Question
A. Find a theta notation for the best-case time required by the following algorithm: is_key(s, n, key) { for i = 1 to n -
A. Find a theta notation for the best-case time required by the following algorithm:
is_key(s, n, key) {
for i = 1 to n - 1
for j = i + 1 to n
if (si + sj ==key)
return 1
else
return 0
}
B. Trace the algorithm of the Insertion Sort for the input
40 18 134 60
Algorithm: the insertion sort
Input: s, n
Output: s (sorted)
insertion_sort(s, n) {
for i = 2 to n {
val = si // save si so it can be inserted into the correct place
j = i - 1
// if val < sj, move sj right to make room for si
while (j 1 val < sj) {
sj+1 = sj
j = j 1
}
sj+1 = val // insert val
}
}
D. Let consider that
1 + 2 + + n = Wn2 + Xn + Y
For all n, and for some constant W, X and Y.
1. Assuming that this is true, plug in n = 1, 2, 3 to obtain three equations in the three unknowns W, X and Y.
2. Solve for W, X and Y with the three equations obtained in the previous question.
3. Prove using the mathematical induction that the statement is true.
E. Use the mathematical induction to show that (Tn is a Fibonacci Sequence):
Tn2 = Tn-1 Tn+1 + (-1)n+1 for all n 2
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