1 Binary Trees and Adaptive Data Compression

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1 University f Illinis at Chicag CS 202: Data Structures and Discrete Mathematics II Handut 5 Prfessr Rbert H. Slan September 18, 2002 A Little Bttle... with the wrds DRINK ME, (r Adaptive data cmpressin frm binary tree rtatins) Due: Tuesday, Octber 1, 10:00 p.m. 1 Binary Trees and Adaptive Data Cmpressin 1.1 Intrductin In this assignment yu are t implement an algrithm fr data cmpressin. The purpse f data cmpressin is t take a file A and transfrm it int anther file B in such a way that (1) B is smaller than A, and (2) it is pssible t recnstruct A given B. It is desirable that the transfrmatin be efficient, and that file B be a lt smaller than A. A prgram that cnverts A int B is called a cmpressr, and ne that undes this peratin is called an uncmpressr. A methd that can cmpress all files is nt pssible. (It is easy t see this: what wuld happen if yu just kept iterating the methd, cmpressing the utput f the cmpressr?) Since we cannt cmpress all files, the gal f data cmpressin must be t cmpress files typically fund n cmputers. A number f pwerful methds have been discvered t d this. 1.2 Prefix cdes Fr simplicity, suppse we have a set f five letters: A, B, C, D, E. Fr each f these letters, a cdewrd will be assigned. These cdewrds will be binary, and have the prperty that n ne is a prefix f any ther. Such a set f cdewrds is called a prefix cde. (Prefix cdes als have the prperty that any sequence f bits can be partitined int cdewrds in a unique way.) There is a ne-t-ne crrespndence between binary prefix cdes fr N letters, and binary trees with N leaves where every internal nde has exactly tw children. Figure 1 is an example f a prefix cde and crrespnding tree fr the five letters abve. Here is the crrespndence: the cdewrd fr a leaf is just a binary string representing the path frm the rt f the tree t the leaf, where left crrespnds t 0 and right crrespnds t 1. Given a tree, any sequence f bits can be turned back int letters in this way: Start at the rt, and walk dwn the tree as yu read the bits, ging left if yu read a 0 and right if yu read a 1. When yu cme t a leaf yu ve cmpleted ne cdewrd (s, print ut its letter). G back t the rt, and cntinue.

2 2 Handut 5: A Little Bttle... with the wrds DRINK ME, A: 00 <--Rt B: 010 C: 011 D: 10 E: 11 <--Internal nde A D E <--Leaf B C Figure 1: A prefix cde binary tree with five letters Nte that the cdewrds vary in length. By cnstructing a cde in which the number f bits needed fr the cmmn letters is very small, we might btain a way t represent a sequence f letters in fewer bits. (A Huffman cde is the binary cde that is the mst efficient fr a given distributin f letters. It is perhaps the single best knwn scheme; years ag I used t teach it in data structures.) 1.3 Adaptive Data Cmpressin In this assignment yu ll be implementing an adaptive data cmpressin algrithm, based n prefix cdes. This algrithm is cntinually changing the cde tree as it reads its input. Amng the advantages f such an adaptive methd are: (1) If ne regin f the input file t be cmpressed has a very different distributin than anther regin, then the adaptive methd can be mre efficient than a static ptimal Huffman cde. (2) It is nt necessary t make a preliminary pass ver the data t cmpute the ptimal Huffman tree, (3) The ptimal static tree des nt have t be transmitted alng with the cdewrds. By the way, mst f the ppular data cmpressin prgrams (e.g., Unix cmpress and gzip ) use adaptive algrithms, althugh they are nt based n prefix cdes as described here. Our adaptive cmpressin algrithm starts ut with a tree that is well balanced. The first letter is sent using the cde fr it in this initial tree. Then the tree is restructured in a way that depends n the letter just sent. The uncmpressr sees the cde fr the first letter, puts ut the first letter, and restructures the tree in exactly the same fashin. The cmpressr and the uncmpressr are in lck-step, and each character f the file is sent using a different prefix cde. The methd fr restructuring the cde tree is based n the premise that if a letter ccurs in the input, then it is likely that that letter will be ccur again in the near future. Thus, each time a letter ccurs, it is mved clser t the rt f the tree (thereby shrtening the cdewrd needed t represent it).

3 Handut 5: A Little Bttle... with the wrds DRINK ME, 3 x v v y ======> u x u w w y Restructuring the tree Figure 2: A right rtatin Let L be the leaf whse cde has just been sent. The restructuring peratin is divided int tw parts. First, a subset f the ndes alng the path frm L t the rt have their left and right children swapped, s that the path frm the rt t L becmes the leftmst path in the tree. Next, this leftmst path is made shrter by a methd called left-splaying. Left-splaying cuts the length f the path frm the rt t L by abut a factr f tw. Left-splaying starting frm a nde x is defined as fllws: If x is a leaf r if x s left child is a leaf, d nthing. Otherwise, d a lcal restructuring peratin, called a right rtatin. (See Figure 2.) Then d a left-splaying peratin starting at the new left child (u in Figure 2). Nte that u, w, and y are arbitrary subtrees (they culd be leaf ndes r have a large number f ndes belw them). Thus the left splay step cntinues frm u n dwn the left path, until it reaches a leaf. In ur algrithm, we are always beginning this left splaying peratin at the rt. Figures 3 and 4 are an example f what happens when the input letter 7 is prcessed by this algrithm. We start with the tree in Figure 3. After making the path t 7 the leftmst path, and then left-splaying 7 we have Figure 4. Nte that tw right rtatins are dne in this particular left-splaying step. 2 The Assignment Yu are t write tw prgrams, shrink and grw, that implement this algrithm n an input alphabet f size 256. This is a very natural alphabet size, since in ASCII there are 8 bits per byte, and thus 256 pssible values fr a byte. The prgram will read the bytes f its input ne by ne, and cnstruct a sequence f bits (by cncatenating the cdewrds generated by the algrithm), pack these bits int bytes, and utput them. There are three useful functins fr dealing with bits in the file bit utils.cc: int get bit(istream& myin) This reads ne bit f the input stream myin (nrmally yu d call it with cin), and returns it. If there are n mre bits, then it returns 1.

4 4 Handut 5: A Little Bttle... with the wrds DRINK ME, Figure 3: Initial tree (the cde fr 7 is 1110) Figure 4: Make 7 the leftmst path, then left-splay

5 Handut 5: A Little Bttle... with the wrds DRINK ME, 5 vid put bit(stream& myut, int b) This puts ne bit int the utput stream myut (nrmally yu d call this with cut). vid put finish(stream& myut) If the number f bits sent s far is nt a multiple f eight, this puts ut enugh zers t make it s. Nte that the cmpressed versin f a file may nt have a length in bits that is a multiple f eight. In this case we will have a prblem packing it int bytes, since the end f any file in Unix must end n a byte bundary. If we fill in the remaining bits with zers (by calling put finish()) this may cnfuse the uncmpressr. It may view thse remaining zers as cdewrds. In rder t deal with this prblem, the cmpressr will send a special terminating cde t indicate the end f the file. S instead f having 256 leaves, the cde tree will have 257. Leaves numbered are fr sending the bytes f the input file, and a special leaf numbered 256 will crrespnd t a special cde indicating the end f the input. The cmpressr writes this special cde just befre it calls put finish() and quits. When the uncmpressr sees this, it realizes the input is finished, and quits. Nte that yur prgrams take their input frm the standard input, and write their utput t the standard utput. Data can be supplied t the standard input in several ways. Yu can use use < t get the input frm a file, and > t put the utput int a file. Fr example: shrink < junk.rig > junk.small Which runs the file junk.rig thrugh the cmpressin prgram and puts the result int anther file called junk.small. 2.1 Data-types and functins yu shuld use T help yu rganize yur wrk, and t make it easier fr us t grade (and give partial credit), yur slutin shuld fllw the fllwing utline. Yur binary tree class shuld nt be templated. Yur tree ndes shuld have the fllwing fields. (Yu can add additinal fields if yu like.): int letter; // -1 if internal nde, else the letter this leaf represents. TreeNde* parent, left, right; Yur Tree class shuld have (at least) the fllwing fur member functins: cnstructr Tree(int size) This builds a balanced tree with size leaves. The left and right subtrees f every internal nde shuld either have the same number f leaves, r the right subtree shuld have ne mre leaf than the left subtree. Yu ll prbably find it cnvenient t have this rutine initialize the letter fields f the tree, either t zer, r their crrect final values. Yu may add mre arguments t this functin if yu want.

6 6 Handut 5: A Little Bttle... with the wrds DRINK ME, vid utput cde(treende start) cnst Given a leaf, this shuld utput the cde fr it, ne bit at a time. (using functins frm bit_utils). vid make path leftmst(treende start) This swaps left and right children f ndes alng the path frm start t the rt in such a way that start becmes a member f the leftmst path dwn frm the rt. vid left splay(treende start) This des the left splay peratin starting frm the specified nde, and cntinuing dwn t the left. 2.2 What yu shuld turn in Yu will write and turn in many files, almst all f which will be pretty small (excluding cmments, which will f curse be vluminus!). At the least, there will be files called shrink.c, grw.c, tree utils.c (and.h), cde and header files fr the bit utilities (described belw) assuming that yu d that part f the assignment, Makefile, and README. shrink.c is the cmpressin prgram, and grw.c is the uncmpressin prgram. tree utils.c will cntain all the functins that are used by bth the shrink and grw prgrams. Each f thse tw has a main; s neither needs a.h file. The file called README shuld list all knwn bugs and what yu knw abut what might be causing them. 2.3 Hints First write the fur member functins f the Tree class mentined abve. Then write the main rutines fr each part (shrink and grw). The main rutines will be pretty shrt. Try t write yur Tree member functins s they can be used fr bth cmpressing and uncmpressing. Yu may, f curse, add additinal methds if yu wish. Leave the bit packing and unpacking fr after yu have mre r less everything else taken care f. Fr initial develpment, just have shrink utput, and grw read plain ld int 1 and 0. (Of curse, at this pint, shrink will in fact blw up the size f the file cnsiderably.) Yu may find it cnvenient t pass in a few mre parameters t the Tree cnstructr. Or yu may find it cnvenient t use sme glbal variables here (they re nt always bad). Many f the functins are very easy if yu use recursin. Indeed, a crrect slutin fr this assignment prbably invlves cnsiderably fewer lines f cde than the first assignment. Remember: fr this assignment, a nde is a leaf if its children are NULL. It is easy t get int truble frgetting t set parent pinters crrectly. Fr cmpressin, yu need, given a character such as a, t be able t find its assciated leaf in the tree. What wuld be an efficient way t implement this? (Miserably inefficient implementatins will lse sme pints.)

7 Handut 5: A Little Bttle... with the wrds DRINK ME, 7 3 Cping with bits as data One prblem that will cnfrnt yu in this assignment is that yu must wrk with bits, as ppsed t, say, ints r chars. I believe that the bit_utils functins will shield yu frm almst all f this issue, but here s sme facts just in case. Bit hackery in C Remember that a char cnsists f exactly 8 bits. The integer cnstant 1 always has the bit pattern f a single 1 preceded by 0 s; the integer cnstant 0 is always all 0s. The peratrs n bits that yu are likely t need are bitwise OR: bitwise AND: & left shift << right shift >> (Operatrs = and &= are als prvided.) Fr the shift peratrs, the first argument is the thing yu want shifted, and the secnd argument is the number f psitins that yu want in shifted by. Fr example, t find ut whether bit psitin p f character c is 0 r 1 we culd use the expressin (c >> p) & 1 If instead, we want t set psitin p f character c t be be 1 if the value f (char r int) flag is 1, and leave c unchanged if flag is 0, we wuld say c = flag << p; 4 Des it wrk? By the way, hw well des this methd f data cmpressin wrk? On what types f files des it wrk well, and n which type des it wrk prly? (Yu dn t need t hand this in; it s just smething t think abut.) 5 Submissin Deadline: Yur cde must be electrnically submitted using the turnin prgram n the departmental machines by 10 p.m. Tuesday, Octber 1. Yu must prvide us with a wrking makefile. Please DO NOT hand in., cre, r executable files they just take up space n the turnin directry.

Using SPLAY Tree s for state-full packet classification

Using SPLAY Tree s for state-full packet classification Curse Prject Using SPLAY Tree s fr state-full packet classificatin 1- What is a Splay Tree? These ntes discuss the splay tree, a frm f self-adjusting search tree in which the amrtized time fr an access,