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Analysis & Design of Algorithms (CSCE 321)

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Analysis & Design of Algorithms (CSCE 321)

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Analysis & Design of Algorithms(CSCE 321)

Prof. Amr Goneid

Department of Computer Science, AUC

Part 2. Types of Complexities

Prof. Amr Goneid, AUC

Types of Complexities

Prof. Amr Goneid, AUC

Types of Complexities

- Rules for Upper Bound
- Comparing Complexities
- Types of Complexities

Prof. Amr Goneid, AUC

1. Rules for Upper Bound

If (k) is a constant, then:

- O(k) O(n)
- O(k f(n)) = O(f(n))

O(kn)

T(n)

O(n)

O(k)

n

Prof. Amr Goneid, AUC

Rules for Big O

- O(f(n)) + O(g(n)) = max(O(f(n)) ,O(g(n)))
- e.g. f(n) = 2n = O(n), g(n) = 0.1 n3 = O(n3)
T(n) = max(O(n) , O(n3)) = O(n3)

2n

O(n3)

0.1 n3

Prof. Amr Goneid, AUC

Rules for Big O

- O(f(n)) * O(g(n)) = O(f(n) * g(n))
e.g. f(n) = n, g(n) = n2

T(n) = n * n2 = O(n3)

Repeat n times

O(n2)

O(n3)

Prof. Amr Goneid, AUC

Rules for Big O

- O(log n) O(n)Prove!
Claim:

for all n ≥ 1, log n ≤ n.

The proof is by induction on n.

The claim is trivially true for n = 1, since 0 < 1.

Now suppose n ≥ 1 and log n ≤ n.

Then,

log(n + 1) ≤ log 2n

= logn + 1

≤ n + 1 (by induction hypothesis)

Prof. Amr Goneid, AUC

Rules for Big O

- For a polynomial of degree m,
- Prove!
- O(nm-1) O(nm) follows from above

Prof. Amr Goneid, AUC

Rules for Big O

Prof. Amr Goneid, AUC

Summary of Rules for Big-O

Prof. Amr Goneid, AUC

2. Comparing Complexities

- Dominance:
If lim(n->) f(n)/g(n) =

then f(n) dominates (i.e. grows faster), but if

lim(n->) f(n)/g(n) = 0

then g(n) dominates.

In the latter case, we say that f(n) = o(g(n))

little oh

Prof. Amr Goneid, AUC

Comparing Complexities

Examples:

- if a > b then na dominates nb since
lim(n->) nb / na = 0

- n2 dominates (3n+2) since
lim(n->) (3n+2) / n2 = 0

- n2 dominates (n log n) since
lim(n->) (n log n) / n2 = lim(n->) log n / n = 0

Prof. Amr Goneid, AUC

Using l’Hopital’s Rule (1696)

Example: Show that

f(n) = n(k+α) + nk (log n)2 and

g(n) = k n(k+α)

grow at the same rate.

Prof. Amr Goneid, AUC

Comparing Complexities

- Which grows faster:
f(n) = n3 or g(n) = n log n

f(n) = n 0.001 org(n) = log n

f(n) = 2n+1 org(n) = 2n

f(n) = 2n org(n) = 22n

Prof. Amr Goneid, AUC

Exercises

Which function has smaller complexity ?

- f = 100 n4g = n5
- f = log(log n3)g = log n
- f = n2g = n log n
- f = 50 n5 + n2 + ng = n5
- f = eng = n!

Prof. Amr Goneid, AUC

3. Types of Complexities

- Constant Complexity
- T(n) = constant independent of (n)
- Runs in constant amount of time O(1)
- Example: cout << a[0][0]

- Polynomial Complexity
- T(n)=amnm+…+ a2n2 + a1n1 + a0
- If m=1, then O(a1n+a0) O(n)
- If m > 1, then O(nm) as nm dominates

Prof. Amr Goneid, AUC

Types of Complexities

- Logarithmic Complexity
- Log2n=m is equivalent to n=2m
- Reduces the problem to half O(log2n)
- Example: Binary Search
T(n) = O(log2n)

Much faster than Linear Search which has T(n) = O(n)

Prof. Amr Goneid, AUC

Types of Complexities

- Exponential
- Example: List all the subsets of a set of n elements {a,b,c}
{a,b,c}, {a,b},{a,c},{b,c},{a},{b},{c},{}

- Number of operations T(n) = O(2n)
- Exponential expansion of the problem O(an) where a is a constant greater than 1

- Example: List all the subsets of a set of n elements {a,b,c}

Prof. Amr Goneid, AUC

Types of Complexities

- Factorial time Algorithms
- Example:
- Traveling salesperson problem (TSP):
Find the best route to take in visiting n cities away from home. What are the number of possible routes? For 3 cities: (A,B,C)

Prof. Amr Goneid, AUC

Possible routes in a TSP

- Number of operations = 3!=6, Hence T(n) = n!
- Expansion of the problem O(n!)

Prof. Amr Goneid, AUC

Execution Time Example

- Example:
- For the exponential algorithm of listing all subsets of a given set, assume the set size to be of 1024 elements
- Number of operations is 21024 about 1.8*10308
- If we can list a subset every nanosecond the process will take 5.7 * 10291 yr!!!

Prof. Amr Goneid, AUC

P and NP – Times

- P (Polynomial) Times:
O(1), O(log n), O(log n)2, O(n), O(n log n),

O(n2), O(n3), ….

- NP (Non-Polynomial) Times:
O(2n) , O(en) , O(n!) , O(nn) , …..

Prof. Amr Goneid, AUC

P and NP – Times

- Polynomial time is
GOOD

Try to reduce the polynomial

power

- NP (e.g. Exponential) Time is
BAD

Prof. Amr Goneid, AUC