2-AnalysisOfAlgs.ppt

Asymptotic Analysis
CLRS, Sections 3.1 – 3.2

Asymptotic Complexity
Running time of an algorithm as a function of
input size n, for large n.
Expressed using only the highest-order
term in the expression for the exact running
time.
Describes behavior of function in the limit.
Written using Asymptotic Notation.
CS 321 - Data Structures 2

Asymptotic Notation
 Q, O, W, o, w
Defined for functions over the natural numbers.
– Ex: f(n) = Q(n2).
• Describes how f(n) grows in comparison to n2.
Defines a set of functions; in practice used to
compare growth rate of two functions.
The notations describe the relationship between
the defining function and a defined set of
functions.

O-notation
O(g(n)) = {f(n) :
 positive constants c and n0,
such that n  n0,we have 0 
f(n)  cg(n) }
For function g(n), we define
O(g(n)), big-O of n, as the set:
g(n) is an asymptotic upper bound for f(n).
Intuitively: Set of all functions whose rate of
growth is the same as or lower than that of g(n).
f(n) = Q(g(n))  f(n) = O(g(n))
Q(g(n))  O(g(n))

W -notation
g(n) is an asymptotic lower bound for f(n).
growth is the same as or higher than that of g(n).
f(n) = Q(g(n))  f(n) = W(g(n))
Q(g(n))  W(g(n))
W(g(n)) = {f(n) :
 positive constants c and n0,
such that n  n0,
we have 0  cg(n)  f(n)}
W(g(n)), big-Omega of n, as the set:

Q-notation
Q(g(n)) = {f(n) :
 positive constants c1, c2, and n0,
such that n  n0, we have 0 
c1g(n)  f(n)  c2g(n)}
Q(g(n)), big-Theta of n, as the set:
g(n) is an asymptotically tight bound for f(n).
Intuitively: Set of all functions that
have the same rate of growth as g(n).

Relationship Between Q, O, W

Relationship Between Q, W, O
Q(g(n)) = O(g(n))  W(g(n))
In practice, asymptotically tight bounds are
obtained from asymptotic upper and lower
bounds.
Theorem : For any two functions g(n) and f(n),
f(n) = Q(g(n)) iff
f(n) = O(g(n)) and f(n) = W(g(n)).

o-notation
g(n) is an upper bound for f(n)that is not
asymptotically tight.
o(g(n)) = {f(n):  c > 0,  n0 > 0 such
that  n  n0, we have 0  f(n) < cg(n)}.
For a given function g(n), the set little-o:
growth is lower than that of g(n).

w(g(n)) = {f(n):  c > 0,  n0 > 0 such
that  n  n0, we have 0  cg(n) < f(n)}.
w -notation
g(n) is a lower bound for f(n) that is not
asymptotically tight.
For a given function g(n), the set little-omega:
growth is higher than that of g(n).

Comparison of Functions
f  g ~ a  b
f (n) = O(g(n)) ~ a  b
f (n) = W(g(n)) ~ a  b
f (n) = Q(g(n)) ~ a = b
f (n) = o(g(n)) ~ a < b
f (n) = w(g(n)) ~ a > b

Limit Definitions of Asymptotic
Notations
o-notation (little-o):
– f(n) becomes insignificant relative to g(n) as n
approaches infinity:
lim [f(n) / g(n)] = 0
n
w -notation (little-omega):
– f(n) becomes arbitrarily large relative to g(n) as
n approaches infinity:
lim [f(n) / g(n)] = 
n

Notations
Q-notation (Big-theta):
– f(n) relative to g(n)equals a constant, c, which is
greater than 0 and less than infinity as n approaches
infinity:
0 < lim [f(n) / g(n)] < 
n

Notations
O-notation (Big-o):
– f(n)  O(g(n))  f(n)  Q(g(n))and f(n)  o(g(n))
– f(n) relative to g(n)equals some value less than infinity as n
lim [f(n) / g(n)] < 
n
W -notation (Big-omega):
– f(n)  W(g(n))  f(n)  Q(g(n))and f(n)  w(g(n))
– f(n) relative to g(n)equals some value greater than 0 as n
0 < lim [f(n) / g(n)]
n

Examples
Use limit definitions to prove:
– 10n - 3n  O(n2)
– 3n4  W(n3)
– n2/2 - 3n  Q(n2)
– 22n Q(2n)

Examples
Use limit definitions to prove:
– 10n - 3n  O(n2) – Yes!
lim [ 10n - 3n / n2 ] = 0
n
– 3n4  W(n3) – Yes!
lim [ 3n4 / n3 ] = 
n
– n2/2 - 3n  Q(n2) – Yes!
lim [ n2/2 - 3n / n2 ] = 1/2
n
– 22n Q(2n) – No!
lim [ 22n / 2n ] = 
n

Why Do We Care?

T(n)
n n n log n n2
n3
n4
n10
2n
10 .01ms .03ms .1ms 1ms 10ms 10s 1ms
20 .02ms .09ms .4ms 8ms 160ms 2.84h 1ms
30 .03ms .15ms .9ms 27ms 810ms 6.83d 1s
40 .04ms .21ms 1.6ms 64ms 2.56ms 121d 18m
50 .05ms .28ms 2.5ms 125ms 6.25ms 3.1y 13d
100 .1ms .66ms 10ms 1ms 100ms 3171y 4´1013
y
103
1ms 9.96ms 1ms 1s 16.67m 3.17´1013
y 32´10283
y
104
10ms 130ms 100ms 16.67m 115.7d 3.17´1023
y
105
100ms 1.66ms 10s 11.57d 3171y 3.17´1033
y
106
1ms 19.92ms 16.67m 31.71y 3.17´107
y 3.17´1043
y
Complexity and Tractability
Assuming the microprocessor performs 1 billion ops/s.

2-AnalysisOfAlgs.ppt

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