P1.
Solve the recurrence relation
without
using Master’s theorem:
T(N) = 3T(N/2) + cN
Ans.:
Assume N = 2
k
T(2
k
)
= 3 T(2
k-1
) + c 2
k
= 3
2
T(2
k-2
) + c ( 2
k
+ 3* 2
k-1
)
…. After k steps
= 3
k
T(1) + c ( 2
k
+ 3* 2
k-1
+ 3
2
* 2
k-2
+ … + 3
k-1
* 2)
= 3
k
d + c 2
k
(1 + 3/2 + (3/2)
2
+ (3/2)
3
+ …. + (3/2)
k-1
)
= 3
k
d + c N [ (3/2)
k
–1] / [3/2 –1]
= 3
k
d + 2c N ( 3
k
/N – 1)
= 3
k
(d + 2c) – 2cN
Now, 3
k
= (2
log 3
)
k
= (2
k
)
log 3
= N
log 3
= N
log 3
(2c + d) – 2cN
= Theta (N
log 3
)
P2.
A
and
B
are playing a guessing game where
B
first thinks up an integer
X
(positive,
negative or zero, and could be of arbitrarily large magnitude) and
A
tries to guess it. In
response to
A
’s guess,
B
gives exactly one of the following three replies:
a)
Try a bigger number
b)
Try a smaller number or
c)
You got it!!
Design an efficient algorithm to minimize the number of guesses
A
has to make. An
example (not necessarily an efficient one) below:
B thinks up the number 35
A’s guess
B’s response
10
Try a bigger number
20
Try a bigger number
30
Try a bigger number
40
Try a smaller number
35
You got it

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