Time Complexity of Algorithms

(Asymptotic Notations)

• The level in difficulty in solving mathematically posed problems as measured by– The time

(time complexity)– memory space required– (space complexity)

What is Complexity?

1. CorrectnessAn algorithm is said to be correct if • For every input, it halts with correct output.• An incorrect algorithm might not halt at all OR• It might halt with an answer other than desired one.• Correct algorithm solves a computational problem2.Algorithm EfficiencyMeasuring efficiency of an algorithm, • do its analysis i.e. growth rate. • Compare efficiencies of different algorithms for the

same problem.

Major Factors in Algorithms Design

• Algorithm analysis means predicting resources such as– computational time– memory

• Worst case analysis– Provides an upper bound on running time– An absolute guarantee

• Average case analysis– Provides the expected running time– Very useful, but treat with care: what is “average”?

• Random (equally likely) inputs• Real-life inputs

Complexity Analysis

Asymptotic Notations Properties• Categorize algorithms based on asymptotic growth

rate e.g. linear, quadratic, exponential• Ignore small constant and small inputs • Estimate upper bound and lower bound on growth

rate of time complexity function• Describe running time of algorithm as n grows to .Limitations• not always useful for analysis on fixed-size inputs. • All results are for sufficiently large inputs.

Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

Asymptotic Notations , O, , o, We use to mean “order exactly”, O to mean “order at most”, to mean “order at least”, o to mean “tight upper bound”, to mean “tight lower bound”,

Define a set of functions: which is in practice used to compare two function sizes.

Asymptotic Notations

.for boundupper

ally asymptotican is function means allfor , 0

such that and constants positiveexist there:functions, ofset theby denoted ,0function given aFor

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o

Big-Oh Notation (O)

Intuitively: Set of all functions whose rate of growth is the same as or lower than that of g(n).

We may write f(n) = O(g(n)) OR f(n) O(g(n))

If f, g: N R+, then we can define Big-Oh as

g(n) is an asymptotic upper bound for f(n).

Big-Oh Notation

c > 0, n0 0 and n n0, 0 f(n) c.g(n)

f(n) O(g(n))

ExamplesExample 1: Prove that 2n2 O(n3)Proof:

Assume that f(n) = 2n2 , and g(n) = n3 f(n) O(g(n)) ?Now we have to find the existence of c and n0

f(n) ≤ c.g(n) 2n2 ≤ c.n3 2 ≤ c.nif we take, c = 1 and n0= 2 OR c = 2 and n0= 1 then 2n2 ≤ c.n3 Hence f(n) O(g(n)), c = 1 and n0= 2

Examples

ExamplesExample 2: Prove that n2 O(n2)Proof:

Assume that f(n) = n2 , and g(n) = n2 Now we have to show that f(n) O(g(n))

Since f(n) ≤ c.g(n) n2 ≤ c.n2 1 ≤ c, take, c = 1, n0= 1

Then n2 ≤ c.n2 for c = 1 and n 1Hence, 2n2 O(n2), where c = 1 and n0= 1

Examples

ExamplesExample 3: Prove that 1000.n2 + 1000.n O(n2)Proof: Assume that f(n) = 1000.n2 + 1000.n, and g(n) = n2 We have to find existence of c and n0 such that 0 ≤ f(n) ≤ c.g(n) n n0 1000.n2 + 1000.n ≤ c.n2 = 1001.n2, for c = 1001 1000.n2 + 1000.n ≤ 1001.n2

1000.n ≤ n2 n2 1000.n n2 - 1000.n 0 n (n-1000) 0, this true for n 1000 f(n) ≤ c.g(n) n n0 and c = 1001

Hence f(n) O(g(n)) for c = 1001 and n0 = 1000

Examples

ExamplesExample 4: Prove that n3 O(n2)Proof:

On contrary we assume that there exist some positive constants c and n0 such that 0 ≤ n3 ≤ c.n2 n n0 0 ≤ n3 ≤ c.n2 n ≤ cSince c is any fixed number and n is any arbitrary constant, therefore n ≤ c is not possible in general.Hence our supposition is wrong and n3 ≤ c.n2, n n0 is not true for any combination of c and n0. And hence, n3 O(n2)

Examples

.for boundlower

ally asymptotican is function that means , allfor 0

such that and constants positiveexist there:functions, ofset theby denote function given aFor

nfngngnf

nnnfncgncnfng

ngng

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o

Big-Omega Notation ()

Intuitively: Set of all functions whose rate of growth is the same as or higher than that of g(n).

We may write f(n) = (g(n)) OR f(n) (g(n))

If f, g: N R+, then we can define Big-Omega as

Big-Omega Notation

g(n) is an asymptotically lower bound for f(n). c > 0, n0 0 , n n0, f(n) c.g(n)

f(n) (g(n))

ExamplesExample 1: Prove that 5.n2 (n)Proof: Assume that f(n) = 5.n2 , and g(n) = n

f(n) (g(n)) ?We have to find the existence of c and n0 s.t.

c.g(n) ≤ f(n) n n0

c.n ≤ 5.n2 c ≤ 5.nif we take, c = 5 and n0= 1 then

c.n ≤ 5.n2 n n0

And hence f(n) (g(n)), for c = 5 and n0= 1

Examples

ExamplesExample 2: Prove that 5.n + 10 (n)Proof: Assume that f(n) = 5.n + 10, and g(n) = n

f(n) (g(n)) ?We have to find the existence of c and n0 s.t.

c.g(n) ≤ f(n) n n0 c.n ≤ 5.n + 10 c.n ≤ 5.n + 10.n c ≤ 15.nif we take, c = 15 and n0= 1 then

c.n ≤ 5.n + 10 n n0And hence f(n) (g(n)), for c = 15 and n0= 1

Examples

ExamplesExample 3: Prove that 100.n + 5 (n2)Proof:

Let f(n) = 100.n + 5, and g(n) = n2 Assume that f(n) (g(n)) ?Now if f(n) (g(n)) then there exist c and n0 s.t.c.g(n) ≤ f(n) n n0 c.n2 ≤ 100.n + 5 c.n ≤ 100 + 5/n n ≤ 100/c, for a very large n, which is not possible

And hence f(n) (g(n))

Examples

Theta Notation ()

.for bound ally tightasymptotican is and factor,constant a within to toequal is function means

allfor 0 such that and , constants positiveexist there:

functions, ofset theby denoted function given aFor

21

21

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ngng

o

o

Intuitively: Set of all functions that have same rate of growth as g(n).

We may write f(n) = (g(n)) OR f(n) (g(n))

If f, g: N R+, then we can define Big-Theta as

We say that g(n) is an asymptotically tight bound for f(n).

f(n) (g(n))

c1> 0, c2> 0, n0 0, n n0, c2.g(n) f(n) c1.g(n)

Theta Notation

Example 1: Prove that ½.n2 – ½.n = (n2)Proof

Assume that f(n) = ½.n2 – ½.n, and g(n) = n2 f(n) (g(n))? We have to find the existence of c1, c2 and n0 s.t.c1.g(n) ≤ f(n) ≤ c2.g(n) n n0

Since, ½ n2 - ½ n ≤ ½ n2 n ≥ 0 if c2= ½ and

½ n2 - ½ n ≥ ½ n2 - ½ n . ½ n ( n ≥ 2 ) = ¼ n2, c1= ¼

Hence ½ n2 - ½ n ≤ ½ n2 ≤ ½ n2 - ½ nc1.g(n) ≤ f(n) ≤ c2.g(n) n ≥ 2, c1= ¼, c2 = ½Hence f(n) (g(n)) ½.n2 – ½.n = (n2)

Theta Notation

Example 1: Prove that 2.n2 + 3.n + 6 (n3)Proof: Let f(n) = 2.n2 + 3.n + 6, and g(n) = n3

we have to show that f(n) (g(n))On contrary assume that f(n) (g(n)) i.e.there exist some positive constants c1, c2 and n0 such that: c1.g(n) ≤ f(n) ≤ c2.g(n) c1.g(n) ≤ f(n) ≤ c2.g(n) c1.n3 ≤ 2.n2 + 3.n + 6 ≤ c2. n3 c1.n ≤ 2 + 3/n + 6/n2 ≤ c2. n c1.n ≤ 2 ≤ c2. n, for large n n ≤ 2/c1 ≤ c2/c1.n which is not possibleHence f(n) (g(n)) 2.n2 + 3.n + 6 (n3)

Theta Notation

Usefulness of Notations• It is not always possible to determine behaviour of

an algorithm using Θ-notation. • For example, given a problem with n inputs, we may

have an algorithm to solve it in a.n2 time when n is even and c.n time when n is odd. OR

• We may prove that an algorithm never uses more than e.n2 time and never less than f.n time.

• In either case we can neither claim (n) nor (n2) to be the order of the time usage of the algorithm.

• Big O and notation will allow us to give at least partial information