Analysis Re Cursive Algorithm

8/8/2019 Analysis Re Cursive Algorithm

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Recursive Algorithm Analysis

Dr. Ying [email protected]

CSCE 310: Data Structures & AlgorithmsCSCE 310: Data Structures & Algorithms

September 8, 2010


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Giving credit where credit is due:Giving credit where credit is due:

Most of the lecture notes are based on the slidesMost of the lecture notes are based on the slides

from the Textbooks companion websitefrom the Textbooks companion website

http://www.awhttp://www.aw--bc.com/info/levitinbc.com/info/levitin

Several slides are from Hsu Wen JingSeveral slides are from Hsu Wen Jing of theof the

National University ofS

ingaporeNational University ofS

ingapore

I have modified them and added new slidesI have modified them and added new slides

CSCE 310: Data Structures & AlgorithmsCSCE 310: Data Structures & Algorithms


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Example:Example: a recursive algorithma recursive algorithm

Algorithm:Algorithm:

ififnn=0 then=0 then FF((nn) := 1) := 1

elseelse FF((nn) :=) :=FF((nn--1) *1) * nn

returnreturn FF((nn))


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Algorithm F(n)

// Compute the nth Fibonacci number recursively

//Input: A nonnegative integer n

//Output: the nth Fibonacci number

ifn e 1return n

else return F(n-1) + F(n-2)

Example:Example: another recursive algorithmanother recursive algorithm


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Example: recursive evaluation ofExample: recursive evaluation of nn !!

Definition:Definition: nn != 1*2!= 1*2**(n**(n--1)*1)*nn

Recursive definition ofRecursive definition ofnn!:!:





M(n): number of multiplications to computeM(n): number of multiplications to compute nn!!

Could we establish aCould we establish a recurrence relationrecurrence relation forforderiving M(n)?deriving M(n)?


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M(n) = M(nM(n) = M(n--1) + 1 for n > 01) + 1 for n > 0

AA recurrencerelationrecurrencerelation is an equation oris an equation or

inequality that describes a function in termsinequality that describes a function in terms

of its value on smaller inputsof its value on smaller inputs

Explicit formula for the sequence (e.g. M(n))Explicit formula for the sequence (e.g. M(n))in terms of n onlyin terms of n only


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Definition:Definition: nn != 1*2!= 1*2**(n**(n--1)*1)*nn






M(n) = M(nM(n) = M(n--1) + 11) + 1

Initial Condition: M(0) = ?Initial Condition: M(0) = ?


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M(n) = M(nM(n) = M(n--1) + 11) + 1 Initial condition: M(0) = 0Initial condition: M(0) = 0

Explicit formula for M(n)?Explicit formula for M(n)?


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Time efficiency of recursive algorithmsTime efficiency of recursive algorithms

Steps in analysis of recursive algorithms:Steps in analysis of recursive algorithms: Decide on parameterDecide on parameter nn indicatingindicating inputsizeinputsize

Identify algorithmsIdentify algorithms basicoperationbasicoperation

DetermineDetermine worstworst,, averageaverage, and, and bestbestcase for inputs of sizecase for inputs of size nn

Set up a recurrence relation and initial condition(s) forSet up a recurrence relation and initial condition(s) forCC((nn))--the number of times the basic operation will bethe number of times the basic operation will beexecuted for an input of sizeexecuted for an input of size nn

Solve the recurrence to obtain a closed form orSolve the recurrence to obtain a closed form orestimate the order of magnitude of the solution (seeestimate the order of magnitude of the solution (seeAppendix B)Appendix B)


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EXAMPLE: tower of hanoiEXAMPLE: tower of hanoi

Problem:Problem: Given three pegs (A, B, C) and n disks of different sizesGiven three pegs (A, B, C) and n disks of different sizes

Initially, all the disks are on peg A in order of size, theInitially, all the disks are on peg A in order of size, thelargest on the bottom and the smallest on toplargest on the bottom and the smallest on top

The goal is to move all the disks to peg C using peg B as anThe goal is to move all the disks to peg C using peg B as anauxiliaryauxiliary

Only 1 disk can be moved at a time, and a larger disk cannotOnly 1 disk can be moved at a time, and a larger disk cannotbe placed on top of a smaller onebe placed on top of a smaller one

A C

B

n disks


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Step 1: Solve simple case when n


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Step 2: Assume that a smaller instance can beStep 2: Assume that a smaller instance can besolved, i.e. can move nsolved, i.e. can move n--1 disks. Then?1 disks. Then?

A C

B

A C

B

A C

B


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A C

BA C

B

A C

B

A C

B


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A C

BA C

B

A C

B

A C

B

TOWER(n, A, B, C)


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A C

BA C

B

A C

B

A C

B

TOWER(n, A, B, C)

TOWER(n-1, A, C, B)

Move(A, C)

TOWER(n-1, B, A, C)


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TOWER(n, A, B, C) {

TOWER(n-1

, A, C, B);Move(A, C);

TOWER(n-1, B, A, C)

}

if n


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TOWER(n, A, B, C) {

TOWER(n-1, A, C, B);

Move(A, C);TOWER(n-1, B, A, C)

}

if n


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TOWER(n, A, B, C) {



}

if n


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TOWER(n, A, B, C) {



}

if n


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InIn--Class ExerciseClass Exercise

P. 76 Problem 2.4.1 (c): solve this recurrence relation:P. 76 Problem 2.4.1 (c): solve this recurrence relation:

x(n) = x(nx(n) = x(n--1) + n1) + n

P. 76 Problem 2.4.4: consider the following recursiveP. 76 Problem 2.4.4: consider the following recursive

algorithm:algorithm:

Algorithm Q(n)Algorithm Q(n)// Input: A positive integer n// Input: A positive integer n

If n = 1 return 1If n = 1 return 1

else return Q(nelse return Q(n--1) + 2 * n1) + 2 * n 11

A.S

et up a recurrence relation for this functions values andA.S

et up a recurrence relation for this functions values andsolve it to determine what this algorithm computessolve it to determine what this algorithm computes

B. Set up a recurrence relation for the number ofB. Set up a recurrence relation for the number of

multiplications made by this algorithm and solve it.multiplications made by this algorithm and solve it.

C. Set up a recurrence relation for the number ofC. Set up a recurrence relation for the number of

additions/subtractions made by this algorithm and solve it.additions/subtractions made by this algorithm and solve it.


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Example: BinRec(n)Example: BinRec(n)

Algorithm BinRec(n)

//Input: A positive decimal integer n

//Output: The number of binary digits in ns binary representation

ifn = 1return 1

else return BinRec( -n/2 ) + 1


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Smoothness ruleSmoothness rule

If T(nIf T(n)) 55(f(n))(f(n))

for values of n that are powers of b, where bfor values of n that are powers of b, where b uu 2,2,

thenthen

T(n)T(n) 55(f(n))(f(n))


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Example: BinRec(n)Example: BinRec(n)

Algorithm BinRec(n)

//Input: A positive decimal integer n

//Output: The number of binary digits in ns binary representation

ifn = 1return 1

else return BinRec( -n/2 ) + 1


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Fibonacci numbersFibonacci numbers

The Fibonacci sequence:The Fibonacci sequence:

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

Fibonacci recurrence:Fibonacci recurrence:

F(F(nn) = F() = F(nn--1) + F(1) + F(nn--2)2)

F(0) = 0F(0) = 0

F(1) = 1F(1) = 1

2nd2nd orderlinearhomogeneousorderlinearhomogeneousrecurrencerelationrecurrencerelation

with constantcoefficientswith constantcoefficients


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Solving linear homogeneous recurrenceSolving linear homogeneous recurrence

relations with constant coefficientsrelations with constant coefficients

Easy first: 1Easy first: 1stst order LHRRCCs:order LHRRCCs:CC((nn) =) =aCaC((nn --1)1) CC(0) =(0) =tt Solution: Solution: CC((nn) =) = tatann

Extrapolate to 2Extrapolate to 2ndnd orderorder

LL((nn) =) =aa LL((nn--1) +1) + bb LL((nn--2) A solution?:2) A solution?: LL((nn) =) =??

Characteristic equation (quadratic)Characteristic equation (quadratic)

Solve to obtain rootsSolve to obtain roots rr11 andand rr22 ((quadratic formulaquadratic formula))

General solution to RR: linear combination ofGeneral solution to RR: linear combination ofrr11nn andand rr22

nn

Particular solution: use initial conditionsParticular solution: use initial conditions


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Solving linear homogeneous recurrenceSolving linear homogeneous recurrence

relations with constant coefficientsrelations with constant coefficients

Easy first: 1Easy first: 1stst order LHRRCCs:order LHRRCCs:CC((nn) =) =aCaC((nn --1)1) CC(0) =(0) =tt Solution: Solution: CC((nn) =) = tatann

Extrapolate to 2Extrapolate to 2ndnd orderorder

LL((nn) =) =aa LL((nn--1) +1) + bb LL((nn--2) A solution?:2) A solution?: LL((nn) =) =??

Characteristic equation (quadratic)Characteristic equation (quadratic)

Solve to obtain rootsSolve to obtain roots rr11 andand rr22 ((quadratic formulaquadratic formula))

General solution to RR: linear combination ofGeneral solution to RR: linear combination ofrr11nn andand rr22

nn

Particular solution: use initial conditionsParticular solution: use initial conditions

Explicit Formula forExplicit Formula for Fibonacci Number: F(n) = F(nFibonacci Number: F(n) = F(n--1) +F(n1) +F(n--2)2)


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1. Definition based recursive algorithm1. Definition based recursive algorithm

Computing Fibonacci numbersComputing Fibonacci numbers

Algorithm F(n)

// Compute the nth Fibonacci number recursively



ifn e 1return n

else return F(n-1) + F(n-2)


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2. Nonrecursive brute2. Nonrecursive brute--force algorithmforce algorithm


Algorithm Fib(n)

// Compute the nth Fibonacci number iteratively



F[0] n 0; F[1] n 1

for i n 2 to n do

F[i] n F[i-1] + F[i-2]

return F[n]


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3. Explicit formula algorithm3. Explicit formula algorithm

integernearestthetorounded)2

51(

5

1)( nnF

!

Special care in its implementation:Special care in its implementation:

Intermediate results are irrational numbersIntermediate results are irrational numbers

Their approximations in the computer are accurate enoughTheir approximations in the computer are accurate enough

Final roundFinal round--off yields a correct resultoff yields a correct result


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InIn--Class ExercisesClass Exercises

Another example:Another example:A(A(nn) = 3A() = 3A(nn--1)1) 2A(2A(nn--2) A(0) = 1 A(1) = 32) A(0) = 1 A(1) = 3

P.83 2.5.4.

Climbing stairs: Find the number of

P.83 2.5.4.

Climbing stairs: Find the number ofdifferent ways to climb an ndifferent ways to climb an n--stair stairstair stair--case if eachcase if each

step is either one or two stairs. (For example, a 3step is either one or two stairs. (For example, a 3--

stair staircase can be climbed three ways: 1stair staircase can be climbed three ways: 1--11--1, 11, 1--

2, and 22, and 2--1.)1.)


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Analysis Re Cursive Algorithm