Recursion Tessema M. Mengistu Department of Computer Science - - PowerPoint PPT Presentation

Recursion

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Tessema M. Mengistu Department of Computer Science Southern Illinois University Carbondale tessema.mengistu@siu.edu Room - 3131

Outline

• What Is Recursion?
• Tracing a Recursive Method
• Recursive Methods that Return a Value
• Recursively Processing an Array
• Recursively Processing a Linked List
• The Time Efficiency of Recursive Methods

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Recursion

• Repetition is a major feature of many algorithms

– Iterative – Recursive

• We often solve a problem by breaking it into

smaller problems – divide and conquer

• When the smaller problems are identical (except

for size)

– This is called “recursion”

• Repeated smaller problems

– Until problem with known solution is reached

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Recursion

• Example – factorial of a positive number n

fact(4) = 4 * fact(3) = 24 3* fact(2) = 6 2 *fact(1) = 2 fact(1) =1

• In general

fact(n) = n* fact(n-1) n-1*fact(n-2) . . . 2*fact(1)

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Recursion

• Recursion is a problem solving process that

breaks a problem into identical but smaller problems.

• A method that calls itself is a recursive

method.

• The invocation is a recursive call or recursive

invocation.

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Recursion

• Example

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Design of Recursive Solution

• What

part

• f

solution can you contribute directly?

• What smaller (identical) problem has solution

that …

– When taken with your contribution – Provides the solution to the original problem

• When does process end?

– What smaller but identical problem has known solution – Have you reached this problem, or base case?

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Design Guidelines

• Method must receive input value
• Must contain logic that involves this input

value and leads to different cases

• One or more cases should provide solution

that does not require recursion

– Base case or stopping case

• One or more cases must include recursive

invocation of method – recursion call

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Recursive Method the returns a value

• Example

– Write a recursive method that displays the sum of the first n positive integers – Compute the sum 1 + 2 + . . . + n

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Design Guidelines

• A recursive method that does not check for a

base case, or that misses the base case is a logic error

– Infinite recursion

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Tracing Recursive Method

• Given recursive countDown method
• For 3

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Tracing Recursive Method

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Tracing Recursive Method

• A recursive method uses more memory than

an iterative method, in general, because each recursive call generates an activation record.

– Results in “stack overflow” – Infinite recursion or large-size problems are the likely causes of this error

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Display an Array Recursively

• Using first element
• Using last element

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Display an Array Recursively

• Using the middle element

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Displaying a Bag Recursively

• When implemented with an array

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Recursively Processing a Linked Chain

• Method display, implemented recursively

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Recursively Processing a Linked Chain

• Recursive method to display the linked chain

in reverse order

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Time Efficiency of Recursive Methods

• Consider

the method countDown() that displays all positive numbers up to n

• It has time complexity of

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Time Efficiency of Recursive Methods

• The equation for t(n) is called a recurrence

relation

– The definition of the function t contains an

• ccurrence of itself
• How can we proof t(n) = n?

– Proof by induction

• Assume t(n-1) = n-1 for n>1
• t(n) = 1+t(n-1) = 1 +n-1 =n
• Therefore t(n) = n for n>1

– So countDown() is O(n)

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Time Efficiency of Computing xn

• Recursive calculation for xn
• Can be shown
• Thus efficiency is O(log n)
• Proof – Reading Assignment

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Simple Solution to a Difficult Problem

• Consider Towers of Hanoi puzzle

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Towers of Hanoi

• Rules
• 1. Move one disk at a time. Each disk you move

must be a topmost disk.

• 2. No disk may rest on top of a disk smaller than

itself.

• 3. You can store disks on the second pole

temporarily, as long as you observe the previous two rules.

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Solution

• Move a disk from pole 1 to pole 3
• Move a disk from pole 1 to pole 2
• Move a disk from pole 3 to pole 2
• Move a disk from pole 1 to pole 3
• Move a disk from pole 2 to pole 1
• Move a disk from pole 2 to pole 3
• Move a disk from pole 1 to pole 3

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Recursive Solution

• To solve for n disks …

– Ask friend to solve for n – 1 disks – He in turn asks another friend to solve for n – 2 – Etc. – Each one lets previous friend know when their simpler task is finished

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Towers of Hanoi

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Towers of Hanoi

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Algorithm Efficiency

• Moves required for n disks
• We note and conjecture

(proved by induction)

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Algorithm Efficiency

• The number of moves required to solve the

Towers of Hanoi problem grows exponentially with the number of disks n.

– m(n) = O(2n)

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Poor Solution to a Simple Problem

• Fibonacci sequence

1, 1, 2, 3, 5, 8, 13, …

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Poor Solution to a Simple Problem

• A recursive solution

Note the two recursive calls

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Tail Recursion

• When the last action performed by a recursive

method is a recursive call

• Example:
• Repeats call with change in parameter or variable

Indirect Recursion

• Consider chain of events

– Method A calls Method B – Method B calls Method C – and Method C calls Method A

• Mutual recursion

– Method A calls Method B – Method B calls Method A

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