Jim Lambers
Math 6A
Spring Quarter 2003-04
Lecture 16 Examples
These examples correspond to Section 5.3 in the text.
Example Suppose that we need to sort nreal numbers x1, x2, . . . , xnin ascending order. We are
using the algorithm known as insertion sort. This algorithm copies the numbers in an array A,
of size n, to another array B, also of size n, one at a time in such a way that Bis always sorted.
It maintains the sorted order in Bby shifting elements as needed when a new one is added. The
following pseudo-code describes how insertion sort can be implemented.
for i= 1 to ndo
j=i−1
while j≥1and B[j]> A[i]do
B[j+ 1] = B[j] (shift element to make room for new element)
j=j−1
end
B[j] = A[i] (place new element in correct position)
end
One way to measure the complexity of this algorithm is in terms of the number of times elements
need to be shifted in order to maintain the sorted order of B. In the best case, when the array A
is already sorted, no shifting is required. In the worst case, when Ais sorted in descending order,
i−1 shifts are required to place the ith element of Ain the correct position within B, so the total
number of shifts is n
X
i=1
i−1 =
n
X
i=1
i−
n
X
i=1
1 = n(n+ 1)
2−n=n2−n
2.
What is the average-case complexity of this algorithm?
Solution Let Sbe the sample space of possible inputs of size n; that is, each input is a list of
nnumbers to be sorted. Assume that each possible ordering is equally likely. If Xis a random
variable whose value for each input is the number of times elements need to be shifted, then the
average-case complexity is given by E(X).
For each i= 1,2, . . . , n, let Xibe the random variable whose value is the number of shifts
required to place the ith element of Ain the proper position in B. Then Xican assume any value
between 0 and i−1, corresponding to the best case and worst case, respectively. If we assume that
1
