CS323: Data Structures and Algorithms Midterm Exam (October 17, 2016) Name:

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1 CS323: Data Structures and Algorithms Midterm Exam (October 17, 2016) Name: You are to honor the Emory Honor Code and the Math/CS SPCA. This is a closed-book and closed-notes exam. You have 70 minutes to complete this exam. Short & Long Answers Σ Perfect Earned Code Completion 7 8 Σ Perfect Earned Algorithm Design 9 Σ Perfect Earned Total points

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3 Short & Long Answers 1. (a) Given an input of size n, a binary heap gives O(log2 n) for both the swim and the sink operations. A k-ary heap is a heap where each node can take k-number children. What are the worst-case complexities of a k-ary heap for the swim and the sink operations? (2 points) (b) Explain how it achieves such complexities. (2 points) (c) Explain how the performance of the add and the remove operations changes as the values of k and n increase. (2 points) 2. Several sequences had been proposed for shell-sort: Shell = b n 2 k c = 1, 2, 4, 8, 16, 32,... Knuth = 3 k 1 2 = 1, 4, 13, 40, 121,... Hibbard = 2 k 1=1, 3, 7, 15, 31, 63,... (a) Among the three sequences above, which sequence is most preferred as the input gets larger? Explain your answer. (2 points) (b) Given any sequence, explain how to determine the initial gap effectively. (1 point) (c) Shell-sort using the Shell sequence gives O(n 2 ). Explain what kind of input would cause this worst-case complexity. Explain how the other two sequences avoid this kind of worstcase complexity. (3 points) 3. (a) Given an input of size n, the original algorithm of merge-sort makes (2 n log n) number of assignments. Explain how this algorithm can be modified to achieve (n log n + e) number of assignments instead, where e 1. (3 points) (b) Intro-sort is a hybrid between quick-sort and heap-sort. Explain how intro-sort guarantees the worst-case complexity of O(n log n). Is it possible to improve intro-sort by using a different combination of sorting algorithms? Explain your answer. (3 points) 4. Describe the processes of balancing for the following two situations in a Red-Black tree. Consider the white nodes are red. The last node inserted is 4 for both trees. (6 points) 3

4 5. (a) Describe the process of the remove operation in a trie. (3 points) (b) Describe a similarity and a difference between a trie and a hash-map. (2 points) 6. (a) Explain how the dynamic table gets pre-populated to find the longest common subsequences using dynamic programming. Explain the intuition behind this population. (3 points) (b) The following code finds the longest common subsequences using recursion. Describe how it can be modified for dynamic programming such that a dynamic table gets populated during recursion. You must define a dynamic table and show how the table is populated during recursion. (3 points) String solve(string a, String b) char[] c = a.tochararray(); char[] d = b.tochararray(); int i = a.length()-1, j = b.length()-1; return solve(c, d, i, j); String solve(char[] c, char[] d, int i, int j) if (i < 0 j < 0) return ""; if (c[i] == d[j]) return solve(c, d, i-1, j-1) + c[i]; String c1 = solve(c, d, i-1, j); String d1 = solve(c, d, i, j-1); return (c1.length() > d1.length())? c1 : d1; 4

5 CS323: Data Structures and Algorithms Midterm, Fall 2016 Code Completion Use only methods in the provided interfaces, classes, and Java built-in APIs for all questions. Extra codes that either hurt or do not contribute to the assigned tasks may be penalized. 7. Given 1) an array representing a binary heap where the first element is null and the root of the heap holds the maximum key, and 2) a key to search, complete the following code such that it returns the index of the key in the binary heap if exists; otherwise, -1. (8 points) /** heap a max-binary heap. key a key to search. the index of the key in the binary heap if exists; * otherwise, -1. */ int indexof(int[] heap, int key) // To be filled. 5

6 8. Given an input array, complete the following code such that the array is sorted in ascending order using the most-significant digit radix sorting algorithm. Your code must not involve any comparisons between keys. (12 points) /** array an array to be sorted. buckets a list of buckets for radix sort. bindex the beginning index to be sorted (inclusive). eindex the ending index to be sorted (exclusive). maxbit the maximum number of digits. */ void MSDRadixSort(Integer[] array, List<List<Integer>> buckets, int bindex, int eindex, int maxbit) // To be filled. int[] getbucketsizes(list<list<integer>> buckets) int[] sizes = new int[10]; for (int i=0; i<10; i++) sizes[i] = buckets.get(i).size(); return sizes; Algorithm Design 9. Write a pseudo-code that takes an acronym (e.g., www ) and returns a list of strings that the acronym can be abbreviated for (e.g., world wide web, what women want ) using a Trie and dynamic programming. Assume that all acronyms, words, or phrases consist of only lower-case characters and whites paces (no symbols), and the Trie already contains all possible words and phrases in the world. Try to design your algorithm as efficient as possible. (15 points) 6