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# Analysis & Design of Algorithms (CSCE 321) - PowerPoint PPT Presentation

Analysis & Design of Algorithms (CSCE 321). Prof. Amr Goneid Department of Computer Science, AUC Part 2. Types of Complexities. Types of Complexities. Types of Complexities. Rules for Upper Bound Comparing Complexities Types of Complexities. 1. Rules for Upper Bound.

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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

Prof. Amr Goneid, AUC

• Rules for Upper Bound

• Comparing Complexities

• Types of Complexities

Prof. Amr Goneid, AUC

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

• 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

• 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

• 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

• For a polynomial of degree m,

• Prove!

• O(nm-1)  O(nm) follows from above

Prof. Amr Goneid, AUC

Prof. Amr Goneid, AUC

Prof. Amr Goneid, AUC

• 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

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

• 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

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

• 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

O(n3)

O(n2)

Log T(n)

O(n)

n

Prof. Amr Goneid, AUC

• 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

T(n)

O(n)

O(log2n)

n

Prof. Amr Goneid, AUC

• 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

Prof. Amr Goneid, AUC

O(2n)

Log T(n)

O(n3)

O(n)

n

Prof. Amr Goneid, AUC

• 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

• Number of operations = 3!=6, Hence T(n) = n!

• Expansion of the problem  O(n!)

Prof. Amr Goneid, AUC

O(nn)

O(n!)

Log T(n)

O(2n)

n

Prof. Amr Goneid, AUC

• 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