Fortgeschrittene Datenstrukturen Vorlesung 11

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1 Fortgeschrittee Datestruture Vorlesug 11 Schriftführer: Marti Weider Succict Data Structures (ctd.) 1.1 Select-Queries A slightly differet approach, compared to ra, is used for select. B represets the bit-vector with B = ad let = log 2. A ew table is defied, which stores the ( i) th occurece of a 1-bit i B. Alteratively, [i] = select 1 (B, i). As we ca see, table [1, ] divides B ito blocs of differet sizes, whereas each bloc cotais 1 s. A example is give i Figure 1, where a abstract bit-vector B is divided ito blocs with 1 s i each. Example 1. Let = [17, 28, 36, 53,...] ad = 8. I this case, the 8 th 1 would be at idex 17 i B, the 16 th 1 at idex 28 ad so o. B =... [1] [2] [3] [4]... The resultig blocs are grouped as follows: Figure 1: Divisio of B ito blocs with 1 s Defiitio 1. A log bloc spas more tha 2 = Θ(log 4 ) positios i B. log 4 Its umber is limited by. Therefore, the aswers for select-queries withi all log blocs ca be stored explicitly i a table: LogBloc[0, log 4 ][1, ], where LogBloc[i][j] = select 1(B, i + j). Moreover, the LogBloc table is idexed by potetial bloc umbers, because we do ot ow how may log blocs there are before a give positio. Therefore, we imagie that a log bloc begis at every 2 positio. Select-queries to log blocs ca be respoded completely based o this structure. Defiitio 2. A short bloc spas 2 positios i B. It cotais 1-bits ad it spas 2 positios i B at most. We divide their rage of argumets ito sub-rages of: = log 2 = Θ(log 2 log ). The, a table [1, ] is defied, whereas the aswers to select-queries for multiples of (relative to the ed of the previous bloc) are stored. 1

2 I table, a symbol idicates if we are i a log bloc. The formal defiitio of is: [i] = select 1 (B, i ) ([ i i ]), with as the bloc before ith 1 ad the subtrahed, represetig the ed of the bloc. Table divides the blocs ito miiblocs, each cotaiig 1-bits. A miibloc is called log if it spas more tha s = 2 = log 2 positios i B, ad short otherwise. Aalogous to the log blocs, the aswers to all select-queries are stored explicitly for all log miiblocs, relative to the begiig of the correspodig short bloc. The table LogMiiBloc[0, s ][1, ] is idexed by the potetial log miibloc umbers, because the umber of log miiblocs up to a give positio is uow. Fially, a looup table is stored for the short miiblocs because of its relative small size. Defiitio 3 (Looup table for small miiblocs). The looup table for short miiblocs is defied as follows: Ibloc[0, 2 s 1][1, ], where: Ibloc[patter][i] = select 1 (patter, i) for all bit-patters of legth s ad 1 i. Based o this table, a select-query withi a short miibloc B[b, i] ca be aswered by looig at Ibloc[B[b, b + s 1]][i]. We should eep i mid that short miibloc could be shorter tha s. I this case, a paddig with arbitrary bits i the ed to match exactly s bits does ot affect the select query aswer. The query procedure follows the descriptio of the data structure. ote that we ca determie if (mii-) blocs are log or short by ispectig adjacet elemets of (or ) ad checig if they differ by more tha 2 (or 2 ). I the followig, it is verified that the required bit space of the defied structures for the select query are succict. Table The table ca have etries as maximum i the case that B oly cotais 1 s. Moreover, oe stored idex requires log bits. We get: = log = O( log ) = log 2 O( log ) = o() Table LogBloc LogBloc cosists of etries, because of oe etry for each potetial log 2 bloc. Moreover, it has colums i order to store the positios of the 1 s, which are iside the bloc. A table cell requires log bits. To sum up, the bit space results i: LogBloc = log = O( 2 log ) = o() Table The aalysis is similar to by just usig the defiitio for : = log 2 = 4 log log log 2 log = O( log log ) = o() LogMiiBloc The table cosists of s potetial miibloc etries ad for each, with idices for 1 s, relative to the edig of the previous bloc. Thus, we get: LogMiiBloc = s log 2 = log 2 log 2 = O( log3 log log ) = o() Ibloc Ibloc as a looup table cotais 2 s differet patters, icludig idices for the positio of the i th 1 (0 < i < ): Ibloc = 2 s log s = O( log 3 log ) = o() As ca be see from the aalysis, all defied structures require a bit space i o() ad thus, they are succict. 2

3 Example 2. A example for the select structures is give i Figure 2. I the upper part, the bit vector B is show, icludig the borders for the differet bloc types ad idices. The three reddashed separators idicate the potetial borders of log blocs. Moreover, there is preseted a log miibloc i darer blue ad three short miiblocs followig. Below, oe ca see the parameters, etc. ad the tables LogBloc, LogMiiBloc ad Ibloc, which are associated by meas of correspodig colors with the blocs i the bit-vector.!!!!!!!!!!!!!!! 0!!!!!!!!!!!!!!!! 1!!!!!!!!!!!!!!!! 2!!!!! 3!!!!!!!!!!! 1!!!!!!!!!! 2!!!!!!!!!! 3!!!!!!!!!! 4!!!!!!!!!! 5!!! 0! 1! 2! 3! 4! 5! 6! 7! 8! 9! 0! 1! 2! 3! 4! 5! 6! 7! 8! 9! 0! 1! 2! 3! 4! 5! 6! 7! 8! 9! 0! 1! 2! 3! 4! 5! 6! 7! 8! 9! 0! 1! 2! 3! 4! 5! 6! 7! 8! 9! 0! 1! 2! 0! 0! 1! 1! 0! 0! 1! 0! 1! 1! 0! 1! 0! 0! 0! 0! 1! 1! 1! 0! 1! 1! 0! 0! 1! 1! 1! 1! 1! 1! 1! 1! 1! 1! 1! 1! 1! 0! 1! 1! 1! 1! 0! 0! 1! 0! 0! 0! 1! 0! 1! 0! 0! 0!!!!!! 1!!!!!! 2!!!!!! 3!!!!!! 4!!!!!! 5!!!!!! 6!!!!!! 7!!!!!! 8!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! log bloc short bloc short bloc "="8," 2 "="16"! "="4," "="6"! ""="17,"28,"36,"53" "=", 7,11 4,8, " (Ibloc) patter' 1' 2' 3' 4' " " " " " " " " " " " 2" 3" 4" 5" " 1" " " " " " " " " " 1" 3" 4" 5" " " " " " " 0" 1" 2" 3" " 0" 1" 2" 3" " " " " " " (LogMiiBloc)! 1! 2! 3! 4! 0" #" 1" #" 2" #" 3" 0" 2" 3" 6" 4" #" 5" #" 6" #" 7" #" 8" #" " (LogBloc) Potetial)bloc)umber) 1" 2" 3" 4" 5" 6" 7" 8" 0) 2" 3" 6" 8" 9" 11" 16" 17" 1) +" 2) 38" 39" 40" 41" 44" 48" 50" 53" 3) +" " Figure 2: Example of select 1.2 Fidclose Agai, let B be a balaced strig of 2 paretheses. Problem 1. We have to defie a succict data structure that requires o() bits of additioal space to aswer fidclose queries i O(1) time for ay balaced strig of legth Example 3. The followig table presets a balaced paretheses strig. A example query for fidclose would be: fidclose(1) = B = ( ( ( ) ( ) ) ( ) ( ( ( ) ( ) ( ) ) ( ) ) ) If = O( ), we ca precompute the aswers for all 2 paretheses, usig O( log 2 log ) = o() bits i total. Otherwise, we costruct the structures as follows: we divide B ito equal-sized blocs of legth s, where: s = log 2. Defiitio 4. Let b(p) = p s deote the bloc i which parethesis p lies. Moreover, we let µ(p) { fidclose(p) B[p] = ( deote the matchig parethesis of p: µ(p) = fidope(p) else 1 A ew variable is itroduced, because the fidclose operatio is applied o the bit-vectors B ad B as well (defied later) ad their legth is. 3

4 Defiitio 5. Let s call p far, if b(µ(p)) b(p) (the matchig parethesis for p is located i aother bloc) or ear, if b(µ(p)) = b(p). ear Paretheses: A looup table earfidclose[0, 2 s 1][0, s 1] is precomputed such that: { fidclose(patter, i) if patter[i] is ear earfidclose[patter][i] = if patter[i] is far patters, i : 1 i s such that patter[i] = (. Far Paretheses: Defiitio 6. Cosider p as a opeig far parethesis ad let q be the immediate predecessor of p which is also a opeig far parethesis. A opeig parethesis p is called a opeig pioeer if: b(µ(p)) b(µ(q)). A closig pioeer is defied symmetrically. A pioeer is either opeig or closig. ote that the match of a pioeer is ot ecessarily a pioeer itself. The root is always a pioeer. Example 4. Figure 3 shows a example of a pioeer p with s = 5. s z} { q p µ(p) s z} { µ(q) ) ( ( ) ( ( ) ) ( ) ( ) ) ( ( pioeer Figure 3: Example of a pioeer The umber of pioeers is size of: #pioeer = B = O( log ), because there ca be at most oe pair (p, µ(p)) per pair of blocs such that p is i the oe bloc ad µ(p) i the other oe, ad p or µ(p) is a pioeer: Imagie a graph, where odes are represeted by blocs ad edges represet a pioeer ad its match. We ca see that the resultig graph is plaar, because matchig paretheses caot cross. Hece, its size is liear i the umber of blocs, which is O( log ). We costruct a data structure B, which represets a substrig of B, but oly cosistig of pioeers ad their matches. To tell whether a parethesis p is stored i B, the pioeers ad their matches are mared i a bitmap piofam[0, 2 1] ad prepared for O(1) ra- ad select-queries. To eep the space withi o(), we eed the followig theorem, which will be proved i a further sectio. Theorem 1. Sparse Bitmap Theorem A bitmap B of legth cotaiig u 1 s ca be represeted i O(u log u ) + o() bits of space, such that subsequet ra- ad select-queries o B ca be aswered i O(1) time. The same structure is stored recursively for B such that B = O( bitmap piofam. I this stage, all aswers ca be precomputed. Additioally, we eed two more looup tables for the fial algorithm. 4 log 2 ) with its correspodig

5 Defiitio 7. Excess[0, 2 s 1][0, s 1][0, s 1] represets a looup table to fid differeces i excess level, defied as follows. Excess[patter][i][j] = excess(patter, j) excess(patter, i) Defiitio 8. Leftmost [patter][ ][i] = mi {j i : excess(j) excess(i) = } with 0 i < s By ow, all relevat data structures for the fidclose operatio are defied. A bit space aalysis will verify that the structures are succict. Vector B We have already show that there exist #pioeers = B = O( stored i o() bits. log ), which ca be Bitmap piofam Based o the sparse bitmap theorem ad for u = O( log ), piofam requires a bit space of: piofam = O( log log log ) + o() = o(). Vector B B, which cotais all pioeer families of B, requires a bit space of: B = O( log 2 ) = o(). Bitmap piofam For the correspodig pioeer bitmap for B, we set u = O( i: piofam = O( log 2 log(log log2 )) + o() = O( log 2 log 2 ), which results log log ) + o() = o(). Table earfidclose The looup table cotais a etry for each of the 2 s patters, oe etry stores O(s) idices for ear paretheses ad each idex requires log s bits. Therefore, a bit space of O(2 s s log s) = o() is required. Excess ad Leftmost Similar to earfidclose, both tables have etries for 2 s patters, O(s 2 ) rows ad a cell with log s bits. If the three tables are combied, they require: O(2 s s 2 log s) = o(). Because both tables ad earfidclose use the same patters, it is eve possible to precompute a combied looup table. As has bee show for all data structures, each requires a bit space i o() ad thus, they are still succict. I the followig, we cotiue with the defiitio of the algorithm for the operatio fidclose(p). Defiitio 9. Operatio fidclose(p) 1. Based o the looup table, determie whether p is far: (a) p is ear The table earfidclose gives the aswer. (b) p is far, the calculate the umber of members i the pioeer family B up to p by q ra 1 (piofam, p) ad the positio of this parethesis i B by p select 1 (piofam, q), which is a opeig parethesis ad the immediate previous pioeer. Usig the recursive structure for B, we fid that j fidclose(b, q 1) 2 is the match of q i B, which ca be mapped bac to a positio i B by µ(p ) = select 1 (piofam, j + 1). 2. Sice the first far parethesis i each bloc is stored i B, b(p) = b(p ). Via a table looup, the excess level differece betwee p ad p is determied. Let b = p s ad set Excess[B[bs, (b + 1)s 1]][p bs][p bs]. 2 q 1 because ra starts at 1. 5

6 3. The chage betwee µ(p) ad µ(p ) must be ad µ(p) is the leftmost positio i µ(p ) s bloc with this property (same excess differece). Thus, we ca use the Leftmost looup table, where b = µ(p ) s is µ(p ) s bloc (hece, also µ(p) s bloc) by µ(p) = b s + Leftmost [B[b s, (b + 1)s 1]][ ][µ(p ) b s] Example 5. Fially, a example of fidclose is preseted. The required data structures are visible i Figure 4. Separators are idicatig bloc borders for s = 5. Our example query is: fidclose(4). As we ca see from a looup i earfidclose, the for our p idicates a far parethesis pair. ext, the algorithm computes the ra 1 i piofam up to p, which results i q = 2. Based o q, the immediate previous pioeer p is calculated. Moreover, the closig parethesis for p is at j = 2 ad µ(p ) = 6 ca be determied. The excess differece betwee p ad p is: = 1 ( Excess, last patter, i = 1 for idex 4 i patter). At the ed, the Leftmost table is accessed for = 1. I the patter, µ(p ) is at idex 1 ad the table returs a 0. We ca fially calculate µ(p) = = 5. excess { B = ( ( ( ) ( ) ) ( ) ( ( ( ) ( ) ( ) ) ( ) ) ) p µ(p ) piofam = B 0 = ( ( ) ( ( ) ) ) piofam 0 = j p B 00 = ( ) Figure 4: Example data structures of fidclose!! earfidclose! ΔExcess! LeftmostΔ!!!!!!!! i!=!0! i!=!1! i!=!2!! Δ!=!1! Δ!=!2!!! patter! 0! 1! 2! 3! 4! 1! 2! 3! 4! 2! 3! 4! 3! 4! 4! 1! 2! 3! 4! 1! 2! 3! 4! etc.!! )!)!)!)!)!!!!!! 21! 22! 23! 24! 21! 22! 23! 21! 22! 21! 0! 1! 2! 3!! 0! 1! 2!!! )!)!)!)!(!!!!!!!!!!!!!!!!!!!!!!!!! bloc!1!! )!)!(!)!(!!! 3!!!!!!!!!!!!! 0! 0!!!!!!!!! )!(!)!(!(!! 2!!!! 1! 0! 1! 2! 21! 0! 1! 1! 2! 1!!!!!!!!!! bloc!0!! (!(!(!)!(!!! 3!!! 1! 2! 1! 2! 1! 0! 1! 21! 0! 1!!! 2!!!!!!!! Figure 5: Example looup table of fidclose 6

7 Refereces R.F. Geary,. Rahma, R. Rama, ad V. Rama. A simple optimal represetatio for balaced paretheses. I Combiatorial Patter Matchig, pages Spriger, J.I. Muro ad V. Rama. Succict represetatio of balaced paretheses ad static trees. SIAM Joural o Computig, 31(3): ,

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