University of Toronto Department of Electrical and Computer Engineering. Midterm Examination. ECE 345 Algorithms and Data Structures Fall 2012

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2 University of Toronto Department of Electrical and Computer Engineering Midterm Examination ECE 345 Algorithms and Data Structures Fall 2012 Print your name and ID number neatly in the space provided below; print your name at the upper right corner of every page. The exam is nine (9) pages including the cover page; if not, report it to the instructor or TA. Name: ID Number: This is an open book exam. You are permitted to use the textbook for the course but nothing else is permissible. You may reference/use textbook page/section/algorithms if you wish so. Non-native English speakers may use a dictionary. Doallthreeproblemsinthisbooklet. Trynottospendtoomuchtimeononeproblem. Useterminology from the textbook. You must define any different terms before you use them. Write clearly and only in the space provided. Ask the proctor if you need more paper. Nothing on the back of the sheets will be graded. You have 110 minutes for this exam. Raise your hand if you have a question. Turn off your mobile phone. Do not give C code! Write pseudocode and analyze time/memory requirements of your algorithms when asked to receive full credit. All logarithms are base 2 unless otherwise noted. Question Points Score Grader Total 100

3 ECE 345 Midterm Fall Name: 1. [Asymptotics, Induction, Recurrences, Hashing, Heaps] Short Answers, points. Answer only in the space provided. (a) Sort the following functions from asymptotically smallest to asymptotically largest, indicating ties if there are any: 8 n, , 3 lgn,n. Show the mathematics to prove your work. (b) Give the best asymptotic ( big-oh ) characterization of the worst case and the best case time complexities of the following algorithm, which takes as input arrays of integers A and B each of size n. Give one sentence in English to briefly explain each characterization. Algorithm i = 0 sum = 0 while i < n if A[i] < 0 for j = 0 to n 1 sum = sum+b[j] i = 2 i else if A[i] < n for j = 0 to lg(a[i]) 2 sum = sum+b[j] endif i = i+1

4 ECE 345 Midterm Fall Name: (c) Use induction to prove that: 2 n > 2n for every positive integer n > 2. Clearly show the induction base/hypothesis/step. (d) Find an asymptotic solution of the following functional recurrences using the Master Theorem. Express your answer using Θ-notation, and clearly show your work. (i) T(n) = 8T( n 2 )+nlg(n) (ii) T(n) = 345T( n 2 )+n 345

5 ECE 345 Midterm Fall Name: (e) Suppose a hash function h maps arbitrary keys to values between 0 and m 1, inclusive. We hash n keys, k 1, k 2,..., k n. If h(k i ) = h(k j ), we say that k i and k j collide. Assuming simple uniform hashing (i.e., a key is equally likely to hash to any one of the hash table slots, independent of any other keys), how should we choose the size of the table m (in terms of n) such that the expected number of collisions is O(1)? Justify your answer. Hint: How many pairs of elements do you have? Use this fact to calculate the total expected number of collisions in order to pick an appropriate table size. (f) The following two questions relate to binary heaps. State if the following is true/false. You need to justify your answer in the space provided in 2-3 lines to receive full credit. (i) We can always find the maximum in a min-heap in O(log n) time. T F (ii) For an n element binary heap with depth d, we have that n 2 d. (Assume the depth of the root is zero). T F

6 ECE 345 Midterm Fall Name: 2. [Search Trees] 25 points. Consider two binary search trees (BST), A and B, each with n elements. Describe a method to merge A and B into a new BST C in O(n)-time, such that C is balanced (i.e. height(c)=o(lg(n))). Answers worst than O(n) will receive less credit. Hint: use an auxiliary array to first sort the keys from the two trees, then figure out how to use this sorted sequence to build C.

7 ECE 345 Midterm Fall Name: 3. Dynamic Programming, points. Imagine starting with a given number n, and repeatedly chopping off a digit from one end or the other (your choice), until only one digit is left. The prime-depth P(n) of n is defined to be the maximum number of prime numbers you can arrange to see along the way of chopping digits from one end or the other. For example, P(1637) = 3 via the sequence: since 1637, 163, and 3 are prime numbers and there is no better sequence of chops giving a larger score (although other ways might be possible). You may assume that you have a function IS-PRIME(x) that returns, in constant time, 1 if n is prime, and 0 if not. Let n i,j = n i...n j, where n i is the i th digit of the number n. That is, n i,j is the number formed by digits i through j of n. (i) We wish to form a dynamic programming solution. As such, let D[i,j] denote a solution to a sub-problem. In one sentence describe what D[i, j] represents and, for the example above (1637), compute D[2, 4]. (ii) Write the recurrence relation for D[i, j], including the base case.

8 ECE 345 Midterm Fall Name: (iii) Compute all relevant sub-problems (D[i, j]) needed to solve the above example (1637). (iv) Describe an efficient algorithm to compute the prime-depth, P(n), of a given number n, written as a d-digit decimal number n 1,n 2,...,n d. Analyze its running time (sub-optimal solutions will receive less credit).

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