CISC 621: Algorithm Design and Analysis
H/W H: Partition
Model Answer
1. A call to 'partition' first calculates the sum of the elements in the set taking ฮ(n) time.
It then calls 'SubsetSum' on the n elements if the sum is even. Thus the time taken by
'partition' in the worst case is:
T(n) = TSubsetSun (n) + ฮ(n)
In the worst case every call to 'SubsetSum' will make two recursive calls to itself on an
input of size (n โ 1). Thus the time taken by 'SubsetSum' on n elements is given by the
following recurrence relation:
TSubsetSun (n) = 2 TSubsetSun (n - 1) + ฮ(1) .1
The above recurrence may be solved in two ways:
1. Using the substitution method:
Since the question tells us that we can prove O(2 n ), we guess that the solution is
TSubsetSun(n) = O(2 n ). To prove by induction we note that the recurrence holds for the
boundary conditions n = 0 and for n = 1 and we assume that it holds for TSubsetSun(n โ
1), that is:
TSubsetSun(n โ 1) โค c2 n โ 1 โ b .2
where b is some constant. Substituting 2 in 1 (replacing ฮ(1) by a constant d), we
get:
TSubsetSun(n) = 2 (c2 n โ 1 โ b) + d
TSubsetSun(n) = c2 n โ 2b + d
TsubsetSun(n) โค c2 n โ b
Where the last inequality holds if (2b โ d) โค b or b โค d for all values of c. Thus we
get:
T(n) = O( 2n )
2. Using the recurrence tree method:
Drawing the recurrence tree (shown overleaf), we get:
TsubsetSun(n) = ฮ(1) (20 + 21 + 22 + 23 + 24 ... ... 2n )
TsubsetSun(n) =
๎ฎ๎ 1๎ ๎ฒ๎i=0๎
n๎2i๎=๎ฎ๎1๎๎2๎n๎1๎โ1๎
๎2โ1๎=๎ฎ๎1๎๎2๎n๎1๎โ1๎=O๎2n๎
