INFO1x05 Tutorial 09

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1 INFO1x05 Tutorial 09 (2,3) Trees, (2,4) Trees and Hash Tables Exercise 1: (INFO1105 and INFO1905) Is the search tree shown below a (2,4) tree? Why or why not? Figure 1: Tree T No. One property of a (2, 4) tree is that all external nodes are at the same depth. The multi-way search tree of the example does not adhere to this property. Exercise 2: (INFO1105 and INFO1905) Your classmate claims that a (2,4) tree storing a set of entries will always have the same structure, regardless of the order in which the entries are inserted. Show that he is wrong. 1

2 First consider insertion order: 4, 6, 12, 15, 3, 5. Then consider insertion order: 12, 3, 6, 4, 5, 15 Exercise 3: (INFO1105 and INFO1905) Consider the sequence of keys (5,16,22,45,2,10,18,30,50,12,1). Draw the result of inserting entries with these keys (in the given order) into an initially empty (2, 4) tree. Exercise 4: (INFO1105 and INFO1905) Consider a tree T storing entries. What is the worst-case height of T in the following cases? 1. T is a (2,4) tree. 2. T is a multi-way search tree, where nodes can have up to 4 children. 3. T is a binary search tree. A (2, 4) tree storing these same number of entries would have a worst-case height of log A binary search tree storing such a set would have a worst-case height of , as would a element multi-way search tree. Exercise 5: (INFO1105 and INFO1905) Consider the (2,4)-tree structure given in the figure below. Remove the following keys from the tree, showing the tree structure after each deletion: (17,22,5,27,23,24) Data Structures Page 2 of 6

3 Figure 2: Tree T Exercise 6: (INFO1105 and INFO1905) Describe in pseudocode a range query for a (2,3) tree and a (2,4) tree. 1 void rangequery(node n, int low, int high) { 2 // Assume values are stored in values[0..(n-1)] 3 // Assume links are stored in nodes[0..n] 4 5 // Check leftmost link 6 if (low < n.values[0]) 7 rangequery(n.nodes[0], low, high); 8 9 for (int i = 0; i < n.numchildren; i++) { 10 // If the value in the node is greater than the high range, 11 // we can stop looking at further children and return. 12 if (values[i] > high) 13 return; 14 // If we are less than high and greater than low (guaranteed 15 // by previous check) print node, and recurse to right child. 16 if (values[i] >= low) { 17 System.out.println(values[i]); 18 rangequery(n.nodes[i+1], low, high); 19 } 20 } 21 } Data Structures Page 3 of 6

4 Exercise 7: (INFO1105 and INFO1905) Write out a sequence of rotations which will transform the following complete binary search tree into a degenerate binary search tree (i.e. a list of elements, such that each node only has one or no children). Can we always find a sequence of rotations which allows us to form a complete binary tree and a degenerate binary tree, given an arbitrary tree T? Figure 3: Tree T 1. Right rotation on 4 2. Right rotation on 2 3. Right rotation on 4 4. Right rotation on 6 Exercise 8: (INFO1105 and INFO1905) The success of a hash-table implementation of the ADT table is related to the choice of a good hash function. A good hash function is one that is easy to compute and that will evenly distribute the possible data. Comment on the appropriateness of the following hash functions. What patterns would hash to the same location? 1. The hash table has size The search keys are English words. The has function is h(key) = (sum of positions in alphabet of key s letters) mod The hash table has size The keys are strings that begin with a letter. The hash function is h(key) = (position in alphabet of first letter of key) mod 2048 Thus, BUT maps to 2. How appropriate is this hash function if the strings are random? What if the strings are English words? Data Structures Page 4 of 6

5 3. The hash table is entries long. The search keys are integers in the range 0 through The hash function is h(key) = (key random)truncated to an integer, where random represents a sophisticated random-number generator that returns a real value between 0 and The hash table is entries long (HASH_TABLE_SIZE is 10000). The search keys are integers in the range 0 through The hash function is given by the following Java method: 1 public int hashindex(int x) { 2 for (int i = 1; i <= ; ++i) { 3 x = (x * x) % HASH_TABLE_SIZE; 4 } // end for 5 return x; 6 } // end hashindex 1. This function will probably not distribute the keys uniformly since the hash function would not distribute strings uniformly, as the function value is highly dependant on length. In addition, many different words would collide, for example given permutations of letters, which is undesirable. 2. This function only uses 26 of a possible 2048 slots and does not even distribute the keys uniformly among the 26 accessible positions. 3. This is not a function; it is not reproducible. 4. This function requires on the order of 10 6 multiplications and is not efficient. Exercise 9: (INFO1905) On the elearning website you will find an implementation for a treap data structure which allows you to specify both the keys for the heap and the keys for the tree data structure. Implement a Main method using this treap data structure and compare the height of the treap when given randomized priorities for balancing with a naïvebinary tree. You should compare the height for: 1. Inserting a monotonically increasing sequence of keys 2. Inserting an almost monotonically increasing sequence of keys 3. Inserting randomized keys In order to do this, you will need to write methods which will allow you to find the height of the treap data structure. HINT: You don t need to write a BST data structure to do this exercise. Consider how you might be able to "select" priorities so that we can simulate a naïvebst using our existing treap implementation. Data Structures Page 5 of 6

6 Exercise 10: (INFO1905) Given a treap structure, is it possible to form a degenerate tree? What circumstances could cause this to occur? Extension: Suggest one method we could use to prevent the circumstances that arose to form the degenerate tree from arising. Exercise 11: (INFO1905) Write a spell-checker class that stores a set of words, W, in a hash table and implements a method, spellcheck(s), which performs a spell check on a string s with respect to the set of words, W. If s is in W, then the call to spellcheck(s) returns an iterable collection that contains only s, as it is assumed to be spelled correctly in this case. Otherwise, if s is not in W, then the call to spellcheck(s) returns an iterable collection of every word in W that could be a correct spelling of s. You program should be able to handle all the common ways that s might be a misspelling of a word in W, including swapping adjacent characters in a word, deleting a single character from a word, and replacing a character in a word with another character. For an extra challenge, consider phonetic substitutions as well. Data Structures Page 6 of 6