analysis recursive algorithm
DESCRIPTION
aTRANSCRIPT
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*Recursive Algorithm Analysis
Dr. Ying [email protected]
RAIK 283: Data Structures & AlgorithmsSeptember 13, 2012
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*Giving credit where credit is due:Most of the lecture notes are based on the slides from the Textbooks companion websitehttp://www.aw-bc.com/info/levitin
Several slides are from Hsu Wen Jing of the National University of Singapore
I have modified them and added new slides
RAIK 283: Data Structures & Algorithms
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*Example: a recursive algorithm
Algorithm:if n=0 then F(n) := 1else F(n) := F(n-1) * nreturn F(n)
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*Algorithm F(n)// Compute the nth Fibonacci number recursively//Input: A nonnegative integer n//Output: the nth Fibonacci numberif n 1 return nelse return F(n-1) + F(n-2)Example: another recursive algorithm
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*Recurrence RelationRecurrence Relation
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*Recurrence RelationRecurrence Relation: an equation or inequality that describes a function in terms of its value on smaller inputs
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*Example: a recursive algorithmAlgorithm:if n=0 then F(n) := 1else F(n) := F(n-1) * nreturn F(n)
What does this algorithm compute?
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*Example: a recursive algorithmAlgorithm:if n=0 then F(n) := 1else F(n) := F(n-1) * nreturn F(n)
What does this algorithm compute?
Whats the basic operation of this algorithm?
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*Example: recursive evaluation of n !Recursive definition of n!:Algorithm:if n=0 then F(n) := 1else F(n) := F(n-1) * nreturn F(n)
M(n): number of multiplications to compute n! with this recursive algorithm
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*Example: recursive evaluation of n !Recursive definition of n!:Algorithm:if n=0 then F(n) := 1else F(n) := F(n-1) * nreturn F(n)
M(n): number of multiplications to compute n! with this recursive algorithm
Could we establish a recurrence relation for deriving M(n)?
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*Example: recursive evaluation of n !Definition: n ! = 1*2**(n-1)*n Recursive definition of n!:Algorithm:if n=0 then F(n) := 1else F(n) := F(n-1) * nreturn F(n)
M(n) = M(n-1) + 1Initial Condition: M(0) = ?
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*Example: recursive evaluation of n !Recursive definition of n!:Algorithm:if n=0 then F(n) := 1else F(n) := F(n-1) * nreturn F(n)
M(n) = M(n-1) + 1Initial condition: M(0) = 0Explicit formula for M(n) in terms of n only?
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*Time efficiency of recursive algorithmsSteps in analysis of recursive algorithms: Decide on parameter n indicating input sizeIdentify algorithms basic operationDetermine worst, average, and best case for inputs of size n
Set up a recurrence relation and initial condition(s) for C(n)-the number of times the basic operation will be executed for an input of size n
Solve the recurrence to obtain a closed form or determine the order of growth of the solution (see Appendix B)
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*EXAMPLE: tower of hanoiProblem:Given three pegs (A, B, C) and n disks of different sizes Initially, all the disks are on peg A in order of size, the largest on the bottom and the smallest on top The goal is to move all the disks to peg C using peg B as an auxiliaryOnly 1 disk can be moved at a time, and a larger disk cannot be placed on top of a smaller one
ACBn disks
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*EXAMPLE: tower of hanoiDesign a recursive algorithm to solve this problem:Given three pegs (A, B, C) and n disks of different sizes Initially, all the disks are on peg A in order of size, the largest on the bottom and the smallest on top The goal is to move all the disks to peg C using peg B as an auxiliaryOnly 1 disk can be moved at a time, and a larger disk cannot be placed on top of a smaller one
ACBn disks
- *EXAMPLE: tower of hanoiStep 1: Solve simple case when n
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*EXAMPLE: tower of hanoiStep 2: Assume that a smaller instance can be solved, i.e. can move n-1 disks. Then?ACBACBACB
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*EXAMPLE: tower of hanoi
ACBACBACBACB
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*EXAMPLE: tower of hanoi
ACBACBACBACBTOWER(n, A, B, C)
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*EXAMPLE: tower of hanoi
ACBACBACBACBTOWER(n, A, B, C)TOWER(n-1, A, C, B) Move(A, C)TOWER(n-1, B, A, C)
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*EXAMPLE: tower of hanoi
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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*EXAMPLE: tower of hanoiAlgorithm analysis: Input size? Basic operation?
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*EXAMPLE: tower of hanoiAlgorithm analysis: Do we need to differentiate best case, worst case & average case for inputs of size n?
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*EXAMPLE: tower of hanoiAlgorithm analysis: Set up a recurrence relation and initial condition(s) for C(n)
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*EXAMPLE: tower of hanoiAlgorithm analysis: C(n) = 2C(n-1)+1
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*In-Class ExerciseP. 76 Problem 2.4.1 (c): solve this recurrence relation: x(n) = x(n-1) + n for n>0, x(0)=0P. 77 Problem 2.4.4: consider the following recursive algorithm: Algorithm Q(n)// Input: A positive integer nIf n = 1 return 1else return Q(n-1) + 2 * n 1A. Set up a recurrence relation for this functions values and solve it to determine what this algorithm computesB. Set up a recurrence relation for the number of multiplications made by this algorithm and solve it. C. Set up a recurrence relation for the number of additions/subtractions made by this algorithm and solve it.
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*Example: BinRec(n) Algorithm BinRec(n)//Input: A positive decimal integer n//Output: The number of binary digits in ns binary representationif n = 1 return 1else return BinRec( n/2 ) + 1
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*Smoothness ruleIf T(n) (f(n)) for values of n that are powers of b, where b 2, then T(n) (f(n))
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*Example: BinRec(n) Algorithm BinRec(n)//Input: A positive decimal integer n//Output: The number of binary digits in ns binary representationif n = 1 return 1else return BinRec( n/2 ) + 1If C(n) (f(n)) for values of n that are powers of b, where b 2, then C(n) (f(n))
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*Fibonacci numbersThe Fibonacci sequence:0, 1, 1, 2, 3, 5, 8, 13, 21,
Fibonacci recurrence:F(n) = F(n-1) + F(n-2) F(0) = 0 F(1) = 1
2nd order linear homogeneous recurrence relation with constant coefficients
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*Solving linear homogeneous recurrence relations with constant coefficientsEasy first: 1st order LHRRCCs:C(n) = a C(n -1) C(0) = t Solution: C(n) = t anExtrapolate to 2nd orderL(n) = a L(n-1) + b L(n-2) A solution?: L(n) = ?
Characteristic equation (quadratic)Solve to obtain roots r1 and r2 (quadratic formula)General solution to RR: linear combination of r1n and r2nParticular solution: use initial conditions
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*Solving linear homogeneous recurrence relations with constant coefficientsEasy first: 1st order LHRRCCs:C(n) = a C(n -1) C(0) = t Solution: C(n) = t anExtrapolate to 2nd orderL(n) = a L(n-1) + b L(n-2) A solution?: L(n) = ?
Characteristic equation (quadratic)Solve to obtain roots r1 and r2 (quadratic formula)General solution to RR: linear combination of r1n and r2nParticular solution: use initial conditions
Explicit Formula for Fibonacci Number: F(n) = F(n-1) +F(n-2)
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*1. Definition based recursive algorithm
Computing Fibonacci numbersAlgorithm F(n)// Compute the nth Fibonacci number recursively//Input: A nonnegative integer n//Output: the nth Fibonacci numberif n 1 return nelse return F(n-1) + F(n-2)
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*2. Nonrecursive brute-force algorithm
Computing Fibonacci numbersAlgorithm Fib(n)// Compute the nth Fibonacci number iteratively//Input: A nonnegative integer n//Output: the nth Fibonacci numberF[0] 0; F[1] 1for i 2 to n doF[i] F[i-1] + F[i-2]return F[n]
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*Computing Fibonacci numbers
3. Explicit formula algorithm
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*In-Class Exercises What is the explicit formula for A(n)?A(n) = 3A(n-1) 2A(n-2) A(0) = 1 A(1) = 3
P.83 2.5.3. Climbing stairs: Find the number of different ways to climb an n-stair stair-case if each step is either one or two stairs. (For example, a 3-stair staircase can be climbed three ways: 1-1-1, 1-2, and 2-1.)
***Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
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M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)**In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.
Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
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M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.
Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
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M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.
Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
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M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.
Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
------------
M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.
Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
------------
M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*In mathematics, a recurrence relation is an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms.
Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
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M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
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M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*Note the difference between the two recurrences. Students often confuse these!
F(n) = F(n-1) nF(0) = 1
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M(n) =M(n-1) + 1M(0) = 0(# recursive calls is the same)*******************http://en.wikipedia.org/wiki/Quadratic_equation
r1=(-b+sqrt(b^2-4ac))/2a; r2=(-b-sqrt(b^2-4ac)/2a.*****