15-122: Principles of Imperative Computation, Summer 2011 Assignment 6: Trees and Secret Codes

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1 15-122: Principles of Imperative Computation, Summer 2011 Assignment 6: Trees and Secret Codes William Lovas Karl Naden Out: Tuesday, Friday, June 10, 2011 Due: Monday, June 13, 2011 (Written part: before lecture, Programming part: 11:59 pm) 1 Written: (25 points) Te written portion of tis week s omework will give you some practice reasoning about variants of binary searc trees. You can eiter type up your solutions or write tem neatly by and, and you sould submit your work in class on te due date just before lecture begins. Please remember to staple your written omework before submission. 1.1 Heaps and BSTs Exercise 1 (3 pts). Draw a tree tat matces eac of te following descriptions. (a) Draw a eap tat is not a BST. (b) Draw a BST tat is not a eap. (c) Draw a non-empty BST tat is also a eap. 1.2 Binary Searc Trees Refer to te binary searc tree code posted on te course website for Lecture 17. Exercise 2 (4 pts). Te eigt of a binary searc tree (or any binary tree for tat matter) is te number of levels it as. (An empty binary tree as eigt 0.) (a) Write a function bst eigt tat returns te eigt of a binary searc tree. You will need a wrapper function wit te following type: 1

2 int bst_eigt(bst B); Tis function sould not be recursive, but it will require a elper function tat will be recursive. See te BST code for examples of wrapper functions and recursive elper functions. (b) Wy is recursion preferred over iteration for tis problem? Exercise 3 (5 pts). In tis exercise you will re-implement bst searc in an iterative fasion. a) Complete te bst searc function below tat uses iteration instead of recursion. elem bst_searc(bst B, key k) //@requires is_bst(b); //@ensures \result == NULL compare(k, elem_key(\result)) == 0; { tree T = B->root; } wile ( ) { if ( ) T = T->left; else T = T->rigt; } if (T == NULL) return NULL; return T->data; b) Using your loop condition, prove tat te ensures clause of bst searc must old wen it returns. 1.3 AVL Trees Exercise 4 (3 pts). Draw te AVL tree tat results after successively inserting te following keys (in te order sown) into an initially empty tree, maintaining and restoring te invariants of a BST and te additional balance invariant required for an AVL tree after every insert Your answer sould sow te tree after eac key is successfully inserted. Also be sure to label eac node wit its balance factor, wic is defined as te eigt of te rigt subtree minus te eigt of te left subtree. 2

3 x x x y y y +1 A B C A +1 B C A B C (a) (b) (c) Figure 1: Te tree ways a tree can violate te AVL balance invariant at te root. Exercise 5 (10 pts). During lecture 17, we sowed briefly ow binary tree rotations could be used to restore AVL invariants after an insertion. We can easily see tat it can take at most O(log n) time to restore te balance invariant, since te invariant can only be broken along te pat from te insertion point to te root, wic can be at longest O(log n) nodes, and rotations take constant time. But in fact, for AVL trees, te balance invariant can be restored in just one (single or double) rotation! 1 In tis exercise we will go troug several steps to prove tat we only need to perform at most one rotation following an insertion into an AVL tree. Te basic idea is to sow tat any rotation of an unbalanced tree created from an insertion of a new element into a balanced AVL tree must reduce te eigt of te rotated tree. After a single insert into an AVL tree, te tree may be sligtly unbalanced: its left and rigt cildren may differ in eigt by 2 instead of 1 or 0. Witout loss of generality, we assume tat te eigt of te left cild is 2 greater tan tat of te rigt cild. Figure 1 depicts all of te ways tis can be te case. (a) Sow ow to restore te balance invariant for te case in Figure 1(a) using a single rotation. Sow also tat te resulting tree is sorter tan te original tree. (b) Sow ow to restore te balance invariant for te cases engendered by Figure 1(b) (all depicted in Figure 2) using a double rotation. Sow also tat in every case, te resulting tree is sorter tan te original tree. (c) Sow ow to restore te balance invariant for te case in Figure 1(c) using a single rotation. But sow tat in tis case, te resulting tree is not sorter tan te original tree. (d) Explain wy case 1(c) cannot arise as te result of a single insertion (witout rebalancing) into a balanced AVL tree. 1 It takes O(log n) time to find te place in te tree to insert te new element, so te insert operation is still O(log n) overall, but fewer rotations mean a smaller constant factor of overead, wic is an important practical consideration. 3

4 x x x y y y A z B 2-1 B 1 C A -1 z B 1 B 2 C A z B 1 B 2 C (a) (b) (c) Figure 2: Te tree ways a tree can violate te AVL balance invariant at te root via an inner cild subtree. (e) Given tat all te invariant-restoring rotations tat can actually arise reduce te eigt of te tree, argue tat after rotation, te parent of te rotated tree cannot be unbalanced. (Hint: Tink about ow te eigt of te subtree as canged between te insertion and te rotation.) 4

5 2 Programming: Huffman Coding (25 points) For te programming portion of tis week s omework, you ll write a C 0 implementation of a popular compression tecnique called Huffman coding. In addition, you will be asked to generate a suite of examples using ideas from testing metodologies presented in lecture 19. You sould submit your code electronically by 11:59 pm on te due date. Detailed submission instructions can be found below. Starter code. Download te file w6-starter.zip from te course website. It contains some code for reading frequency tables (freqtable.c0), described below, as well as some sample inputs. Compiling and running. For tis omework, you will use te standard cc0 command wit command line options. Don t forget to include te -d switc if you d like to enable dynamic annotation cecking. Write all non-testing code in te file uffman.c0. For details on ow we will compile your code, see te file COMPILING.txt included in te starter code. Warning: You will lose credit if your code does not compile. Submitting. te command Once you ve completed some files, you can submit tem by running andin -a w6 <file1>.c0... <filen>.c0 You can submit files as many times as you like and in any order. Wen we grade your assignment, we will consider te most recent version of eac file submitted before te due date. If you get any errors wile trying to submit your code, you sould contact te course staff immediately. Annotations. Be sure to include appropriate //@requires, //@ensures, //@assert, and //@loop invariant annotations in your program. You sould write tese as you are writing te code rater tan after you re done: documenting your code as you go along will elp you reason about wat it sould be doing, and tus elp you write code tat is bot clearer and more correct. Annotations are part of your score for te programming problems; you will not receive maximum credit if your annotations are weak or missing. Style. Strive to write code wit good style: indent every line of a block to te same level, use descriptive variable names, keep lines to 80 caracters or fewer, document your code wit comments, etc. We will read your code wen we grade it, and good style is sure to earn our good graces. Feel free to ask on te course bboard (academic.cs ) if you re unsure of wat constitutes good style. 5

6 Caracter Code c 000 e 001 f 010 m 011 o 100 r 101 Figure 3: A custom fixed-lengt encoding for te non-witespace caracters in te string "more free coffee". 2.1 Data Compression Primer Wenever we represent data in a computer, we ave to coose some sort of encoding wit wic to represent it. Wen representing strings in C 0, for instance, we use ASCII codes to represent te individual caracters. Under te ASCII encoding, eac caracter is represented using 7 bits, so a string of lengt n requires 7n bits of storage, wic we usually round up to 8n bits or n bytes. Oter encodings are possible as well, toug. Te UNICODE standard defines a variety of caracter encodings wit a variety of different properties. Te simplest, UTF-32, uses 32 bits per caracter. Sometimes wen we wis to store a large quantity of data, or to transmit a large quantity of data over a slow network link, it may be advantageous to seek a specialized encoding tat requires less space to represent our data by taking advantage of redundancies inerent in it. For example, consider te string "more free coffee"; ignoring spaces, it can be represented in ASCII as follows wit 14 7 = 98 bits: Tis encoding of te string is rater wasteful, toug. In fact, since tere are only 6 distinct caracters in te string, we sould be able to represent it using a custom encoding tat uses only lg 6 = 3 bits to encode eac caracter. If we were to use te custom encoding sown in Figure 3, te string would be represented wit only 14 3 = 42 bits: In bot cases, we may of course omit te separator between codes; tey are included only for readability. If we confine ourselves to representing eac caracter using te same number of bits, i.e., a fixed-lengt encoding, ten tis is te best we can do. But if we allow 6

7 Caracter Code e 0 o 100 m 1010 c 1011 r 110 f 111 Figure 4: A custom variable-lengt encoding for te non-witespace caracters in te string "more free coffee". ourselves a variable-lengt encoding, ten we can take advantage of special properties of te data: for instance, in te sample string, te caracter e occurs very frequently wile te caracters c and m occur very infrequently, so it would be wortwile to use a smaller bit pattern to encode te caracter e even at te expense of aving to use longer bit patterns to encode c and m. Te encoding sown in Figure 4 employs suc a strategy, and using it, te sample string can be represented wit only 34 bits: Since tis encoding is prefix-free no code word is a prefix of any oter code word te separators are redundant ere, too. It can be proven tat tis encoding is optimal for tis particular string: no oter encoding can represent te string using fewer tan 34 bits. Moreover, te encoding is optimal for any string tat as te same distribution of caracters as te sample string. In tis assignment, you will implement a metod for constructing suc optimal encodings due to David Huffman. 2.2 Huffman Coding A Brief History Huffman coding is an algoritm for constructing optimal encodings given a frequency distribution over caracters. It was developed in 1951 by David Huffman wen e was a P.D student at MIT taking a course on information teory taugt by Robert Fano. It was towards te end of te semester, and Fano ad given is students a coice: tey could eiter take a final exam to demonstrate mastery of te material, or tey could write a term paper on someting pertinent to information teory. Fano suggested a number of possible topics, one of wic was efficient binary encodings: wile Fano imself ad worked on te subject wit is colleague Claude Sannon, it was not known at te time ow to efficiently construct optimal encodings. 7

8 Figure 5: Te custom encoding from Figure 3 as a binary tree. Huffman struggled for some time to make eadway on te problem and was about to give up and start studying for te final wen e it upon a key insigt and invented te algoritm tat bears is name, tus outdoing is professor, making istory, and attaining an A for te course. Today, Huffman coding enjoys a variety of applications: it is used as part of te DEFLATE algoritm for producing ZIP files and as part of several multimedia codecs like JPEG and MP Huffman Trees Recall tat an encoding is prefix-free if no code word is a prefix of any oter code word. Prefix-free encodings can be represented as binary trees wit caracters stored at te leaves: a branc to te left represents a 0 bit, a branc to te rigt represents a 1 bit, and te pat from te root to a leaf gives te code word for te caracter stored at tat leaf. For example, te encodings from Figures 3 and 4 are represented by te binary trees in Figures 5 and 6, respectively. Recall tat te variable-lengt encoding represented by te tree in Figure 6 is an optimal encoding. Te tree representation reflects te optimality in te following property: frequently-occuring caracters ave sort pats to te root. We can see tis property clearly if we label eac subtree wit te total frequency of te caracters occurring at its leaves, as sown in Figure 7. A frequency-annotated tree is called a Huffman tree. Huffman trees ave a recursive structure: a Huffman tree is eiter a leaf containing a caracter and its frequency, or an interior node containing te combined frequency of two cild Huffman trees. Since only te leaves contain caracter data, we draw tem as rectangles to distinguis tem from te interior nodes, wic we draw as circles. We represent bot kinds of Huffman tree nodes in C 0 using a struct tree: 8

9 Figure 6: Te custom encoding from Figure 4 as a binary tree. Figure 7: Te custom encoding from Figure 4 as a binary tree annotated wit frequencies, i.e., a Huffman tree. 9

10 typedef struct tree* tree; struct tree { car caracter; int frequency; tree left; tree rigt; }; // \0 except at leaves Te caracter field of an tree sould consist of a NUL caracter \0 everywere except at te leaves of te tree, and every interior node sould ave exactly two cildren. Tese criteria give rise to te following recursive definitions: An tree is a valid tree if it is non-null, its frequency is strictly positive, and it is eiter a valid tree leaf or a valid tree interior node. An tree is a valid tree leaf if its caracter is not \0 and its left and rigt cildren are NULL. An tree is a valid tree interior node if its caracter is \0, its left and rigt cildren are valid trees, and its frequency is te sum of te frequency of its cildren. Task 1 (6 pts). Write a specification function bool is tree(tree H) tat returns true if H is a valid tree and false oterwise Finding Optimal Encodings Huffman s key insigt was to use te frequencies of caracters to build an optimal encoding tree from te bottom up. Given a set of caracters and teir associated frequencies, we can build an optimal Huffman tree as follows: 1. Construct leaf Huffman trees for eac caracter/frequency pair. 2. Repeatedly coose two minimum-frequency Huffman trees and join tem togeter into a new Huffman tree wose frequency is te sum of teir frequencies. 3. Wen only one Huffman tree remains, it is an optimal encoding. Tis is an example of a greedy algoritm since it makes locally optimal coices tat neverteless yield a globally optimal result at te end of te day. Selection of a minimum-frequency tree in step 2 can be accomplised using a priority queue based on a eap. A sample run of te algoritm is sown in Figure 8. (Implementation Note: wen joining two Huffman trees put te tree removed from te priority queue first as te left cild of te new Huffman tree). 10

11 Figure 8: Building an optimal encoding using Huffman s algoritm. 11

12 Task 2 (8 pts). Write a function tree build tree(car[] cars, int[] freqs, int n) tat constructs an optimal encoding for an n-caracter alpabet using Huffman s algoritm. Te cars array contains te caracters of te alpabet and te freqs array teir frequencies, were cars[i] occurs wit frequency freqs[i]. Use te code in te included eaps.c0 as your implementation of priority queues. To test your implementation, you may use te code in freqtable.c0, wic reads a caracter frequency table from a file in te following format: <caracter 1> <frequency 1> <caracter 2> <frequency 2> Decoding Bitstrings Huffman trees are a data structure well-suited to te task of decoding encoded data. Given an encoded bitstring and te Huffman tree tat was used to encode it, decode te bitstring as follows: 1. Initialize a pointer to te root of te Huffman tree. 2. Repeatedly read bits from te bitstring and update te pointer: wen you read a 0 bit, follow te left branc, and wen you read a 1 bit, follow te rigt branc. 3. Wenever you reac a leaf, output te caracter at tat leaf and reset te pointer to te root of te Huffman tree. If te bitstring was properly encoded, ten wen all of te bits are exausted, te pointer sould once again point to te root of te Huffman tree. If tis is not te case, ten te decoding fails for te given inputs. As an example, we can use te encoding from Figure 7 to decode te following message: r o o m f o r c r e m e Room for creme? Bitstrings can be represented in C 0 as ordinary strings composed only of caracters 0 and 1. typedef string bitstring; bool is_bitstring(bitstring s) { int len = string_lengt(s); int i; for (i = 0; i < len; i++) { car c = string_carat(s, i); 12

13 } if (!(c == 0 c == 1 )) return false; } return true; You will implement tis algoritm in a function string decode(tree encoding, bitstring bits) tat decodes bits based on te encoding. Te function sould return te decoded string if te bitstring can be decoded and sould signal an error oterwise. But first, recall te metodology of black box testing via equivalence classes, in wic we attempt to reduce te number of test cases by dividing all te possible inputs to te function into equivalence classes on wic te function sould beave in te same way and ten testing only one input from eac class. Before you implement te decode function, you will use tis metodology to implement a suite of examples to test your implementation. Task 3 (5 pts). For tis task, write a series of tests tat you will use to be confident tat your implementation of decode is correct based on your reading of te spec. Following black box testing metodology, write tests tat are representative of a large class of inputs and tink about te different kind of results you expect from different inputs. For eac test you will submit a test file and potentially some input files. Eac individual test sould be given a unique number used in te naming conventions below. Tere sould be two primary components of eac test: Inputs: Te main portion of eac test is a well cosen set of inputs tat represents te beavior of te function on a large class of possible inputs. You sould eiter ard-code tese inputs into te testing function (see below) or store tem in files tat you submit along wit your tests. If you coose to place te inputs in files, put te specification of te Huffman Tree and te bitstring in different files. Name tem using te convention: input#-<inputtype>.txt were # is te test number and <inputtype> is FT for te frequency table and BS for te bitstring (For example, te bitstring input file for your second test sould be named input2-bs.txt). Main function: You sould write a main function tat runs te test on your inputs. Te main function sould do tree tings: 1. Test Description: Before doing anyting else, you sould print a sort description of te test to te screen including te class of inputs it is testing and te expected result. 2. Build/Import Inputs: Next you sould build or read in te inputs to decode for tis test. If you import te inputs from files, read in te bitstring wit 13

14 te read words function defined in te readfile.c0 file and import te frequency table used to build your Huffman Tree using te read freqtable function defined in te file freqtable.c0. Call tese functions wit te bare file name (For example, load te bitstring for your second test by calling read words("input2-bs.txt")). 3. Call decode: Call decode on your inputs. 4. Verify Results: If te test sould result in an error, tere will be no output to test. Oterwise compare te output to te output you expect for te test and return 0 if te test succeeded and 1 if it did not (recall tat 0 is te return value of coice wen a function succeeds). You sould name te file containing te main function for te test using te following format: test#-<expectedoutput>.c0 were # is te test number and <expectedoutput> is F if te function sould end in an assertion or annotation failure or S if te function sould succeed (return 0). (For example, if your second test sould succeed, you would name te file test2-s.c0) Submit all test and input files. We will be looking for a set of tests tat ceck all different possible outputs of te decode function witout duplicate tests for inputs in te same equivalence class. A few ints to get you started: You sould be writing a relatively small number of tests. Your examples don t ave to be long or even form complete sentences, but tey sould test te functionality explained in te spec. As a start, you migt consider using te provided example above. Task 4 (6 pts). Write a function string decode(tree encoding, bitstring bits) tat decodes bits based on te encoding. Use any functions from te string library to build te result string. Use te tests you developed to ceck your program. Add additional tests if you notice new classes of inputs tat were not tested by your original suite. 14