CMSI 282 Problem Set #1
Due January 29, 2009
1) Prove:
a) logba = 1 / ( logab )
b) logca = ( logcb ) ( logba )
c) logb(xy) = ( logbx ) + ( logby )
d) a
log b
= b
log a
e) for any integer n ≥ 0, 1
2
+ 2
2
+ ... + n
2
= n(n+1)(2n+1) / 6
2) Which of these recurrences can be classified using the Master Theorem?
a) Ta(n) = if n = 1 then 1 else 2Ta(n-1) + (n-2)
b) Tb(n) = if n = 1 then 1 else 2Tb(n/3) + (n-2)
c) Tc(n) = if n = 1 then 1 else 3Tc(n/2) + 5n
2
d) Td(n) = if n = 1 then 1 else 2Td(n-1) + 3Td (n-2)
e) Te(n) = if n = 1 then 5 else 16Te(n/3) + 5n
3
3) Classify these recurrences using big-theta notation. In each case, indicate how you
arrived at your conclusion (e.g., MASTER THEOREM, or, REPEATED
SUBSTITUTION, etc.):
a) Ta(n) = if n = 1 then 1 else Ta(n-1) + (n-1)
b) Tb(n) = if n = 1 then 1 else 2Tb(n/3) + (n-2)
c) Tc(n) = if n = 1 then 1 else 3Tc(n/2) + 5n
2
d) Td(n) = if n = 1 then 1 else 16Td(n/3) + 5n
2
e) Te(n) = if n = 1 then 1 else 16Te(n/3) + 5n
3
f) Tf(n) = if n = 1 then 1 else 16Tf(n/4) + 5n
4
g) Tg(n) = if n = 1 then 1 else 3Tg(n/2) + 5
