Assignment 3 (concept) Solutions

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1 CS10b Data Structures and Algorithms Assignment 3 (concept) Solutions Due: Friday, February 18th 1. (0%) Consider the following algorithm: Algorithm Power(x, n) Input: A number x and integer n 0 Output: The value x n if n = 0 then return 1 if n is odd then y Power(x,(n 1)/) return x y y else y Power(x,n/) return y y 1.1 Write down the recurrence equation which describes the running time of Algorithm Power(x,n) as a function of n. Answer (10 Mars): T(n) = c, if n = 1, T(n) = T( n )+, if n > 1, where c and are constants, and n generates the smallest integer greater than or equal to n. 1. Assuming that n is a power of, solve this recurrence equation using either the substitution method or the recursion tree technique. Give the asymptotic (Big-Oh) complexity of Algorithm Power(x, n). Solution 1 (by substitution method) (10 mars): When n = m, the above recurrence equation becomes: T(n) = c, if n = 1, T(n) = T( n )+, if n > 1. We can solve this recurrence equation using substitution method as follows. T(n) = T( n )+ = T( n )+ + = T( n 3 )+ + + = = T( n i )+i 1

2 When T( n i ) = T(1), we get to the base case of T(1). Then, we get n = i. Solving it for i, we obtain i = log(n). Thus, T(n) = T(1)+log(n) = log(n)+c. This shows that T(n) is O(log(n)). Solution (by recursion tree technique) (10 mars): T(n) = T(n/) = =... = T(n/) log(n)... We draw the recursion tree in levels, each level of the tree representing a level of recursion. When we reach the base case of size 1, the total number of levels will be log(n), since at each recursive call of Power, the size of the problem is reduced by one half. At each recursive call, it performs a constant number of operations. Thus, the total number of operations is log(n), which is O(log(n)).. (0%) Consider a hash table of size 7 with hash function h() = mod 7. Draw the table that results after inserting, in the given order, the following values: 35, 40, 1, 33, 6 for each of the three scenarios below: Hash values: ey h() h When collisions are handled by separate chaining; Answer (6 mars):

3 . When collisions are handled by linear probing; Answer (7 mars): Hash Table index Hash Table Entries When collisions are handled by double hashing using a second hash function h () = 5 ( mod 5). Answer (7 mars): Hash Table index Hash Table Entries (0%) The path length of a tree T is the sum of the depths of all the nodes in T. Assume that the depth of any node in T is not nown beforehand. 3.1 Describe an algorithm in pseudo-code for computing the path length of a tree T (which is not necessarily binary). Solution (15 mars): Call Algorithm PathLength with v = root and d = 0. Algorithm PathLength(T, v, d) Input: Node v at depth d in tree T Output: Path length of T if v is a leaf node then return d else currentdepth d for each child c of v do currentdepth currentdepth + PathLength(T, c, currentdepth + 1) return currentdepth 3

4 3. Compute the worst case running time of your algorithm as a function of the total number n of nodes in the tree and give the asymptotic (Big-Oh) characterization. You must explain how you compute it. Answer (5 mars): This algorithm will perform a postorder traversal of the tree T. In each visit of a node, it does a constant c number of operations. Assume the total number of nodes in the tree is n, the total running time T(n) will be cn. Therefore, T(n) is O(n). 4. (0%) How many different trees T can be built with three internal nodes having eys 1, and 3 in each of the following cases? Please illustrate your answers. (Remember that the leaves do not store eys in binary search trees or in AVL trees.) 4.1 T is a proper binary tree; Answer (6 mars): 4. T is a binary search tree; Answer (7 mars): 4

5 4.3 T is an AVL tree. Answer (7 mars): 5. (0%) Consider the following AVL tree. 5.1 Insert the ey 36 into the tree and re-balance if needed. Draw the final tree and all intermediate trees that you need. You must use the algorithms studied in class for inserting and re-balancing. Answer (10 mars): 5

6 5. Remove the value 15 from the original tree and re-balance if needed. Draw the final tree and all intermediate trees that you need. You must use the algorithms studied in class for removing and re-balancing. Answer (10 mars): 6

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