Discrete Mathematics 2. Exercise Sheet

Algorithmic

Discrete Mathematics 2. Exercise Sheet

Department of Mathematics SS 2012

PD Dr. Ulf Lorenz 2. and 3. May 2012

Dipl.-Math. David Meffert Version of April 26, 2012

Groupwork

Exercise G1 (Master-Theorem)

Determine, if possible, fixed bounds for the complexities of the recurrences (a) T(n) =4T(ⁿ₂) +n³,

(b) T(n) =4T(ⁿ₂) +n, (c) T(n) =4T(ⁿ₂) +n²logn, (d) T(n) =4T(ⁿ₂) +n². Hint:

Exercise G2 (Complexity)

(a) Let f,t:N→Rbe functions with f ∈O(t). ProveO(f) +O(t)⊆O(t)andO(f) +O(f)⊆O(t). (b) Does3³⁺ⁿ∈O(3ⁿ)hold?

(c) Does3³ⁿ∈O(3ⁿ)hold?

(d) Show thatO(f)·O(g) =O(f ·g)holds for f,g:N→R₊.

Remark: For real valued functions f,g:N →R one just substitutes f(n), g(n)with|f(n)|,|g(n)|in the definition of O(g).

Exercise G3 (Algorithms)

(a) Given two algorithmsAandB:

• AlgorithmAhas complexityO(f).

• AlgorithmBhas complexityO(g).

We want to look at two new algorithms usingAandB.

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Algorithm 1 INPUT :n∈N fori=1, ..., 100do

run algorithm A end for

fori=1, ...,ⁿ

2 do run algorithm B end for

Algorithm 2 if n≥30then

run algorithm A else

run algorithmus B end if

We already knowf ∈Ω(g). Determine the best possible estimates for the runtime of both algorithms.

(b) Take a look at algorithm3and determine the best possible estimate for its runtime. Justify you answer.

Algorithm 3 INPUT : n∈N m = n

whilem > 1do forj = 1,...,ⁿ

2do a=3·b c = a +b end for m =¹₂·m end while

Exercise G4 (Sets) Order the functions

n²,p

n,n!,nⁿ,n

by their complexity. Start with lowest complexity and use theo-notation. Determinen₀dependend onc>0in every of those cases, too.

Remark:

f ∈o(g):⇐⇒ ∀c>0∃n₀∈N∀n≥n₀: 0≤f(n)<c g(n)

Homework

Exercise H4 (Asymptotics) (14 points)

(a) Prove that forr₁,r₂∈R₊we haven^r1∈O(n^r2)andr₁ⁿ∈O(r₂ⁿ)iffr₁≤r₂. (b) Prove the following statements for functions f,t:N→R:

i. O(f) +O(f)⊆O(f).

ii. O(f)·O(t)⊆O(f ·t).

iii. max{f,t} ∈Θ(f +t)for f,t≥0.

Exercise H5 (A sorting algorithm) (10 points)

The algorithmSortListsorts a sequence of numbers in ascending order.

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Algorithm 4SortList(list)

INPUT: sequence of numbers,list=a₁, ...,a_n,a_i∈N ifn <=1then

returnlist else

leftlist=a₁, ...,a_dⁿ

2e

rightlist=a_dⁿ

2e+1, ...,a_n

return Sort(SortList(lelftlist),SortList(rightlist)) end if

Algorithm 5Sort(rightlist,leftlist) INPUT: two sequences of numbers:

rightlist=a₁, ...,a_l,leftlist=b₁, ...,b_k,a_i,b_i∈N newlist

whilerightlistandleftlistnot emptydo

iffirst element ofleftlist<= first element ofrightlistthen

append first element ofleftlisttonewlistand delete it fromleftlist else

append first element ofrightlisttonewlistand delete it fromrightlist end if

end while

whileleftlistnot emptydo

append first element ofleftlisttonewlistand delete it fromleftlist end while

whilerightlistnot emptydo

append first element ofrightlisttonewlistand delete it fromleftlist end while

returnnewlist

(a) Sort the sequence9, 10, 7, 3, 1, 2, 12, 9, 23in ascending order by using the algorithmSortList. Make sure to include detailed steps for the algorithm in your solution to indicate that you understand how it works.

(b) What is the runtime of the algorithmSortList?

Exercise H6 (6 points)

Given algorithm 6. What does the algorithm? Determine its runtime.

Algorithm 6 INPUT :n∈N K1 = 2;

K2 = n;

whileK2 > K1do K2 = n/K1

ifdK2e==K2then return K1 else

K1=K1+1 end if end while return 0

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