University of Waterloo CS240 Winter 2018 Assignment 3

Transcription

1 University of Waterloo CS240 Winter 2018 Assignment 3 version: :37 Due Date: Wednesday, Feb. 28th, at 5pm Please read the guidelines on submissions: w18/guidelines.pdf. This assignment contains written questions and a programming question. Submit your written solutions electronically as a PDF with files named a03.pdf using MarkUs. Problem 1 [5+3=8 marks] Let T be a binary tree with a set of n nodes in which each node has a pointer to its parent node. Choosing two arbitrary nodes u and v from the tree, we execute the following algorithm: Algorithm Nameless (u, v) 1: P u, P v empty stack of references to nodes 2: do 3: P u.push(u) 4: u u.parent 5: while (u!= null ) 6: do 7: P v.push(v) 8: v v.parent 9: while (v!= null ) 10: create a new node c 11: while (!P u.isempty() and!p v.isempty() and P u.top()=p v.top()) do 12: c P u.top() 13: P u.pop() 14: P v.pop() 15: return (c) a) Describe what Nameless returns. b) Analyze the run-time of Nameless in both worst case and best case using asymptotic notation. Briefly justify your answer. 1

2 Problem 2 [4+4+6=14 marks] I. Consider the following AVL trees T 1 and T 2, Figure 1: T 1 and T 2 a) Show the result of inserting 2, 37, 7, 12, 3, 36 into the tree T 1. Draw the tree before and after every insertion that results in rebalancing (similar to slide 25 of Module 4). b) Draw your final AVL tree after removing the keys 22 and 58, in the order given, from the tree T 2. II. Consider an AVL tree containing n distinct keys. Design an algorithm to compute a function called IncKey(k, d) whose purpose is to increase the key k (if a node containing key k is present) by the given value d 0 (we assume the key k + d is unique in the tree). At the end, the resulting tree must still be AVL tree. Analyze the run-time of your algorithm using asymptotic notation. For full credit, the asymptotic run-time of your algorithm must be as small as possible. Problem 3 [ =20 marks] I. Given an empty skip list L, a) Show the result of inserting the following keys, Keys: 44, 9, 26, 50, 12, 37, 51, 52, 53 Coin flip sequence: THTTHTHTTHTTHHHHT 2

3 b) Count the number of key-comparisons needed to successfully search the key 79 in the following skip list and maintain the track of your searches in a stack called P (similar to slide 4 of Module 5). S 3 + S S S II. Consider a skip list L that contains n distinct keys. Associated with each key x at each level i is a field dist[x, i] containing the number of keys skipped by the pointer at x to the next element, including the key x itself. For example, in the above skip list the key 23 at level S 0 has dist[23, 0] = 1 and at level S 2 has dist[23, 2] = 2. a) Design an algorithm in pseudocode using the minimum number of visiting nodes (i.e., after/below calls) to compute Select(L, k). The task of Select(L, k) is to return the kth smallest key in the skip list L. If no such key exists, it return null. For example in the above skip list, 14 is the 1st smallest key, 23 is the 2nd, and etc. You may assume the skip list L has a variable L.depth that stores the number of levels in L. b) Describe the update operations that are necessary after insertion or deletion. Problem 4 [5+5+4=14 marks] Suppose a linked list (or an array) with dynamic ordering contains the items ABDCEF GH, a) Show a possible sequence of searches using the Move-To-Front (MTF) heuristic that leads to the sequence EHGDABCF. b) Using the Transpose heuristic, give a sequence of searches that leads to the sequence ADBCEGHF. c) Having the sequence of searches like DHHGHEGH, compute the total number of operations (key-pair comparison and swap) for both the Move-To-Front (MTF) and the Transpose heuristics. 3

4 Problem 5 [ =20 marks] Consider the following ordered array of size n = 10, and the pseudocode of the Interpolation Search algorithm given below. InterpolationSearch(A[l, r], k) A: an array l: index of the left boundary r: index of the right boundary k: key to search for 1. if A[l] > k A[r] < k then 2. return false 3. if l = r then 4. if A[l] = k then 5. return true 6. else 7. return false 8. i l + (r l) k A[l] A[r] A[l] 9. if A[i] = k then 10. return true 11. else if A[i] < k then 12. return InterpolationSearch(A[i + 1, r], k) 13. else 14. return InterpolationSearch(A[l, i 1], k) a) Apply the InterpolationSearch algorithm to search for the keys 5 and 46, showing the behavior of the algorithm until the key is found. b) Determine the number of key equality comparisons (lines 4 and 9) needed in both cases and discuss whether it is consistent with the expected complexity. c) Provide an example of an ordered array that contains n = 10 distinct keys, including keys 5 and 46, such that the InterpolationSearch algorithm performs at least n/2 key equality comparisons for searching both keys 5 and 46. For full credit, exactly n/2 key equality comparisons in both searches are required. Show the behavior of the pseudocode on your solution (for each recursive call to InterpolationSearch show the values of l, r, i, A[i] after the calculation at line 8). d) Generalize part c) to an arbitrary n. That is, given n, explain how to construct an ordered array with n distinct keys, where there exists at least one key whose search through the InterpolationSearch algorithm has time complexity Ω(n). 4

5 Problem 6 [3+3+8=14 marks] a) Draw your final trie after inserting binary keys S = {0001$, 0011$, 1010$, 11$, 110$, 10011$} into an initially empty (uncompressed) binary trie. b) Draw your final trie after removing the keys 0101$, 001$, in the order given, from the following trie. c) Consider a trie T that contains n strings. Design an algorithm called Look(T, x, l), which returns the list of all strings with length l (without $) that begin with a given string x of length m, where m l. Assume x does not end with $. The worst-case running time of your algorithm should be O(m + 2 (l m+1) ). 5