关于算法:COSC-21231285算法与分析

COSC 2123/1285 Algorithms and Analysis

Assignment 1: Word Completion

1 Objectives

There are a number of key objectives for this assignment:
• Understand how a real-world problem can be implemented by different data structures and/oralgorithms.
• Evaluate and contrast the performance of the data structures and/or algorithms with respect todifferent usage scenarios and input data.In this assignment, we focus on the word completion problem.

2 Background

Word/sentence (auto-)completion is a very handy feature of nowadays text editors and email browsers(you must have used it in yourOutlook). While sentence completion is a much more challenging taskand perhaps requires advanced learning methods, word completion is much easier to do as long asyou have a dictionary available in thememory. In this assignment, we will focus on implementing a
dictionary comprising of words and their frequencies that allows word completion. We will try severaldata structures and compare their performances, which are array (that is, Python list), linked list,
and trie, which are described one by one below. Please read them very carefully. Latest updates andanswers for questions regarding this assignment will be posted on the Discussion Forum. It is your
responsibility to check the post frequently for important updates.
Array-Based DictionaryPython’s built-in‘list’is equivalent to‘array’in other language. In fact, it is a dynamic array in the
sense that its resizing operation (when more elements are inserted into the array than the originalsize) is managed automatically by Python. You can initialize an empty array in Python, add elements
at the end of the array, remove the first instant of a given value by typing the following commands(e.g., on Python’s IDLE Shell).

>>> array = []
>>> array.append(5)
>>> array.append(10)
>>> array.append(5)
>>> array
[5, 10, 5]
>>> array.remove(5)
>>> array
[10, 5]

In the array-based dictionary implementation, we use the Python list (a data structure) to implement common operations for a dictionary (an abstract data type). We treat each element of thearray as a pair (word, frequency) (defined as an object of the simple class WordFrequency), whereword is an English word (a string), e.g.,‘ant’, and frequency is itsusage frequency (a non-negative
integer), e.g., 1000, in a certain context, e.g., in some forums or social networks.The array must be sorted in the alphabetical order, i.e.,‘ant’appears before‘ape’, to facilitatesearch. A new word, when added to the dictionary, should be inserted at a correct location thatpreserves the alphabetical order (using the module bisect is allowed – but you need to know what

it does). An example of a valid array is [(‘ant’, 1000), (‘ape’, 200), (‘auxiliary’, 2000)].
Adding (‘app’, 500) to the array, we will have [(‘ant’, 1000), (‘ape’, 200), (‘app’, 500),
(‘auxiliary’, 2000)]. Note that the pair (‘ant’, 1000) in our actual implementation should be

an object of the class WordFrequency and not a tuple.A Search for‘ape’from the array above should return its frequency 200. If the word doesn’t exist,0 is returned.A Delete for a word in the dictionary should return True and remove the word from the dictionary
if it exists, and return False otherwise.An Autocompletion for a string should return a sorted list (most frequent words appear first) ofthe three words (if any) of highest frequencies in the dictionary that have the given string as a prefix.For the array above, an autocompletion for‘ap’should return the list [(‘app’, 500),(‘ape’, 200)].Notice that while both‘app’and‘ape’have‘ap’as a prefix,‘app’has a larger frequency and appearsfirst in the returned list of autocompletion.

Linked-List-Based Dictionary

A linked list is precisely what it is called: a list of nodes linked together by references. In a singlylinked list, each node consists of a data item, e.g., a string or a number, and a reference that holds thememory location of the next node in the list (the reference in the last node is set to Null). Each linkedlist has a head, which is the reference holding memory location of the first node in the list. Once weknow the head of the list, we can access all nodes sequentially by going from one node to the nextusing references until reaching the last node.In the linked-list-based implementation of dictionary, we use an unsorted singly linked list. Youcan use the implementation of the linked list in the workshop as a reference foryourimplementation.Each node stores as data a pair of (word, frequency) (an object of the class WordFrequency) and
a reference to the next node. A word and its frequency are added as a new node at the front of thelist by updating the head reference. Apart from the fact that they are carried out in the linked list,
Search, Delete, and Autocomplete work similarly as in the array-based implementation. Note thatunlike the array-based dictionary, the words in the linked list are not sorted.

Trie-Based Dictionary

Trie (pronounced as either‘tree’or‘try’) is a data structure storing (key, value) pairs where keysare strings that allows fast operations such as spell checking and auto-completion. Introduced in thecontext of computer decades ago, it is no longer the most efficient data structure around. However,our purpose is to focus more on the understanding of what data structures mean, how they can beused to implement an abstract data type, and to empirically evaluate their performance. Thus, westick to the original/simple idea of‘trie’. You are strongly encouraged to read about more advanced
data structures evolving from trie.Each node of a trie contains the following fields:
• a lower-case letter from the English alphabet (‘a’to‘z’), or Null if it is the root,
• a boolean variable that is True if this letter is the last letter of a word in the dictionary and Falseotherwise,
• a positive integer indicating the word’s frequency (according to some dataset) if the letter is thelast letter of a word in the dictionary,
• an array of A = 26 elements (could be Null) storing references pointing to the children nodes.In our implementation, for convenience, we use a hashtable/Python’s dictionary to store the
children.As an example, consider Figure 1.Figure 1: An example of a trie storing six words and their frequencies. The boolean value True
indicates that the letter is the end of a word. In that case, a frequency (an integer) is shown, e.g., 10for‘cut’. Note that a word can be a prefix of another, e.g.,‘cut’is a prefix of‘cute’.
Construction. A trie can be built by simply adding words to the tree one by one (order of wordsbeing added is not important). If a new word is the prefix of an existing word in the trie then we cansimply change the boolean field of the node storing the last letter to True and update its frequency.For instance, in the example in Figure 1, if (‘cut’, 10) is addedafter (‘cute’, 50), then one can simplychange the boolean field of the node containing the letter‘t’to True and set the frequency to 10,signifying that now (‘cut’, 10) is part of the tree. In another case, when (‘cup’, 30) is added, a newnode has to be constructed to store‘p’, which has a True in the boolean field and 30 as its frequency.In this case,‘cup’can reuse the two nodes storing‘c’and‘u’from the previous words.
Searching. To search for a word (and to get its frequency) in a trie,we use its letters to navigatethe tree by following the corresponding child node. The search is successful if we can reach a nodestoring the last letter in the word and has the boolean field True. In that case, the frequency storedin this node is returned. The search fails, that is, the word is not in the tree, if either a) the searchalgorithm couldn’t find a child node that matches a letter in the word, or b) it finds all the nodesmatching all the letters of the words but the boolean field of the node corresponding to the last letteris False.Deletion. The deletion succeeds if the word is already included in the tree. If the word is a prefixof another word then it can be deleted by simply setting the boolean field of the node storing the lastletter to False. For example, if (cut, 10) is to be deleted from the trie in Figure 1, then we only needto change the boolean field of the node storing‘t’to False. Otherwise, if the word has a unique suffix,then (only) nodes corresponding to the suffix are to be deleted. For example, if (cup, 30) is to beremoved, then the node storing the last letter‘p’must be deleted from the trie to save space but not‘c’and‘u’because these still form part of other words.Auto-completion. Auto-completion returns a list of three words (if any) in the dictionary (trie)of highest frequencies that have the given string as a prefix. For example, in the trie given in Figure 1,• the auto-completion list for‘cu’is: [(cute, 50), (cup, 30), (cut, 10)],• the auto-completion list for‘far’is: [(farm, 40)].Suppose we add one more word (curiosity, 60) into this tree, then the auto-completion list of‘cu’willbe changed to [[(curiosity, 60), (cute, 50), (cup, 30)]. In this example, although‘cut’contains‘cu’asa prefix, it is not in the top three of the most common words having‘cu’as a prefix. In general, theauto-completion list contains either three, two, one, or no words. They must be sorted in decreasingfrequencies, that is, the first word has the highest frequency and the last has the lowest frequency.

3 Tasks

The assignment is broken up into a number of tasks, to help you progressively complete the assignment.3.1 Task A: Implement the Dictionary and Its Operations Using Array, LinkedList, and Trie (12 marks)In this task, you will implement a dictionary of English words that allows Add, Search, Delete,and Auto-completion, using three different data structures: Array (Python’s list), Linked List, andTrie. Each implementation should support the following operations:

• Build a dictionary from a list of words and frequencies: create a dictionary that stores wordsand frequencies taken from a given list. This operation is not tested.
• (A)dd a word and its frequency to the dictionary. Return True if successful or False if it alreadyexists in the dictionary.
• (S)earch for a word in a dictionary and return its frequency. Return 0 if not found.