Winter 2016 COMP-250: Introduction to Computer Science. Lecture 24, April 7, 2016

Transcription

1 Winter 2016 COMP-250: Introduction to Computer Science Lecture 24, April 7, 2016

2 Tries

3 Tries Atrie is tree-sed dt dte structure for storing strings in order to mke pttern mtching fster. Tries cn e used to perform prefix queries for informtion retrievl. Prefix queries serch for the longest prefix of given string X tht mtches prefix of some string in the trie. A trie supports the following opertions on set S of strings: insert(x): Insert the string X into S Input: String Ouput: None remove(x): Remove string X from S Input: String Output: None prefixes(x): Return ll the strings in S tht hve longest prefix of X Input: String Output: Enumertion of strings

4 Let S e set of strings from the lphet Σ such tht no string in S is prefix to nother string. A stndrd trie for S is n ordered tree T tht: - Ech edge of T is leled with chrcter from Σ - The ordering of edges out of n internl node is determined y the lphet Σ - The pth from the root of T to ny node represents prefix in Σ tht is equl to the concntention of the chrcters encountered while trversing the pth. For exmple, the stndrd trie over the lphet Σ = {, } for the set {,,,, }

5 Tries (cont.) An internl node cn hve 1 to d children when d is the size of the lphet. Our exmple is essentilly inry tree. A pth from the root of T to n internl node v t depth i corresponds to n i-chrcter prefix of string of S. We cn implement trie with n ordered tree y storing the chrcter ssocited with n edge t the child node elow it.

6 Compressed Tries Acompressed trie is like stndrd trie ut mkes sure tht ech trie hd degree of t lest 2. Single child nodes re compressed into n single edge. Acriticlnode is node v such tht v is leled with string from S, v hs t lest 2 children, or v is the root. To convert stndrd trie to compressed trie we replce n edge (v 0, v 1 ) y ech chin of on nodes (v 0, v 1...v k ) for k 2 such tht -v 0 nd v 1 re criticl ut v 1 is criticl vi for 0<i<k - ech v 1 hs only one child vi Ech internl node in compressed tire trie hs t lest two children nd ech externl is ssocited with string. The compression reduces the totl spce for the trie from O(m) where m is the sum of the the lengths of strings in S to O(n) where n is the numer of strings in S.

7 An exmple:

8 Prefix Queries on Trie Algorithm prefixquery(t, X): Input: Trie T for set S of strings nd query string X Output: The node v of T such tht the leled nodes of the sutree of T rooted t v store the strings of S with longest prefix in common with X v T.root() i 0 {i is n index into the string X} repet for ech child w of v do let e e the edge (v,w) Y string(e) {Y is the sustring ssocited with e} l Y.length() {l=1 if T is stndrd trie} Z Z X.sustring(i, i+l-1) {Z holds the next l chrc ters of X} if Z = Y then v w i i+1{move to W, incrementing i pst Z} rek out of the for loop else if proper prefix of Z mtched proper prefix of Y then v w rek out ot the repet loop until v is externl or v w return v

9 Insertion nd Deletion Insertion: We first perform prefix query for string X. Let us exmine the wys prefix query my end in terms of insertion. - The query termintes t node v. Let X 1 e the prefix of X tht mtched in the trie up to node v nd X 2 e the rest of X. If X 2 is n empty string we lel v with X nd the end. Otherwise we crete new externl node w nd lel it with X. - The query termintes t n edge e=(v, w) ecuse prefix of X mtch prefix(v) nd proper prefix of string Y ssocited with e. Let Y 1 e the prt of Y tht X mtched mthed to nd Y 2 the rest of Y. Likewise for X 1 nd X 2. Then X=X 1 +X 2 = prefix(v) +Y 1 +X 2. We crete new node u nd split the edges(v, u) nd (u, w). If X2 is empty then w we lel u with X. Otherwise we crete node z which is externl nd lel it X. Insertion is O(dn) when d is the size of the lphet nd n is the length of the string t insert.

10 Insertion nd Deletion (cont.) serch stops here 6 insert()

11 serch stops here insert()

12 Lempel Ziv Encoding Constructing the trie: - Let phrse 0 e the null string. - Scn through the text - If you come cross letter you hven t seen efore, dd it to the top level of the trie. - If you come cross letter you ve lredy seen, scn down the trie until you cn t mtch ny more chrcters chrcters, dd node to the trie representing the new string. - Insert the pir (nodeindex, lstchr) into the compressed string. Reconstructing the string: - Every time you see 0 in the compressed string dd the next chrcter in the compressed string directly to the new string. - For ech non-zero nodeindex, put the sustring corresponding to tht node into the new string, followed y the next chrcter in the compressed string.

13 Lempel Ziv Encoding (contd.) A grphicl exmple: Uncompressed text: phrses: Compressed text: how now rown cow in town. (nil) h0o0w0_0n2w40r6n4c6_0i5_0t9. Trie: 0 h o w _ n r i t w c _ n 6 _

14 File Compression text files re usully stored y representing ech chrcter with n 8-it ASCII code (type mn scii in Unix shell to see the ASCII encoding) the ASCII encoding is n exmple of fixe d- le ngth encoding, where ech chrcter is represented with the sme numer of its in order to reduce the spce required to store text file, we cn exploit the fct tht some chrcters re more likely to occur thn others v ri le-length encoding uses inry codes of different lengths for different chrcters; thus, we cn ssign fewer its to frequently used chrcters, nd more its to rrely used chrcters. Exmple: - text: jv - encoding: = 0, j = 11, v = 10 - encoded text: (6 its) How to decode? - = 0, j = 01, v = 00 - encoded text: (6 its) - is this jv, jvv, j...

15 Encoding Trie to prevent miguities in decoding, we require tht the encoding stisfies the prefix rule, tht is, no code is prefix of nother code - = 0, j = 11, v = 10 stisfies the prefix rule - = 0, j = 01, v= 00 does not stisfy the prefix rule (the code of is prefix of the codes of j nd v) we use n encoding trie to define n encoding tht stisfies the prefix rule - the chrcters stored t the externl nodes - left edge mens 0 - right edge mens D B C 0 1 A R A = 010 B = 11 C = 00 D = 10 R = 011

16 Exmple of Decoding trie: D B C 0 1 A R A = 010 B = 11 C = 00 D = 10 R = 011 encoded text: text: A B R A C A D A B R A A 010 B 11 R 011 A 010 C 00 A 010 D 10 A 010 B 11 See? Decodes like mgic... R 011 A 010

17 Trie this! E N K C S B T W R O R 10 O 00 B 0111 E 1100 R 10 T 0110 O 00 K 1110 N 1111 O 00 W 0101 S 0100 C 1101 S 0100

18 Optiml Compression An issue with encoding tries is to insure tht the encoded text is s short s possile: C R ABRACADABRA its A A 0 1 D C 0 1 D ABRACADABRA its B B R

19 Huffmn Encoding Trie ABRACADABRA chrcter A B R C D frequency A B R C 2 1 D 1 5 A 4 2 B 2 R 2 C 1 D A B R C D

20 A B R A C A D A B R A its Another Huffmn Encoding Trie A 2 B R C D 1

21 Construction Algorithm with Huffmn encoding trie, the encoded text hs miniml length Algorithm Huffmn(X): Input: String X of length n Output: Encoding trie for X Compute the frequency f(c) of ech chrcter c of X. Initilize priority queue Q. for ech chrcter c in X do Crete single-node tree T storing c Q.insertItem(f(c), T) while Q.size() > 1 do f 1 Q.minKey() T 1 Q.removeMinElement() f 2 Q.minKey() T 2 Q.removeMinElement() Crete new tree T with left sutree T 1 nd right sutree T 2. Q.insertItem(f 1 + f 2 ) return tree Q.removeMinElement() runing time for text of length n with k distinct chrcters: O(n + k log k)

22 Imge Compression we cn use Huffmn encoding lso for inry files (itmps, executles, etc.) common groups of its re stored t the leves Exmple of n encoding suitle for /w itmps

23 Dt Representtion/ Lossy Compression Sound formts Imge formts Movie formts

24 Dt Representtion sound formts

25 Sound formts

26 AIFF Sound formt ech smple is signed 15 (or 23 or 31) its vlue 176 smples 4 ms ( smples = 1 s)

27 AIFF Sound formt smples / second 16 = 2 B / smple (or 24 = 3 B / smple or 32 = 4 B / smple) stereo = two chnnels 2 x 2 x = 176,4 kb/s CD 700 MB 75 minutes

28 AIFF Sound formt why smples / second? ecuse it is in the correct rnge... ecuse is divisile y 2,3,4,5,6,7,9,10

29 MP3 Sound formt Bsed on Fourrier trnsform. 576 smples of mplitude / time re converted to 576 smples of distinct frequencies. Bss Trele

30 In humn ers, the cochle is mechniclly performing process nlog to the Fourrier Trnsform. The erdrum virtes ck nd forth ccording to the wve-like representtion of the sound. The frequency informtion stimultes specific re in the cochle.

31 MP3 Sound formt Bss Trele Frequencies with smll coefficients removed Wveform reconstructed is close to originl

32 MP3 Sound formt HIGH qulity low Bss Trele

33 Dt Representtion Imge formts

34 TIFF imge formt

35 TIFF imge formt n 8x8 su-region of lrge imge: ech individul pixel uses 24 ites: 8 for red, 8 for lue, 8 for green. totl size = numer of pixels x 3 Bytes.

36 Animl eyes focus light on the retin where n imge of the environment is produced. This imge is nlysed ccording to 3 types of colour sensitive cones, mostly triggered ner the red, green nd lue nds. A perceived colour is triplet (x,y,z) of excittions of the 3 types of cones. Two comintions of colours yielding the sme triplet (x,y,z) re indistinguishle.

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38 JPEG imge formt Using trnsformtion similr to Fourier trnsform (used for udio), so clled Discrete Cosine Trnsform is pplied to ech su-loc of size 8x8. Notice: colours re used for strct dt. Drk mens smll, right mens lrge.

39 JPEG imge formt If no dt is removed, the resulting imge is nerly identicl to the originl. Imprecision in the trnsform cuses smll errors.

40 JPEG imge formt If ll dt very close to zero is removed, the resulting imge is only slightly different from the originl Notice: colours re used for strct dt. Drk mens smll, right mens lrge.

41 JPEG imge formt If ll dt close to zero is removed, the resulting imge is somewht different from the originl

42 JPEG imge formt If ll dt of smll mgnitude is removed, the resulting imge is still very similr to the originl Notice: colours re used for strct dt. Drk mens smll, right mens lrge.

43 JPEG imge formt If only dt of lrge mgnitude is kept, the resulting imge is similr ut quite different from the originl. Most detils re wiped out.

44 Dt Representtion movie formts

45 RAW movie formt pixels per frme 24 its (colour) per pixel 30 frmes per second 30 x 3 x 720 x MB/s 135 GB/hour typiclly 200 GB per movie!!! ( 50 DVDs)

46 MPEG2 Movie Formt

47 MPEG2 Movie Formt

48 MPEG2 Movie Formt

49 MPEG2 Movie Formt

50 MPEG2 formt Fixed Bckground imges

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53 sving is out 96%

54 Trvelling MPEG2 formt

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59 MPEG2 formt Ech imge is encoded with JPEG or similr. Sound is encoded with MP3 or similr. Most frmes use only smll mount of info to construct from previous frmes. A complete frme is displyed every so often to mke sure the fix prt or trvelling prt hs not sustntilly chnged.

60 Winter 2016 COMP-250: Introduction to Computer Science Lecture 24, April 7, 2016