Transcript Algorithm Complexity - monjur-ul

Algorithm Complexity
LEVEL II, TERM II
CSE – 243
MD. MONJUR-UL-HASAN
LECTURER
DEPT OF CSE, CUET
EMAIL: [email protected]
A motivation for complexity
2
 Talking about running time and memory space of
algorithms is difficult.



Different computers will run them at different speeds and
memory
Different problem sizes run for a different amount of time and
memory(even on same computer).
Different inputs of the same size run for a different amount of
time and memory.
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE
CUET
Definition: Complexity
3
 Efficiency or complexity of an algorithm is stated
as a function relating the input length to the number
of steps (time complexity) or storage locations
(space complexity).
log n
n
n log n
n2
n3
2n
0
1
0
1
1
2
1
2
2
4
8
4
2
4
8
16
64
16
3
8
24
64
512
256
4
16
64
256
4096
65536
5
32
160
1024
32768
4294967296
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
4
exp (n)
More Curve
n
f(n)
log n
n
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Fig 1.3
Fundamental
Computer
Algorithm
By,
E. Horowitz
S. Sahni
S. Rajasekaran
Using Upper and Lower Bounds to talk about running time.
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Running
time (and
memory)
on some
computer
Upper Bound
Different inputs
Lower Bound
5
10
Md. Monjur-ul-hasan. Lecturer, Dept of CSE CUET
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Problem Size
Bounds on running time
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 We want to bound the running time:
 “for arrays of size n it takes my computer 45·n2
milliseconds at most to sort the array with bubblesort”
 But what about other computers? (faster/slower)
 Instead we say:
 “On any computer, there is a constant c such that it takes
at most c·n2 time units to sort an array of size n with
bubblesort”

How is this sentence useful?
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Big O notation.
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 We say that
running time of an alg. is O(f(n))
If there exists c such that for large enough n:
RunningTime(n) < c·f(n)
Translation: From some point c·f(n) is an upper bound
on the running time.
 Why only for “large enough n”?
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Big-O notation
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 Do not assume that because we analyze running time
asymptotically we no longer care about the
constants.
 If you have two algorithms with different asymptotic
limits – easy to choose.

(may be misleading if constants are really large).
 Usually we are happy when something runs in
polynomial time and not exponential time. O(n2) is
much much better than O(2n).
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Use of big-O notation
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 Example:
The running time of bubble sort on arrays of size n is
O(n2).
Translation: for a sufficiently large array, the running
time is less than c·n2 for some c (No matter how bad
the input is).
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Big Omega
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 For lower bounds we use Ω(f(n)).
 I.e., if we say the running time of an algorithm is
Ω(f(n)), we mean that
RunningTime(n)>c·f(n) for some c.
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
An Example:
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 Checking a the number n is prime.
 We can try and divide the number n by all integers in the range
2,…,n1/2.
 Best case:
sometimes we find it isn’t prime really quickly (e.g. it divides
by 2).
 Worst case:
sometimes the number is prime and we try n1/2 different
divisions.
 Running time is O(n1/2), and is also Ω(1).
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Bounds might not be tight
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 We saw that finding out if a number n is prime takes
us O(n1/2) time.
 It also takes us O(n5) time.
 The second bound guarantees less. It is not tight.
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Big Theta Notation
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 The function f(n) = Θ(g(n)) iff there exists positive
constants c1 and c2 such that c1Xg(n) < f(n) <
c2Xg(n).
 Average time
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
Measurement: Tabular Method
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Algorithm
Significant
Frequency
Total
Algorithm Sum(a,n)
{
sum := 0.0
for i:= 1 to n do
sum := sum +a[i];
return sum
}
0
0
1
1
1
1
0
1
n+1
n
1
-
0
0
1
n+1
n
1
0
Total
Complexity: O(n)
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
2n+3
Bounds might not be tight
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Algorithm
Significant
Frequency
Total
Algorithm Add(a,b,c,m,n)
{
for i:= 1 to m do
for j:=1 to n do
c[I,j] := a[i,j] + b[i,j];
}
0
0
1
1
1
0
m+1
n+1
n
0
0
m+1
mn+m
mn
Total
Complexity: O(mn)
Md. Monjur-ul-hasan. Lecturer, Dept. of CSE CUET
2mn+2m+1