Combinatorial optimization

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Combinatorial optimization

Papp László <lazsa@cs.bme.hu>

2022. spring 1st practice

1. Determine whether these functions are inO(n²), Ω(n²) orΘ(n²). (a) 10n²+ 20n+|sin(n)|

(b) 8n²log(n) 2. Show that:

(a) 8n²log₂n∈O(n³) (b) 2n−√

n∈Ω(n)

(c) (n+ 32)(2n²+ 12n)∈Θ(n³) (d) (nlog₂n+n²)(n³ + 2)∈O(n⁵) 3. Show that8nlog₂n /∈Θ(n).

4. Sort these function to increasing order according to their growing speed. So iff_i preceeds f_j that means that f_i ∈O(f_j) and f_j ∈/O(f_i).

f₁(n) = 8n³ f₂(n) = 5√

n+ 1000n f₃(n) = 2^(log²ⁿ⁾² f₄(n) = 1514n²log₂(n)

5. We have two algorithms which solve the same problem. The time complexity of algorithm A is the function f_A(n) and similarly the time complexity ofB is f_B(n). We know that f_A(n)∈O(f_B(n)). Are these statements true?

(a) Algorithm A is faster than algorithm B on all possible inputs?

(b) Algorithm A is faster than algorithm B on all suciently large inputs?

6. LetG= (V(G), E(G))be the following graph: V(G) = {1,2,3,4,5,6,7,8}

E(G) ={{1,2},{1,4},{2,3},{3,4},{3,5},{7,8}

LetH = (V(H), E(H)) be the following graph: V(H) = {1,2,3,4}

E(H) ={{1,2},{1,4},{2,3}. (a) Draw a diagram of G.

(b) Determine the degree of vertex 3. (c) Does G contain an isolated vertex?

(d) Is Gsimple?

(e) How many cycles does G contain?

(f) Draw a diagram of H. (g) Is H a subgraph of G?

(h) Is H an induced subgraph of G?

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7. Consider the functions f(n) = 1.5n! and g(n) = 200(n − 1)!. Prove or disprove the following statements:

(a) f(n)∈O(g(n)) (b) g(n)∈O(f(n)) (c) f(n)∈Ω(g(n)) (d) g(n)∈Ω(f(n)) (e) f(n)∈Θ(g(n)) (f) g(n)∈Θ(f(n))

8. Which a, b >1integers satisfy the following?

(a) log_an∈O(log_bn) (b) 2^an ∈O(2^bn)

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