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Answer the questions below before the end of class and turn it in. The results you give will not
count toward your grade in the class. The purpose of this test is for me to get a feel for how
well the class overall understands this material, and for you to see how well you understand
this material.
1) Set up the formula for calculating the average case complexity for Sequential Search
(AKA Linear Search) on an unsorted array. Assume that the "target" element is
definitely in the array, but that there is a 50% chance it is in the first position (and
equally likely in all other positions). Just set up the formula - you don't have to solve it.
For position 1, we do one comparison so T(I) = 1. P(I) = 0.5.
For each position i from 2 to n, we T(I) = i and P(I) is 0.5/(n-1).
So A(n) = 1
x
0.5 + Sum_{i=2 to n} (0.5
x
i / (n-1))
2) Argue that one cannot find the minimum element in an unsorted array of size n without
doing at least n-1 comparisons.
To find the minimum you must show that the other n-1 items are not the minimum,
which implies that they win when compared to some other item. One comparison
produces exactly one winner, so you need n-1 comparisons.
3) Selection sort works by finding the largest element in an array from position 0 to
position n-1, moving it to the end of the array, and then repeating this process with the
sub-array from 0 to n-2, etc.
(A) Given what you know about FindMax, how many comparisons does selection sort
do? Sum_{i=1 to n-1} (n-i) = Sum{i=1 to n-1} i = n(n-1)/2
(B) FindMax is optimal: does this imply selection sort is optimal? No!
4) Show that any polynomial grows more slowly than any exponential (see the slide)
using the method shown in class using limits and L'Hopital's rule.
This is problem O5 of HW1. See solution posted for that. (Doh! I didnโt mean to put
this hard a problem on this quiz! I meant you to compare log n to n here.)
5) How often is x incremented in the following code?
for i = 1 to n
for j = i downto 1
x = x + 1
Sum_{i=1 to n} (i) = n(n+1)/2
6) How often is x incremented in the following code?
j = n
while (j >= 1) {
for i = 1 to j
x = x + 1
j = j / 2
}
See page 49-50 in the textbook. If n is a power of 2, this is just 2(n-1)