Đề thi cấu trúc dữ liệu và giải thuật dsa ch3 recursion (2)

Recursion Dr Nguyen Ho Man Rang Recursion and the basic components of recursive algorithms Properties of recursion Designing recursive algorithms Recursion and backtracking Recursion implementation in[.]

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Dr Nguyen HoMan Rang

Recursion and thebasic components

of recursivealgorithmsProperties ofrecursionDesigning recursivealgorithmsRecursion andbacktrackingRecursionimplementation inC/C++

3.1

Chapter 3

Recursion

Data Structures and Algorithms

Dr Nguyen Ho Man Rang Faculty of Computer Science and Engineering

University of Technology, VNU-HCM

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Outcomes

• L.O.8.1 - Describe the basic components of recursive

algorithms (functions).

• L.O.8.2 - Draw trees to illustrate callings and the value

of parameters passed to them for recursive algorithms.

• L.O.8.3 - Give examples for recursive functions written

in C/C++.

• L.O.8.5 - Develop experiment (program) to compare

the recursive and the iterative approach.

• L.O.8.6 - Give examples to relate recursion to

backtracking technique.

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3 Designing recursive algorithms

4 Recursion and backtracking

5 Recursion implementation in C/C++

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Recursion and the basic

components of recursive

algorithms

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F actorial(n) =

"

n × F actorial(n − 1) if n > 0

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Basic components of recursive algorithms

Two main components of a Recursive Algorithm

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3.7

Recursion

Factorial (3) Recursively (source: Data Structure - A pseudocode

Approach with C++)

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Recursion

Factorial: Iterative Solution

Algorithm iterativeFactorial(n)

Calculates the factorial of a number using a loop.

Pre: n is the number to be raised factorially

Post: n! is returned - result in factoN

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3.9

Recursion

Factorial: Recursive Solution

Algorithm recursiveFactorial(n)

Calculates the factorial of a number using a recursion.

Pre: n is the number to be raised factorially

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Recursion

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3.11

Properties of recursion

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Properties of all recursive algorithms

problem by using its solution to a simpler

sub-problem

enough that it can be solved without

applying the algorithm to it recursively.

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3.13

Designing recursive

algorithms

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The Design Methodology

Rules for designing a recursive algorithm

case)

3 Combine the base case and the general

cases into an algorithm.

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3.15

Limitations of Recursion

slowly than its nonrecursive

implementation.

more understandable.

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Print List in Reverse

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3.17

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Algorithm printReverse(list)

Prints a linked list in reverse.

Pre: list has been built

Post: list printed in reverse

if list is null then

return

end

printReverse (list -> next)

print (list -> data)

End printReverse

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Greatest Common Divisor

Algorithm gcd(a, b)

Calculates greatest common divisor using the Euclidean

algorithm.

Pre: a and b are integers

Post: greatest common divisor returned

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+

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1 2

3

Result

0, 1, 1, 2, 3, 5, 8, 13, 21, 34,

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Fibonacci Numbers

Algorithm Fibo(n)

Calculates the n th Fibonacci number.

Pre: n is postive integer

Post: the n th Fibonnacci number returned

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The Towers of Hanoi

Move disks from Source to Destination using Auxiliary:

1 Only one disk could be moved at a time.

2 A larger disk must never be stacked above a smaller one.

3 Only one auxiliary needle could be used for the

intermediate storage of disks.

3

2

1

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The Towers of Hanoi

Moved disc from pole 1 to pole 2.

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Destination Moved disc from pole 3 to pole 2.

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The Towers of Hanoi

2 1

Destination 3

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The Towers of Hanoi

Source

1

3 2

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3.33

The Towers of Hanoi

3 2 1

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The Towers of Hanoi

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The Towers of Hanoi

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3.37

The Towers of Hanoi

Algorithm move(val disks <integer>, val source

<character>, val destination <character>, val auxiliary

<character>)

Move disks from source to destination.

Pre: disks is the number of disks to be moved

Post: steps for moves printed

print("Towers: ", disks, source, destination, auxiliary)

if disks = 1 then

print ("Move from", source, "to", destination)

else

move(disks - 1, source, auxiliary, destination)

move(1, source, destination, auxiliary)

move(disks - 1, auxiliary, destination, source)

end

return

End move

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Recursion and

backtracking

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Eight Queens Problem

Place eight queens on the chess board in such a way that no

queen can capture another.

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3.41

Algorithm putQueen(ref board <array>,

r is the row to place queens onwards

Post: all the remaining queens are safely

placed on the board; or backtracking to

the previous rows is required

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for every column c on the same row r do

if cell r,c is safe then

place the next queen in cell r,c

if r < n-1 then

putQueen (board, r + 1) else

output successful placement end

remove the queen from cell r,c

end

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3.43

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Recursion

implementation in

C/C++

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The Towers of Hanoi

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Tiêu đề	Recursion in Data Structures and Algorithms
Người hướng dẫn	Dr. Nguyễn Ho Man Rang
Trường học	University of Technology, VNU-HCM
Chuyên ngành	Data Structures and Algorithms
Thể loại	lecture notes

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