Outline Data Structures and Algorithms. Data compression. Data compression. Lossy vs. Lossless. Data Compression
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1 5-2 Dt Strutures n Algorithms Dt Compression n Huffmn s Algorithm th Fe 2003 Rjshekr Rey Outline Dt ompression Lossy n lossless Exmples Forml view Coes Definition Fixe length vs. vrile length Huffmn s Algorithm The lgorithm Prtil onsiertions Dt ompression Dt Compression Is one of the funmentl tehnologies of the Internet. Is neessry for fster t trnsmission. Useful even lolly to keep smller files or kup t. Dt ompression Types of ompression Lossless enoes the originl informtion extly. Lossy pproximtes the originl informtion. Uses of ompression Imges over the we: JPEG Musi: MP3 Generl-purpose: ZIP, GZIP, JAR, Lossy vs. Lossless If lossless ompression represents extly the sme t, in ompresse form, why use lossy t ll? Mye you n get exellent ompression without too muh loss of informtion? Let s look t n exmple
2 Compre two imges So where is the ifferene? One imge is 400K the other is 00K. Whih is whih? Wht n we onlue? There is efinitely tre-off. Lossless my not perform so well, ut it retins 00% of the informtion. Lossy n perform extremely well, ut is the ompression worth the loss of informtion? So how o we eie whih one to use? Some Consiertions Wht types of files woul you use lossless lgorithm on? Wht types of files woul you use lossy lgorithm on? Some Consiertions Wht types of files woul you use lossless lgorithm on? 2 gres. Complete works of Shkespere. Wht types of files woul you use lossy lgorithm on? Imges, musi. Files where you n get wy with n pproximtion to the t. Another Exmple - SVD Hyri Algorithm. You ompress n imge to ertin rnk So epening on the rnk, you hve either lossy or lossless informtion. But mking this lgorithm lossless tully oules the size of the file! In wht kins of situtions might it e useful? 2
3 Another Exmple - SVD Suppose we sen root to explore the moon. It oesn t know wht informtion is useful to us. We n sk it to first sen us smll rnk n then, if we re intereste, we n sk for lrger rnks. Ultimtely we get the ll the informtion, ut only if we relly wnt/nee it. Another Exmple - SVD Oky, so the root on the moon seems it ontrive. But wht out surfing the we on your hnhel? There is so muh nonsense on the we, we lerly on t wnt to ownlo everything sine nwith is t premium. Another Exmple - SVD Question Is there lossless ompression lgorithm tht n ompress ny file? Rnk Rnk 8 Rnk 6 Originl Answer Asolutely not! Why not? How oes ompression work? Lossy lgorithms re generlly mthemtilly se. They work y pplying trnsforms. Eg. JPEG isrete osine trnsform By pplying trnsform, they ttempt to pproximte the originl t. Lossless lgorithms nnot o tht sine they nee to mintin the originl t. So wht n they o? 3
4 How oes ompression work? They nee to nlyze the file n tke vntge of ertin properties it might hve. Or its struture. For exmple, if you wnte to ompress the first 0000 igits of Pi, wht oul you o? In se the slippe your min, here they re Pi A Forml View of Compression We nee to know some istriution over hrter frequenies. We n then tke vntge of tht to enoe more frequently ourring hrters with fewer its. E.g. Huffmn s Algorithm. Alterntely, we n look hrter ptterns rther thn just hrters. We n then reple frequently ourring ptterns with smll oe. E.g. Lempel- Ziv-Welh or LZW (next leture). Interlue: Bit-level Representtion of Dt All t is store on omputer s sequene of 0 s n s. I.e. its. This is very nturl wy to represent t, for the following reson: A omputer nnot, in generl, infer 0 ifferent vlues from the intensity of signl. It n however infer 2 ifferent vlues very esily. I.e. whether the signl is high or low. The prolem: If we use sequenes of just 0 s n s inste of 0 9 to represent t, regrless of the onveniene, ren t we using lot more spe? To ress this issue, let s look t n exmple 4
5 Suppose you h text file (sy, the omplete works of Shkespere) n you know tht it hs 32 ifferent symols n totl of hrters. How muh spe woul e nee to represent this in se 0? How out se 2? Suppose you h text file (sy, the omplete works of Shkespere) n you know tht it hs 32 ifferent symols n totl of hrters. How muh spe woul e nee to represent this in se 0? * Log0 32 How out se 2? * Log2 32 But Log2 32 = Log2 0 * Log0 32 So we re only onstnt ftor off. Rell symptoti nlysis: Wht o we o with the onstnts? Rememer, we re more intereste in the sling ftor. Oky, so we ve estlishe tht s it s esiest to store t s sequene of 0 s n s, ut how oes tht help us? In prtiulr, how o I tke text file n store it on the omputer? To o this we nee to invent oe. Coes Coes We n think of t s lrge sequene of its whih n e prtitione into smller meningful sequenes. A oe then is simply mpping from sequenes of its to hrters (or something meningful) For exmple, the ASCII system is oe. It mps single ytes (8 its) to unique hrters. 5
6 Coes You n think of oe s funtion mpping hrters to it strings. We woul like it to e ijetion. Wht if it is mny-to-one? Wht if it is one-to-mny? In the one-to-mny se nothing spetulrly hppens, ut it is pin to use the oe. Coes A oewor is simply inry string n oe is olletion of oewors n their menings. Must eh oewor in oe neessrily hve the sme length? I.e. is every oe fixe length oe? If not, we n then onstrut oes. But if ll the oewors in oe re the sme length, then Huffmn s lgorithm wouln t ompress t t ll! Prefix Free Coes Enoing strings A prefix free oe is one where no oewor is prefix of nother oewor. Wht gret ie! We n now onstrut oes whose oewors re of vrying lengths. Known s vrile length oes. Let s see how they help us Symols Fixe-length oe Vrile-length oe 50 We n enoe e s: Relly n 8 it string: Totl 205 hrs its e its Enoing strings Enoing strings Symols e Totl Symols e Totl Frequeny hrs Frequeny hrs Fixe-length oe its Fixe-length oe its Vrile-length oe its Vrile-length oe (optiml) its We n enoe e s: Relly 4 it string: Vrile-length oes Exploit sttistis of symols. More frequently ourring symols enoe using fewer its. Wht mkes goo vrile-length oe? It shoul e prefix free! 6
7 Tree representtion Why full inry tree? Represent prefix free oes s full inry trees Full: every noe Is lef, or Hs extly 2 hilren. The enoing is then (unique) pth from the root to lef. 0 =, =00, =000, =0 0 0 A noe with no siling n e move up level, improving the oe. An optiml oe for string n lwys e represente y full inry tree. 0 Enoing ost Huffmn s Algorithm Alphet: C Symol: Symol Frequeny: f() Depth in tree T: () (() is lso numer of its to enoe ) Enoing ost: K = C ( ) f ( ) Q: How to onstrut full inry tree tht minimizes K? Huffmn s Algorithm Huffmn s lgorithms will give you n optiml prefix free oe y onstruting n pproprite tree. Dt struture use: A Priority Queue. insert(element, priority) inserts n element with given priority into the queue. eletemin() returns the element with lest priority. Huffmn s Algorithm. Compute f() for every symol C 2. insert(, f()) into priority queue Q 3. for i = to C - (while Q is not empty) 4. z = new TreeNoe() 5. x = z.left = Q.eleteMin() 6. y = z.right = Q.eleteMin() 7. f(z) = f(x) + f(y) 8. Q.insert(z, f(z)) 9. return Q.eleteMin() 7
8 Exmple Exmple Exmple Exmple Exmple Exmple 8
9 Huffmn s Algorithm Is greey lgorithm tht onstruts n optiml prefix free oe for given piee of t Does it relly generte n optiml prefix free oe? Yes, ut the proof is eyon the sope of this ourse! Greey Algorithms At every step greey lgorithm mkes lolly optiml eision hoping tht it will up to glol optimum. This strtegy works surprisingly well for lot of lgorithms. Some exmples: Huffmn s for t ompression. Kruskl s for lulting minimum spnning tree s in grphs. Hill Climing Suppose you wnte to reh the summit of mountin ut oul only see 0 metres in ny iretion (ue to fog). Whih wy woul you go? Hill Climing Mking the lolly-est guess is effiient n esy, ut oesn t lwys work. Huffmn s Algorithm Why is it greey? Beuse t eh itertion in the loop, it pike the two optiml trees in the priority queue with whih to rete new noe without onsiering their implitions from glol stnpoint. Hw4 Hw4 is the t ompression l Prts n 2 re lossless ompression using Huffmn n LZW Prt 3 is tritionlly some lossy lgorithm, ut this semester it will e ompetition. Coneptully, ll you nee to know is in the letures. But this l n get very triky sine you will e eling with its n ytes n some low level stuff. 9
10 Notie tht Huffmn s lgorithm, in the setting we stuie it, n only ompress files of hrters sine it nees to know wht the lphet is in orer to ount the frequenies. Do we nee to moify the lgorithm in orer to ompress ritrry files? Tke minute to think out this. No, we on t! Suppose we hve file F to ompress. We n tret F s strem of its. So we re the first yte n onsier it in the ontext of our preefine lphet. ASCII in this se. Impliitly, we then en up treting every file s text file. Is tht goo ie? Wht out imges? It oesn t mtter! So long s we reproue the originl it sequene fter eompression. We n tret the file s ontining just the hrters {,,,} if we wnt, it won t ffet the orretness of our lgorithm. It will, however, ffet the performne. Why? 0
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