 1*1 = 1
return 2*1 = 2
return 3*2 = 6
return 4*6 = 24 final answer
call
Goldwasser

Recursion 5
Example: English Ruler
 Print the ticks and numbers like an English ruler:
Goldwasser

6
Using Recursion
drawTicks(length)
Input: length of a ‘tick’
Output: ruler with tick of the given length in
the middle and smaller rulers on either side
Recursion
drawTicks(length)
if( length > 0 ) then
drawTicks( length  1 )
draw tick of the given length
drawTicks( length  1 )
Slide by Matt Stallmann
included with permission.

Recursion 7
Recursive Drawing Method
 The drawing method is
based on the following
recursive definition
 An interval with a
central tick length L >1
consists of:
 An interval with a central
tick length L1
 An single tick of length L
 An interval with a central
tick length L1
drawTicks (3) Output
drawTicks (0)
(previous pattern repeats )
drawOneTick (1)
drawTicks (1)
drawTicks (2)
drawOneTick (2)
drawTicks (2)
drawTicks (1)
drawTicks (0)
drawTicks (0)
drawTicks (0)
drawOneTick (1)
drawOneTick (3)
Goldwasser

Recursion 8
A Recursive Method for Drawing
Ticks on an English Ruler
Note the two
recursive calls

Binary Search
 Search for an integer, target, in an
ordered list.
© 2013 Goodrich, Tamassia, Goldwasser 9
Recursion

Visualizing Binary Search
 We consider three cases:
 If the target equals data[mid], then we have found the target.
 If target < data[mid], then we recur on the first half of the
sequence.
 If target > data[mid], then we recur on the second half of the
sequence.
Recursion

Analyzing Binary Search
 Runs in O(log n) time.
 The remaining portion of the list is of size
high – low + 1.
 After one comparison, this becomes one of
the following:
 Thus, each recursive call divides the search
region in half; hence, there can be at most
log n levels.
Recursion

Recursion 12
Linear Recursion
 Test for base cases
 Begin by testing for a set of base cases (there should be
at least one).
 Every possible chain of recursive calls must eventually
reach a base case, and the handling of each base case
should not use recursion.
 Recur once
 Perform a single recursive call
 This step may have a test that decides which of several
possible recursive calls to make, but it should ultimately
make just one of these calls
 Define each possible recursive call so that it makes
progress towards a base case.

Recursion 13
Example of Linear Recursion
Algorithm LinearSum(A, n):
Input:
A integer array A and an integer
n = 1, such that A has at least
n elements
Output:
The sum of the first n integers
in A
if n = 1 then
return A[0]
else
return LinearSum(A, n - 1) +
A[n - 1]
Example recursion trace:
LinearSum(A,5)
LinearSum(A,1)
LinearSum(A,2)
LinearSum(A,3)
LinearSum(A,4)
call
call
call
call return A[0] = 4
return 4 + A[1] = 4 + 3 = 7
return 7 + A[2] = 7 + 6 = 13
return 13 + A[3] = 13 + 2 = 15
call return 15 + A[4] = 15 + 5 = 20

Recursion 14
Reversing an Array
Algorithm ReverseArray(A, i, j):
Input: An array A and nonnegative integer
indices i and j
Output: The reversal of the elements in A
starting at index i and ending at j
if i < j then
Swap A[i] and A[ j]
ReverseArray(A, i + 1, j - 1)
return

Recursion 15
Defining Arguments for Recursion
 In creating recursive methods, it is important to define
the methods in ways that facilitate recursion.
 This sometimes requires we define additional
paramaters that are passed to the method.
 For example, we defined the array reversal method as
ReverseArray(A, i, j), not ReverseArray(A).
 Python version:

Recursion 16
Computing Powers
 The power function, p(x,n)=xn, can be
defined recursively:
 This leads to an power function that runs in
O(n) time (for we make n recursive calls).
 We can do better than this, however.
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Recursion 17
Recursive Squaring
 We can derive a more efficient linearly
recursive algorithm by using repeated squaring:
 For example,
24 = 2(4/2)2 = (24/2)2 = (22)2 = 42 = 16
25 = 21+(4/2)2 = 2(24/2)2 = 2(22)2 = 2(42) = 32
26 = 2(6/ 2)2 = (26/2)2 = (23)2 = 82 = 64
27 = 21+(6/2)2 = 2(26/2)2 = 2(23)2 = 2(82) = 128.

Recursion 18
Recursive Squaring Method
Algorithm Power(x, n):
Input: A number x and integer n = 0
Output: The value xn
if n = 0 then
return 1
if n is odd then
y = Power(x, (n - 1)/ 2)
return x · y ·y
else
y = Power(x, n/ 2)
return y · y

Recursion 19
Analysis
Algorithm Power(x, n):
Input: A number x and
integer n = 0
Output: The value xn
if n = 0 then
return 1
if n is odd then
y = Power(x, (n - 1)/ 2)
return x · y · y
else
y = Power(x, n/ 2)
return y · y
It is important that we
use a variable twice
here rather than calling
the method twice.
Each time we make a
recursive call we halve
the value of n; hence,
we make log n recursive
calls. That is, this
method runs in O(log n)
time.

Recursion 20
Tail Recursion
 Tail recursion occurs when a linearly recursive
method makes its recursive call as its last step.
 The array reversal method is an example.
 Such methods can be easily converted to non-
recursive methods (which saves on some resources).
 Example:
Algorithm IterativeReverseArray(A, i, j ):
Input: An array A and nonnegative integer indices i and j
Output: The reversal of the elements in A starting at
index i and ending at j
while i < j do
Swap A[i ] and A[ j ]
i = i + 1
j = j - 1
return

Recursion 21
Binary Recursion
 Binary recursion occurs whenever there are two
recursive calls for each non-base case.
 Example from before: the DrawTicks method for
drawing ticks on an English ruler.

Recursion 22
Another Binary Recusive Method
 Problem: add all the numbers in an integer array A:
Algorithm BinarySum(A, i, n):
Input: An array A and integers i and n
Output: The sum of the n integers in A starting at index i
if n = 1 then
return A[i ]
return BinarySum(A, i, n/ 2) + BinarySum(A, i + n/ 2, n/ 2)
 Example trace:
3, 1
2, 2
0, 4
2, 1
1, 1
0, 1
0, 8
0, 2
7, 1
6, 2
4, 4
6, 1
5, 1
4, 2
4, 1

Recursion 23
Computing Fibonacci Numbers
 Fibonacci numbers are defined recursively:
F0 = 0
F1 = 1
Fi = Fi-1
+ Fi-2 for i > 1.
 Recursive algorithm (first attempt):
Algorithm BinaryFib(k):
Input: Nonnegative integer k
Output: The kth Fibonacci number Fk
if k = 1 then
return k
else
return BinaryFib(k - 1) + BinaryFib(k - 2)

Recursion 24
Analysis
 Let nk be the number of recursive calls by BinaryFib(k)
 n0 = 1
 n1 = 1
 n2 = n1 + n0 + 1 = 1 + 1 + 1 = 3
 n3 = n2 + n1 + 1 = 3 + 1 + 1 = 5
 n4 = n3 + n2 + 1 = 5 + 3 + 1 = 9
 n5 = n4 + n3 + 1 = 9 + 5 + 1 = 15
 n6 = n5 + n4 + 1 = 15 + 9 + 1 = 25
 n7 = n6 + n5 + 1 = 25 + 15 + 1 = 41
 n8 = n7 + n6 + 1 = 41 + 25 + 1 = 67.
 Note that nk at least doubles every other time
 That is, nk > 2k/2. It is exponential!

Recursion

Recursion 26
A Better Fibonacci Algorithm
 Use linear recursion instead
Algorithm LinearFibonacci(k):
Input: A nonnegative integer k
Output: Pair of Fibonacci numbers (Fk , Fk1)
if k = 1 then
return (k, 0)
else
(i, j) = LinearFibonacci(k  1)
return (i +j, i)
 LinearFibonacci makes k1 recursive calls

Ch4-recursion.ppt

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