Similarity Search. The Binary Branch Distance. Nikolaus Augsten.
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1 Similrity Srh Th Binry Brnh Distn Nikolus Augstn Dpt. of Computr Sins Univrsity of Slzurg Vrsion Jnury 11, 2017 Wintrsmstr 2016/2017 Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
2 Outlin 1 Binry Brnh Distn Binry Rprsnttion of Tr Binry Brnhs Lowr Boun for th Eit Distn Complxity Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
3 Outlin Binry Brnh Distn Binry Rprsnttion of Tr 1 Binry Brnh Distn Binry Rprsnttion of Tr Binry Brnhs Lowr Boun for th Eit Distn Complxity Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
4 Binry Tr Binry Brnh Distn Binry Rprsnttion of Tr In inry tr h no hs t most two hilrn; lft hil n right hil r istinguish: no n hv right hil without hving lft hil; Nottion: T B = (N, E l, E r ) T B nots inry tr N r th nos of th inry tr E l n E r r th gs to th lft n right hilrn, rsptivly Full inry tr: inry tr h no hs xtly zro or two hilrn. Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
5 Exmpl: Binry Tr Binry Brnh Distn Binry Rprsnttion of Tr Two iffrnt inry trs: T B = (N, E l, E r ) T B1 = ({,,,,, f, g}, {(, ), (, ), (, ), (, f )}, {(, ), (, g)}) T B2 = ({,,,,, f, g}, {(, ), (, ), (, f )}, {(, ), (, ), (, g)}) T B1 T B2 f g f g A full inry tr: h i f g Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
6 Binry Brnh Distn Binry Rprsnttion of Tr Binry Rprsnttion of Tr Binry tr trnsformtion: (i) link ll nighoring silings in tr with gs (ii) lt ll prnt-hil gs xpt th g to th first hil Trnsformtion mintins ll informtion strutur informtion Originl tr n ronstrut from th inry tr: lft g rprsnts prnt-hil rltionships in th originl tr right gs rprsnts right-siling rltionship in th originl tr Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
7 Binry Brnh Distn Binry Rprsnttion of Tr Exmpl: Binry Tr Trnsformtion Rprsnt tr T s inry tr: T inry rprsnttion of T Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
8 Binry Brnh Distn Binry Rprsnttion of Tr Normliz Binry Tr Rprsnttion W xtn th inry tr with null nos s follows: null no for h missing lft hil of non-null no null no for h missing right hil of non-null no Not: Lf nos gt two null-hilrn. Th rsulting normliz inry rprsnttion is full inry tr ll non-null nos hv two hilrn ll lvs r null-nos (n ll null-nos r lvs) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
9 Binry Brnh Distn Binry Rprsnttion of Tr Exmpl: Normliz Binry Tr Trnsforming T to th normliz inry tr B(T): T B(T) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
10 Outlin Binry Brnh Distn Binry Brnhs 1 Binry Brnh Distn Binry Rprsnttion of Tr Binry Brnhs Lowr Boun for th Eit Distn Complxity Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
11 Binry Brnh Binry Brnh Distn Binry Brnhs A inry rnh BiB(v) is sutr of th normliz inry tr B(T) onsisting of non-null no v n its two hilrn Exmpl: BiB() = ({,, }, {(, )}, {(, )}) BiB() = ({, 1, 2 }, {(, 1 )}, {(, 2 )}) Although th two null nos hv intil lls (), thy r iffrnt nos. W mphsiz this y showing thir IDs in susript. Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
12 Binry Brnh Distn Binry Brnhs Binry Brnhs of Trs n Dtsts Binry rnhs n sriliz s strings: BiB(v) = ({v,, }, {(v, )}, {(v, )}) λ(v) λ() λ() w n sort ths strings ( > λ(v) for ll non-null nos v) Binry rnh sts: BiB(T) is th st of ll inry rnhs of B(T) BiB(S) = T S BiB(T) is th st of ll inry rnhs of tst S BiB sort (S) is th vtor of sort sriliz strings of BiB(S) Not: nos r uniqu in th tr, thus inry rnhs r uniqu lls r not uniqu, thus th sriliz inry rnhs r not uniqu Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
13 Binry Brnh Distn Binry Brnhs Exmpl: Binry Brnhs of Trs n Dtsts 1 3 T 1 T BiB( 1 ) BiB( 4 ): BiB( 1 ) = ({ 1, 2, 3 }, {( 1, 2 )}, {( 1, 3 )}) BiB( 4 ) = ({ 4, 5, 6 }, {( 4, 5 )}, {( 4, 6 )}) Sriliztion of oth, BiB( 1 ) n BiB( 2 ), is intil: Sort vtor of sriliz strings of BiB(S), whr S = {T 1, T 2 }: BiB sort (S) = (,,,,,,,,, ) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
14 Binry Brnh Vtor Binry Brnh Distn Binry Brnhs Th inry rnh vtor BBV (T) is rprsnttion of th inry rnh st BiB(T) Constrution of th inry rnh vtor BBV (T): omput BiB sort (S) (sriliz n sort BiB(S)) i is th i-th sriliz inry rnh in sort orr ( i = BiB sort (S)[i]) BBV (T)[i]) is th numr of inry rnhs in B(T) tht sriliz to i Not: BBV (T)[i] is zro if i os not ppr in BiB(T) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
15 Binry Brnh Distn Exmpl: Binry Brnh Vtors Binry Brnhs T 1 T2 S = {T 1, T 2 } is th t st BiB sort (S) is th vtor of sort sriliz strings of BiB(S) BBV (T i ) is th inry rnh vtor of T i th vtor of sriliz strings n th inry rnh vtors r: BiB sort (S) BBV (T 1 ) BBV (T 2 ) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
16 Outlin Binry Brnh Distn Lowr Boun for th Eit Distn 1 Binry Brnh Distn Binry Rprsnttion of Tr Binry Brnhs Lowr Boun for th Eit Distn Complxity Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
17 Binry Brnh Distn Binry Brnh Distn [YKT05] Lowr Boun for th Eit Distn Dfinition (Binry Brnh Distn) Lt BBV (T) = ( 1,..., k ) n BBV (T ) = ( 1,..., k ) inry rnh vtors of trs T n T, rsptivly. Th inry rnh istn of T n T is k δ B (T, T ) = i i. i=1 Intuition: W ount th inry rnhs tht o not mth twn th two trs. Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
18 Binry Brnh Distn Exmpl: Binry Brnh Distn Lowr Boun for th Eit Distn W omput th inry rnh istn twn T 1 n T 2 : T 1 T 2 Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
19 Binry Brnh Distn Lowr Boun for th Eit Distn Exmpl: Binry Brnh Distn Th normliz inry tr rprsnttions r: B (T 1 ) B (T 2 ) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
20 Binry Brnh Distn Exmpl: Binry Brnh Distn Lowr Boun for th Eit Distn Th inry rnh vtors of T 1 n T 2 r: BiB sort (S) BBV (T 1 ) BBV (T 2 ) Th inry rnh istn is δ B (T 1, T 2 ) = 10 i=1 1,i 2,i = = 9, whr 1,i n 2,i r th i-th imnsion of th vtors BBV (T 1 ) n BBV (T 2 ), rsptivly. Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
21 Binry Brnh Distn Lowr Boun Thorm Lowr Boun for th Eit Distn Thorm (Lowr Boun) Lt T n T two trs. If th tr it istn twn T n T is δ t (T, T ), thn th inry rnh istn twn thm stisfis δ B (T, T ) 5 δ t (T, T ). Proof (Skth Full Proof in [YKT05]). Eh no v pprs in t most two inry rnhs. Rnm: Rnming no uss t most two inry rnhs in h tr to mismth. Th sum is 4. Similr rtionl for insrt n its omplmntry oprtion lt (t most 5 inry rnhs mismth). Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
22 Binry Brnh Distn Lowr Boun for th Eit Distn Proof Skth: Illustrtion for Rnm trnsform T 1 to T 2 : rn(, x) f g inry trs B(T 1 ) n B(T 2 ) f g x f g Two inry rnhs (, g) xist only in B(T 1 ) Two inry rnhs (x, xg) xist only in B(T 2 ) δ t (T 1, T 2 ) = 1 (1 rnm) δ B (T 1, T 2 ) = 4 (4 inry rnhs iffrnt) f x g Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
23 Binry Brnh Distn Proof Skth: Illustrtion for Insrt trnsform T 1 to T 2 : ins(x,, 2, 3) f g inry trs B(T 1 ) n B(T 2 ) f g x f g f x g Lowr Boun for th Eit Distn Two inry rnhs (, f g) xist only in B(T 1 ) Tr inry rnhs (x, f, xg) xist only in B(T 2 ) δ t (T 1, T 2 ) = 1 (1 insrtion) δ B (T 1, T 2 ) = 5 (5 inry rnhs iffrnt) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
24 Proof Skth Binry Brnh Distn Lowr Boun for th Eit Distn In gnrl it n shown tht Rnm hngs t most 4 inry rnhs Insrt hngs t most 5 inry rnhs Dlt hngs t most 5 inry rnhs Eh it oprtion hngs t most 5 inry rnhs, thus δ B (T, T ) 5 δ t (T, T ). Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
25 Outlin Binry Brnh Distn Complxity 1 Binry Brnh Distn Binry Rprsnttion of Tr Binry Brnhs Lowr Boun for th Eit Distn Complxity Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
26 Binry Brnh Distn Complxity Complxity: Binry Brnh Distn Comput th istn twn two trs of siz O(n): (S = {T 1, T 2 }, n = mx{ T 1, T 2 }) Constrution of th inry rnh vtors BBV (T 1 ) n BBV (T 2 ): 1. BiB(S) omput th inry rnhs of T 1 n T 2 : O(n) tim n sp (trvrs T 1 n T 2 ) 2. BiB sort (S) sort sriliz inry rnhs of BiB(S): O(n log n) tim n O(n) sp 3. onstrut BBV (T 1 ) n BBV (T 2 ): () trvrs ll inry rnhs: O(n) tim n sp () for h inry rnh fin position i in BiB sort (S): O(n log n) tim (inry srh in BiB sort (S) for n inry rnhs) () BBV (T)[i] is inrmnt: O(1) Computing th istn: th two inry rnh vtors r of siz O(n) omputing th istn hs tim omplxity O(n) (sutrting two inry rnh vtors) Th ovrll omplxity is O(n log n) tim n O(n) sp. Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
27 Binry Brnh Distn Complxity Improving th Tim Complxity with Hsh Funtion Not: Improvmnt using hsh funtion: w ssum hsh funtion tht mps th O(n) inry rnhs to O(n) ukts without ollision w o not sort BiB(S) position i in th vtor BBV (T) is omput using th hsh funtion O(n) tim (inst of O(n log n)) n O(n) sp In th following w ssum th sort lgorithm with O(n log n) runtim. Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
28 Binry Brnh Distn Complxity for Similrity Joins Complxity Join two sts with N trs h (tr siz: n): Comput Binry Brnh Vtors (BBVs): O(Nn log(nn)) tim, O(N 2 n) sp BBVs r of siz O(Nn) tim: sort O(Nn) inry rnhs / O(Nn) inry srhs in BBVs sp: O(N) BBVs must stor Comput Distns: O(N 3 n) tim omputing th istn twn two trs hs O(Nn) tim omplxity (sutrting two inry rnh vtors) O(N 2 ) istn omputtions rquir Ovrl Complxity: O(N 3 n + Nn log n) 2 tim n O(N 2 n) sp 2 O(N 3 n + Nn log(nn)) = O(N 3 n + Nn log N + Nn log n) = O(N 3 n + Nn log n) Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
29 Rui Yng, Pnos Klnis, n Anthony K. H. Tung. Similrity vlution on tr-strutur t. In Proings of th ACM SIGMOD Intrntionl Confrn on Mngmnt of Dt, pgs , Bltimor, Mryln, USA, Jun ACM Prss. Augstn (Univ. Slzurg) Similrity Srh Wintrsmstr 2016/ / 28
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