asymptotic analysis.pdf

7/30/2019 Asymptotic Analysis.pdf

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As m totic Anal sis


2/23

How many resources will it take to solve a problem ofa given size?

time space

Expressed as a function of problem size (beyond

some minimum size) how do requirements grow as size grows?

Problem size

number of elements to be handled size of thing to be operated on


3/23

Constant

f(n) = c

ex: ettin arra element at known locationtrying on a shirt

callin a friend for advice before an interview

Linearf n = cn + ossible lower order terms

ex: finding particular element in array (sequential

search)trying on all your shirts

callin all our n friends for dress advice


4/23

Growth Functions (cont)

Quadratic

= 2

ex: sorting all the elements in an array (using bubblesort tr in all our shirts n with all our ties(n)

havin conference calls with each air of n

friendsPolynomial

f(n) = cnk [ + possible lower order terms]

ex: looking for maximum substrings in arraytrying on all combinations of k separates (n of

each)

having conferences calls with each k-tuple of nfriends


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Growth Functions (cont)

Exponential

f(n) = cn [+ ossible lower order termsex: constructing all possible orders of array elements

Logarithmicf n = lo n + ossible lower order terms

ex: finding a particular array element (binary search)

tr in on all combinationschoosing movie with n friends using phone tree


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What happens as problem size grows really, really

large? (in the limit)

constants dont matter lower order terms dont matter


7/23

What particular input (of given size) gives

worst/best/average complexity?

Mileage example: how much petrol does it take to go20 kms ?

Worst case: all uphill Best case: all downhill, just coast

Average case: average terrain


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Growth of Various Functions

Function Growth

10000

f(n)

5n 0.1n^2+2n+40 2 n log n 0.00001 2^n

8000

9000

6000

7000

4000

5000

2000

3000

0

1000

n

10 210 410 610 810 1010


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Consider sequential search on an unsorted array of

length n, what is time complexity?

Best case:

Worst case:

Average case:


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Complexity Bounds

Upper bound (big O):

=

T(n) cf(n) for some constants c, n0

and n n0

T(n) = (g(n)) if

, 0 0Exact bound (theta):

=

T(n) = O(h(n)) and T(n) = (h(n))

T(n) = o(p(n)) if

n = p n an n p n


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1. If f n = O n and n = O h n then f n =

O(h(n))

. n = g n or any > , en n = g n

3. If f1(n) = O(g1(n)) and f2(n) = O(g2(n)),

then f1(n) + f2(n) = O(max (g1(n), g2(n)))

. 1 n = g1 n an 2 n = g2 n ,

then f1(n) * f2(n) = O(g1(n) * g2(n))


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Code:

a = b;

Complexity:


13/23

Code:

sum = 0;

f or ( i =1; i


14/23

Code:

sum = 0;

f or ( j =1; j


15/23

Code:

sum1 = 0;

= =

f or ( j =1; j


16/23

Code:sum2 = 0;

f or i =1 i


17/23

Code:

sum1 = 0;

= = *=

f or ( j =1; j


18/23

Code:sum2 = 0;

f or k=1 k


19/23

1. Is upper bound the same as worst case?

2. Does lower bound ha en with shortest in ut?

3. What if there are multi le arameters?

Ex: Rank order of p pixels in c colorsf or ( i = 0; i < c; i ++)

count [ i ] = 0;

f or ( i = 0; i < p; i ++)count [ val ue( i ) ] ++;

sor t ( count )


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Does it matter?

What determines s ace com lexit ?

How can ou reduce it?

What tradeoffs are involved?


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Theorem:

O(cf(x) = O(f(x))

Proof: T(x) = O(cf(x)) implies that there are constants c0

and n0 such that T(x) c0(cf(x)) when x n0

Therefore, T(x) c1(f(x)) when x n0 where c1 =c0c

Therefore, T(x) = O(f(x))


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Theorem:

Let T1(n) = O(f(n)) and T2(n) = O(g(n)).

Then T1(n) + T2(n) = ???


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Theorem:

Let T1(n) = O(f(n)) and T2(n) = O(g(n)).

Then T1(n) * T2(n) = ???

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