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# Show that if f ( n ) is O ( h ( n )) and g ( n ) is O (i( n )), then f ( n ) + g ( n ) is O ( h ( n ) + i(n)). Show that 3( n + 1) 7 + 2 n log n is...

1.     Show that if f(n) is O(h(n)) and g(n) is O(i(n)), then f(n) + g(n) is O(h(n) + i(n)).

2.    Show that 3(+ 1)7 + 2log is O(n7). Hint: Try applying the rules of Theorem 1.7. You will have to use the insert equations to answer this question.

3.    Give an O(n)-time algorithm for computing the depth of each node of a tree T, where is the number of nodes of T. Assume the existence of methods setDepth (v,d) and getDepth(v) that run in O(1) time.

4.    What does the following algorithm do? Analyze its worst-case running time and express it using "Big-Oh" notation.

Algorithm Foo (a,n):

Input: two integers, a and n

Output: ?

k  ß 0

b  ß 1

while k < n do

k  ßk + 1

b  ß b *a

return b

5.    a.  Describe (in pseudo-code) a findAll Elements (k) method of an AVL tree T. It should run in O(logn + s) time where n is the size of T and s is the number of elements returned (i.e., the number of nodes in T whose key is k).

b.  Analyze the running time of your algorithm.

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