CS502 Fundamentals of Algorithms assignment no 1 spring 25 October 2011

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[B]Guidelines
RULES FOR CALCULATING TIME COMPLEXITY AND BIG-OH
Rule 00
Normally these formulas are very handy:
If x y = z then y z x = log
Also,
( )
2 1
1
n
n
i
i a a n a + = Σ=
( 1)
1 2
+ = Σ=
i n n
n
i r
r r
m m
k
k
−
−
=
+
= Σ
1
1 1
0
( for n 1)
6
( 1)(2 1)
1
2 >=
+ +
= Σ=
i n n n
n
i
Rule 0
The condition that stops a loop executes ONE MORE time than the loop itself
(the last time is when it is evaluated false)
Rule 1
for (i=0;i<n;i=i+k) Anything inside the loop will run approximately n/k times
Rule 2
for (i=n;i>0;i=i-k) Anything inside the loop will run approximately n/k times
Rule 3
for (i=1;i<n;i=i*k) Anything inside the loop will run approximately logkn times
Rule 4
for(i=1;i<=n;++i)
for (j=1;j<=i;++j)
The above nested loop approximately runs ½ n(n+1) times.
The variable j depends upon the value of i
Rule 5
for(i=1;i<=n;i=i*2)
for (j=1;j<=i;++j)
The statements in the above nested loop approximately run
2n-1 times.
The variable j depends upon the value of i
Rule 6
If the loop variables are independent then the total times a statement inside a
nested loop is executed is equal to the product of the times the individual loops
run
e.g. for (i=0;i<n;++i)
for (j=0;j<m;++j)
A statement inside the above nested loop will run n*m times
Other Guidelines
While loop related information
Complexity of “while” loop depend upon the initial entrance condition if it remains
true for “n” iterations it will be “n+1”; Note here “1” will be added for the last time
check of the condition .Here this will be clear to you if the some logical conditions
are checked other then counters then all complexity will be based on scenario of
the problem and nature of the logical condition.
Function Growth rate concept
If some function f1(x)>f2(x) for positive values of x then the function f1(x) is said to have
greater growth rate then f2(x). For example f1(x)=x5 and f2(x)= x6 it is obvious that f1(x)
has greater growth rate ( 26 > 25).This concept relate to complexity of algorithm ,an
algorithm having greater growth rate function means the algorithm has greater
complexity here f2(x) is more complex then f1(x).
Estimated Time 2.5 hour
For all parts of the question to understand maximum time is 1.25 hours and for
solution maximum time is 1.25 hour. It all depends upon your sheer
concentration.
Question (4*5)
Write piece of pseudo codes having the following time
complexities:
a) T1 (n) =n+
0
n
i
i
= Σ
b) T2 (n) =6n (log12n) +1+ (m/7) log (m)
c) O (z+m)
d) T3 (n) =
1
5 (1/)
n
i
n i
= Σ
Note: Where the base of “log “has not been mentioned take it as base “2”.[/B]