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Unformatted text preview: 433-521 Algorithms and ComplexitySolutions to Project 1First Semester 2009Posted on LMS: Wednesday, 11 March 2009Due: Wednesday, 18 March 2009 [9.05am]Marks released: Friday, 20 March 2009Solutions to Questions1 [2 marks]First we write the simplest expression for the order of growth of each of these functions.(n-2)!∈Θ((n-2)!)5 log2(n+ 100)10= 50 log2(n+ 100)≤50 log2(2n)(forn≥100)∈Θ(logn)22n= 4n∈Θ(4n).001n4+ 2n3+ 1∈Θ(n4)(lnn)2∈Θ((logn)2)3√n∈Θ(n1/3)3n∈Θ(3n)If we recall that:•nagrows more slowly thannb, if< a < b;•factorials grow faster than exponentials;•logn∈O(na)for alla >;we obtain the correct sequence:5 log2(n+ 100)10(lnn)23√n.001n4+ 2n3+ 13n22n(n-2)!2 [2 marks](i)The first sum can be processed thusS=2·2+3·21+. . .+ (n+ 1)·2n-1,therefore2S=+2·21+. . .+n·2n-1+(n+ 1)·2nSubtracting the first line from the second, we see thatS= (n+ 1)2n-[2n-1+ 2n-2+. . .+ 21+ 2·2] = (n+ 1)2n-2n=n2n∈Θ(n2n)....
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This note was uploaded on 05/11/2010 for the course COMPUTER S 301 taught by Professor . during the Spring '10 term at Kadir Has Üniversitesi.
- Spring '10