Update Conscious Bitmap Indices

Update Conscious Bitmap Indices Guadalupe Canahuate,Michael Gibas,Hakan Ferhatosmanoglu

The Ohio State University

Department of Computer Science

canahuat,gibas,hakan@https://www.wendangku.net/doc/4011198011.html,

Abstract

Bitmap indices have been widely used in several domains such as data warehousing and scienti?c applications due to their ef?ciency in answering certain query types over large data sets.However,their utilization has been largely limited to read-only data sets or to static snapshots of data due to the cost associated with the update and append of new data. Typically,several bitmaps are associated with each indexed attribute in a table,i.e.one for each attribute value,bin, or range.Each one of these bitmaps needs to be updated to re?ect a new,appended row.Since a given table could be represented by hundreds or even thousands of bitmaps, the insertion of a single record can be prohibitively costly. In order to transfer the fast query response times offered by bitmap indices to dynamic database domains,we propose an update conscious bitmap index that provides a mecha-nism to quickly update bitmaps to re?ect dynamic database changes.For an insert operation only the bitmaps that rep-resent the values being inserted need to be updated.We formalize the insert and delete operations of the proposed technique and provide a cost model for bitmap updates. We compare the update conscious bitmaps to traditional bitmaps in terms of storage space,update performance,and query execution time.

1Introduction

Bitmap indices are widely used in data warehouses and scienti?c applications to ef?ciently query large-scale data repositories.They are successfully implemented in com-mercial Database Management Systems,e.g.,Oracle[1,2], IBM[15],Informix[7,13],Sybase[6,8],and have found many other applications such as visualization of scienti?c data[10,18,19].Bitmap indices can provide very ef?cient performance for point and range queries thanks to fast hard-ware supported bit-wise operations over the bitmaps.

Traditionally,the use of bitmaps has been restricted to read-only or read-mostly(data warehouses)data environ-ments.Even though commercial RDBMS,such as Oracle, support updates to bitmap indexed tables,the general con-census is that updating bitmapped indexes take a lot of re-sources and time and the recommendation is to use nightly batch updates,drop the index,apply the changes,and re-build the bitmaps[5,3,12].In Oracle,“the bitmap index is an extreme case of resource consumption single row DML on tables with bitmap indexes can result in massive amounts of undo and redo being generated,combined with a huge space wastage inside the index that results in lots of un-necessary I/O as the index is both written or read”[11]. The reason is that bitmaps were not originally designed to handle updates ef?ciently.The main goal of a bitmap in-dex is to execute queries fast and ef?ciently for large vol-umes of data.Within a bitmap index,a single bitmap is a bit vector that identi?es the tuples that have attribute-values that match the value,category,or range associated with the bitmap.The entire bitmap index is made up of multiple bitmaps for each indexed attribute.The main reason why updates are so costly is that all the bitmaps,often hundreds of them,need to be updated when a new row is inserted. Insertion is the most common update operation.

As a sample application,consider High Energy Physics (HEP)experiments consisting of accelerating sub-atomic particles to nearly the speed of light and forcing their colli-sion.Sequences of events notable for physicists are stored with all the details.The number of events stored and an-alyzed in one year is on the order of several millions and the number of attributes per each event is above100[17]. Bitmap indices provide ef?cient query support for such data[18].New events are periodically appended to the database as more experiments are performed.Bitmap up-date is done periodically in batches.This means that the new data is not accessible from the index structure until the next scheduled update of the bitmap index is done.In or-der to update the bitmap index,the new data needs to be encoded into the bitmap,compressed,and appended to the existing bitmaps.

For the most commonly used bitmap encoding,where bits are only set for tuples that match the bitmap value,and

in the case where updates were performed in real time,the insertion process would need to access all the bitmaps for the table,add a‘0’for those bitmaps that are not relevant for the value(s)inserted and a‘1’on the relevant bitmaps. Considering this domain,the number of bitmaps per table makes this insertion process prohibitively costly.

However,if it were possible to propagate database changes to only those relevant bitmaps and still maintain consistency between index and database,then the cost of bitmap updates would be greatly reduced.By updating bitmaps ef?ciently,we can expand the domain of applica-tions for which bitmap indices can be useful.We could es-sentially make the same number of changes to the bitmap index structure as would be necessary for an equivalent set of B+-tree indices over the attributes and maintain index-data consistency.

In this paper,we propose a bitmap index design for equality encoded bitmap indices for which new records can be added by updating a single bitmap per indexed attribute. Using compressed bitmaps,we add an extra word as the last word of each bitmap.This extra word is a compressed large run of0’s that pads the number of rows encoded in the bitmap to a?xed number,much bigger than the actual number of rows in the table.When we insert a new row, we set the bit in the corresponding position of the relevant bitmaps,and compress the last words accordingly.Dele-tions are handled by maintaining a compressed existence bitmap(EB)[14].When a record is deleted,the appropriate bit is set to0in the EB.Query execution is performed in the same manner as traditional bitmaps with one additional AND bit operation with the existence bitmap.Updates of existing data are handled by performing a delete operation followed by an insertion.

These changes expand the applicable domain of bitmap indices.For record insertions,the number of changes that need to be made to the index structure are reduced by a fac-tor equal to the average cardinality of the indexed attributes of the dataset.These changes make bitmap indices feasible for some dynamic databases and makes them more attrac-tive for high cardinality data domains.

The rest of this paper is organized as follows.Sec-tion2provides some background on bitmap indices and bitmap compression.Section3presents the update con-scious bitmap approach.Section4provides the cost model for bitmap updates and query execution.Section5shows the experimental results and the performance comparison with traditional bitmap indices.We conclude in Section6.

2Background

Bitmap tables are a special type of bit matrices.Each binary row in the bitmap table represents one tuple in the database.The bitmap columns are produced by quantizing

Attribute1Attribute2

Tuple b1b2b3b1b2b3

t1100001

t2010100

t3100100

t4001001

t5010010

t6001100

Figure 1.Simple bitmap example for a ta-

ble with two attributes and three bins per at-

tribute.

the attributes in the database into categories or bins.Each tuple in the database is then encoded based on which bin each attribute value falls into.

For the simple bitmap encoding(also called equality en-coding)[14],if a value falls into a bin,this bin is marked “1”,otherwise“0”.Since a value can only fall into a sin-gle bin,only a single“1”can exist for each row of each attribute.After binning,the whole database is converted into a huge0-1bitmap,where rows correspond to tuples and columns correspond to bins.Figure1shows an exam-ple of the equality encoded bitmap using a table with two attributes,each partitioned into three bins.The?rst tuple t1falls into the?rst bin in attribute1,and the third bin in attribute2.

Bitmap indices can provide very ef?cient performance for point and range queries thanks to fast bit-wise opera-tions over the bitmaps,which are ef?ciently supported by hardware.

With equality encoded bitmaps a point query is executed by ANDing together the bit vectors corresponding to the values speci?ed in the search key.For example,?nding the data points that correspond to a query where Attribute 1is equal to3and Attribute2is equal to5is only a mat-ter of ANDing the two bitmaps together.Equality Encoded Bitmaps are optimal for point queries[4].Range queries are executed by?rst ORing together all bit vectors speci?ed by each range in the search key and then ANDing the answers together.If the query range for an attribute queried includes more than half of the cardinality then we execute the query by taking the complement of the ORed bitmaps that are not included in the range query.

No matter which bitmap encoding we use,the bitmap in-dex table is a0-1table.This table needs to be compressed to be effective on a large database.General purpose text compression techniques are clearly not suitable for this pur-pose since they signi?cantly reduce the ef?ciency of queries [9,21].Specialized bitmap compression schemes have been proposed to overcome this problem.These schemes are based on run-length encoding,i.e.,they replace runs of0’s or1’s in the columns by a single instance of the symbol and

133bits1,20*0,4*1,78*0,30*1

31-bit groups1,20*0,4*1,6*062*010*0,21*19*1 groups in hex400003C00000000000000000001FFFFF000001FF WAH(hex)400003C080000002001FFFFF000001FF Figure2.An example of a WAH compressed bit vector.

a run count.These methods not only compress the data but also enable fast bitwise logical operations,which translates into faster query processing.

Run length encoding[16]can therefore be used over ev-ery column to compress the data when long runs of“0”or “1”blocks become available.Pure run length encoding is not a good strategy because of its accessing inef?ciency. The two most popular compression techniques for bitmaps are the Byte-aligned Bitmap Code(BBC)[1]and the Word-Aligned Hybrid(W AH)code[22].Unlike traditional run length encoding,these schemes mix run length encoding and direct storage.BBC stores the compressed data in Bytes while WAH stores it in Words.W AH is more CPU ef?cient because it only has two types of words:literal words and ?ll words.In our examples it is the most signi?cant bit that indicates the type of word we are dealing with.Let w de-note the number of bits in a word,the lower(w?1)bits of a literal word contain the bit values from the bitmap.If the word is a?ll,then the second most signi?cant bit is the?ll bit,and the remaining(w?2)bits store the?ll length.W AH imposes the word-alignment requirement on the?lls.This requirement is key to ensure that logical operations only ac-cess words.

Figure2shows a W AH compressed bit vector represent-ing133bits.The?rst line is the original bit vector.The vector starts with one1,followed by twenty0’s,four1’s seventy eight0’s,and thirty1’s.In this example,we as-sume32bit words.Under this assumption,each literal word stores31bits from the bitmap,and each?ll word represents a multiple of31bits.The second line in Figure2shows how the bit vector is divided into31-bit groups,and the third line shows the hexadecimal representation of the groups.

The last line shows the values of W AH words.Since the ?rst and third groups do not contain greater than a multiple of31of a single bit value,they are represented as literal words(a0followed by the actual bit representation of the group).The fourth group is less than31bits and thus is stored as a literal.The second group contains a multiple of 310’s and therefore is represented as a?ll word(a1,fol-lowed by the0?ll value,followed by the number of?lls in the last30bits).The?rst three words are regular words, two literal words and one?ll word.The?ll word80000002 indicates a0-?ll of two-word long(containing62consecu-tive zero bits).The fourth word is the active word,and it stores the last few bits that could not be stored in a regular word.Another word with the value9,not shown,is needed to store the number of useful bits in the active word.Log-ical operations are performed over the compressed bitmaps resulting in another compressed bitmap.

3Update Conscious Bitmaps(UCB)

3.1The General Idea

Let us consider the update operations over a traditional bitmap indexed attribute A of table T.We denote as B= {b1,b2,...,b C}the set of bitmaps over the domain of A and as n the number of tuples in T.The simple uncompressed bitmap encoding would have n bits in each bitmap.To in-sert a new tuple with value corresponding to b i,one would need to add a‘1’at the end of bitmap b i and a‘0’at the end of all other bitmaps.

To delete a record,we use the Existence Bitmap (EB)[14].Originally,the EB was proposed to handle non-existent rows.The bits corresponding to non-existing tuples are set to zero in all bitmaps.Therefore,the EB needed to be used(ANDed together with another bitmap)after a NOT operation to make sure that only existing rows are reported as answers.

In the uncompressed bitmaps,to update a row we need to change two bitmaps.The old value bitmap to unset the corresponding bit and the new value bitmap to set the corre-sponding bit to1.However,given the amount of space they require,it is not feasible to store the bitmaps in an uncom-pressed form.Several compression techniques have been proposed for bitmap indices.While these techniques effec-tively reduce the space required by the bitmaps,they have the disadvantage that there is no longer a direct mapping between the bit position in a bitmap and the position of the row in the table.In this case,for example,updates become even more expensive because we need to scan(decompress) the bitmap in order to locate the corresponding bit in the two bitmaps.This is the reason why we handle updates by performing a delete followed by an insert operation.

We modify the current equality encoded bitmaps in the following ways:

?To each bitmap we add a number of non-set bits,called pad-bits,that would be used for the new tuples.By padding the bitmaps with zeros we can insert a new tu-ple by setting the corresponding bit only in the bitmap that corresponds to the inserted value.

?We maintain a total bit counter,TBC,for the number of non-pad bits currently in the bitmaps.

?We also pad the Existence Bitmap(EB)with non-set bits to indicate non-existing rows.As in the original approach,the EB is ANDed with the resulting bitmap after a NOT operation to make sure that only existing rows are reported as answers.

In the following section we detail the update conscious bitmap index approach.From here on,we refer to the equal-ity encoded WAH-compressed bitmaps as original or tradi-tional bitmaps.

3.2Bitmap Creation

We based our implementation of the Update Conscious Bitmaps on equality encoded bitmaps using Word-Aligned Hybrid(W AH)compression.W AH has been proven to provide faster query execution(2X-20X faster)than Byte-aligned Bitmap Code(BBC)with a smaller compression ra-tio(compressed bitmaps are typically40-60%larger)[20]. WAH is currently considered the start-of-the-art in bitmap compression due to its simplicity and ef?ciency.

Bitmaps are created as usual and compressed using WAH.For the rest of this paper the word size is set to w=32bits.The number of words encoded in each bitmap is T0= n/(w?1) ,where n is the number of rows in the table.We divide by w?1because with W AH com-pression,one bit is reserved to indicate the type of word, whether a literal or?ll word.The compressed size of each bitmap can be smaller than T0words,however,a total of T0words are encoded in each bitmap.We then add a?-nal word,called a pad-word,which is the?ll representation of words of0s.With one pad-word,each bitmap encodes T1=2w?2?1words.The pad-word would be the0-?ll word of T2=T1?T0words.In case more bits are needed, one could add one or more pad-words to encode T1more words for each pad-word.For the rest of paper we assume that T1words are enough to encode all the rows in the table, i.e.at any given time we have less T1?(w?1)rows in the indexed table(including the rows that would be inserted in the future).

Figure3shows an example of a compressed bit vector for both the original W AH compressed bitmap(3rd line) and the update conscious bitmap(4th line).It is worth not-ing that we changed the encoding of the active word in the W AH implementation so incoming bits of zeros do not re-quire a bitmap update.In the original implementation the active word would be represented as000001FF,in which case,adding a zero would update this word to be000003FE. However,by encoding the rows from the most signi?cant bits,the active word7FC00000in the UCB would remain unchanged when we add a zero.The last word in the update conscious bitmap(BFFFFFFA),represents the pad-word, i.e.a0-?ll word for a run of33,285,996,358zeros.The total bit counter(TBC),not shown in this example,is set to 133.

3.3Insertions

Let us consider the insertion of a new row where attribute A is appended with the value encoded by bitmap b i.In this case,only b i needs to be accessed to perform the insertion. The basic idea is to use the TBC and the number of words in the pad word to decide how many zeros we need to put before the new1in the current bitmap.We also update the number of words encoded by the pad word so the total num-ber of words in the bitmap remains the same.We change the EB to include the new row and increment the TBC by1.

We have two cases when inserting a new row for attribute A depending on whether the inserted value is encoded by an existing bitmap.In the?rst case,when the value is already encoded by a bitmap b i,we compute the number of words encoded by the non-pad words of b i,T0=T1?T2.Note that even when T0is initially the same for all the bitmaps, as we insert new rows,T0remains the same for all the un-changed bitmaps.If all the bitmaps were updated on an insertion then the number of words encoded by the bitmaps would be T 0= TBC/(w?1) and T0would always be equal to T 0.In the case when T0=T 0we only need to change the bit corresponding to position(TBC mod(w?1)) in the active word.In the case when T0

As an illustration,consider the following example.In Figure4we present the last7compressed words for the bitmap index over the attribute sex,which only have two possible values M or F,for a table with200rows.Both bitmaps have6regular words,1active word and1pad-word (BFFFFFF8).The TBC is200,and the number of words needed is7(T 0= 200/31 ).Now consider the insertion of a new row where sex=M.TBC is updated to201.We only access the bitmap corresponding sex=M and update the active word(W1)to be3FCF0000(since the number of words needed remains to be7,i.e.T0=T 0).After16more insertions of rows where sex=M,TBC=217,and W1= 3FCFFFFF.When a new insertion comes where sex=M, TBC=218and the words needed are now8(T0

133bits1,20*0,4*1,78*0,30*1

31-bit groups1,20*0,4*1,6*062*010*0,21*19*1

Orig(hex)400003C080000002001FFFFF000001FF

UCB(hex)400003C080000002001FFFFF7FC00000BFFFFFFA

Literal Word Fill Word Literal Word Last Word Pad Word Figure3.WAH compressed bit vector for original bitmaps and Update Conscious Bitmaps(UCB).

Symbol Description

W0The pad word in a UCB

W1The last non-pad word in a UCB

W2The previous to the last non-pad word in UCB

W i.value or just W i The literal value of the i th word

W i.isFill Indicates whether the i th word is a?ll or not

W i.?ll The?ll value of the i th word

W i.nWords The number of words encoded by the i th?ll word

w The word size,we set it to32

TBC The global total bit counter

word[i]The literal word with only the i th bit set.

T1The total number of words encoded in each bitmap,including the pad word,in our examples T1=2w?1?1 T2The number of words encoded by the pad word,T2=W0.nW ords

T0The number of words encoded by the non-pad words in this bitmap,T0=T1?T2

T 0The number of literal words needed to encode TBC rows, TBC/(w?1)

Table1.Notation list and their meaning.

so we need to add one word and subtract1from the total words in W0.Since(218mod31=1),the new word is 40000000and W0is updated to be BFFFFFF7.

Now consider the insertion of a new row where sex=F. TBC=219and the number of words needed is8.Since there are only7words encoded in the bitmap(T0

For the second and special case of insertions,when the value being inserted is not encoded by an already existing bitmap,i.e.the new row is the?rst row with such a value, we do not need to access any of the bitmaps for attribute A, we just add a new bitmap with at most two non-pad words. The?rst word corresponds to the?ll words for(almost)all the previous rows and the second one is the active word with a1in the position corresponding to(TBC mod(w?1)). We also add as many pad-words as needed,e.g.1in our experiments.

3.4Deletions

We propose to handle deletes by unsetting the corre-sponding bit in the EB and leaving the rest of the bitmaps unchanged.Since the bit corresponding to the deleted row remains set in some bitmaps,when we execute the query we need to AND the EB with the?nal result to ensure that no deleted row is retrieved as an answer.4Bitmap Index Cost Model

The cost to modify a set of indices associated with a database update is the cost to propagate all database changes to the set of indices such that all possible operations over the indices exactly re?ect the state of current database.This section describes the bitmap modi?cations that need to oc-cur so that the indices are up to date after a record inser-tion for both traditional bitmaps and our update conscious bitmaps.

In our?nal update cost estimates,we assume that reads and writes dominate the cost of updates compared to the bit comparisons and bit operations that need to be performed. We also assume that I/O operations have a consistent cost, but they may in fact differ based on disk locality.The exper-imental results will show to what extent these assumptions are valid.

4.1Insertions

4.1.1Traditional Bitmaps

For traditional bitmaps,when a new row is inserted every bitmap for each indexed attribute needs to be updated in order to maintain consistency with the database.

Assuming the bitmap index is not in memory and bitmap B i is a pointer to a binary?le that contains the compressed bitmap,the insertion routine involves reading at most the

Bitmap

Value W6W5W4W3W2W1W0 Initially M7C3B773F6D75B7FF38F427F87E95B5FA6FFFE3AF3FCE0000BFFFFFF8 (200rows)F03C488C0128A4800470BD807016A4A0510001C5040300000BFFFFFF8 Insert1row M7C3B773F6D75B7FF38F427F87E95B5FA6FFFE3AF3FCF0000BFFFFFF8 (sex=M)F03C488C0128A4800470BD807016A4A0510001C5040300000BFFFFFF8 Insert16rows M7C3B773F6D75B7FF38F427F87E95B5FA6FFFE3AF3FCFFFFF BFFFFFF8 (sex=M)F03C488C0128A4800470BD807016A4A0510001C5040300000BFFFFFF8 Insert1row M6D75B7FF38F427F87E95B5FA6FFFE3AF3FCFFFFF*40000000*BFFFFFF7 (sex=M)F03C488C0128A4800470BD807016A4A0510001C5040300000BFFFFFF8 Insert1row M6D75B7FF38F427F87E95B5FA6FFFE3AF3FCFFFFF40000000BFFFFFF7 (sex=F)F128A4800470BD807016A4A0510001C5040300000*20000000*BFFFFFF7 Figure4.Insertion operations for Update Conscious Bitmap Index over Attribute sex.

UpdateBitmap(Bitmap Index B,Total Bit Counter TBC) 1:read W2,W1,and W0from B

2:if T0=T 0

3:W1=W1|word[TBC mod(w?1)]

4:if(TBC mod(w?1)=0)and(W1is all1s)

5:if W2.isFill and W2.?ll=1

6:W2.nWords++

7:delete W1

8:else if W2is all1s

9:W2.isFill=true,W2.?ll=1,W2.nWords=2

10:delete W1

11:else if T0

12:W0.nWords=(T1?T 0)

13:nWords=T 0?T0

14:if nWords=1

15:add word[TBC mod(w?1)]before W0

16:else if nWords=2

17:add words0and word[TBC mod(w?1)]

before W0

18:else

19:W.isFill=true,W.?ll=0,W.nWords=(nWords-1) 20:add words W and word[TBC mod(w?1)]

before W0

Bitmap Indices.

last two words of the bitmap?le,making the appropriate modi?cations,and writing the updated?le to disk.All the bitmaps,including the existence bitmap(EB),need to be accessed and updated.

The cost to perform this update is:

t ins=(tbm+1)?(rt+ut+wt)

where t ins is the estimated time to update the complete bitmap index upon a record insert,tbm is the total num-ber of bitmaps excluding the EB,rt is the time to read the last words of the bitmap?le,ut is the time to update the last words of the bitmap to be consistent with the record insertion,and wt is the time to write the bitmap?le.

4.1.2Update Conscious Bitmap

For update conscious bitmaps,when a new row is inserted, only one bitmap for each indexed attribute needs to be up-dated to maintain consistency with the database.

With the same assumptions as with the traditional bitmaps,insertions using UCB involve reading at most the last three words of the bitmap?le,making the appropri-ate modi?cations,and writing the updated?le to disk.One bitmap per indexed attribute,as well as the existence bitmap (EB),needs to be accessed and updated.

The estimated cost associated with a record insert is:

t ins=(atts+1)?(rt+ut+wt)

where atts is the number of attributes.

Assuming that the read,write,and bit manipulation op-erations require similar times for the traditional and update conscious bitmaps,the speedup for the insertion of a new row is:

speedup=(tbm+1)/(atts+1)

4.2Deletions

4.2.1Traditional Bitmaps

The brute force schema for handling row deletes in tradi-tional bitmaps would be to remove the bit corresponding to the deleted row in all bitmaps.However,this is clearly very inef?cient since it involves decompressing all the bitmaps and altering the(w-1)-bit groups created to compress the bitmaps.A more clever way would be to use the EB to?ag the deleted row as inexisting.There would then be two al-ternatives.The?rst one is to keep the rest of the bitmaps unchanged in which case we would need to exclude the deleted rows during query execution by AND-ing the?nal

bitmap with the EB.The second one is to change the cor-responding bits to0in the relevant bitmaps in which case query execution is not affected since deleted rows are0s in all bitmaps.In order to maintain the query execution performance of traditional bitmaps,we chose the second al-ternative for traditional bitmaps.When a row is deleted, the bitmaps corresponding to the values of each indexed at-tribute in the row need to be updated,i.e.the set bit needs to be unset,and then,the bit corresponding to the deleted row in the existence bitmap needs to be unset as well.Now, recall that since the bitmaps are compressed,the bitmaps need to be scanned(decompressed)and the change in the bit value could mean increasing or decreasing the size of the bitmap by at most three words.

The estimated cost to perform a delete is:

t del=atts?(rt bc+dt+ut+ct+wt bc)

where t del is the estimated time required to update the entire index upon a record deletion,rt bc is the time it takes to read the bitmap?le,dt is the time required to decompress/scan the bitmaps to locate the relevant bit,ct is the time to update the?le,and wt bc is the time needed to write the?le to disk. These times will vary depending on the compressed length of the bitmap.

Note that one could choose the?rst alternative to handle deletes with traditional bitmaps,in which case the deletion cost and the query execution cost would be the same for both,traditional and update-conscious bitmaps.

4.2.2Update Conscious Bitmaps

For the update conscious bitmaps,we decided to only mod-ify the existence bitmap and handle the extra set bits during query execution by ANDing the?nal results with the exis-tence bitmap in case there are deletions.Assuming the exis-tence bitmap is not in memory,the operation requires read-ing the existence bitmap,scanning it,changing the bit asso-ciated with the deleted row to‘0’,compressing the bitmap, and writing it to disk.The estimated cost is:

t del=rt bc+dt+ut+ct+wt bc

Assuming similar times associated with the compressed ex-istence bitmap and the average bitmap size,the estimated speedup for deleting a row is:

speedup=atts

4.3Query Execution

For both approaches,when a bitmap is negated during query execution the?nal answer needs to be ANDed with the EB in order to prune nonexistent rows.However,for update conscious bitmaps,this additional operation needs to be performed for every query if there are deletions.

The following analysis refers to queries where no bitmap is negated and there are deletions,i.e.the update conscious bitmaps require one bit-wise operation more than the tra-ditional bitmaps.Note that in other cases,the UCB will exhibit the same query execution performance as the tradi-tional bitmap.For simplicity,and without loss of general-ity,we will express the queries in terms of the number of bitmaps that need to be accessed to answer the query.

For point queries,the number of bitmaps that need to be accessed is equal to the number of attributes queried.

For range queries,the number of bitmaps that need to be accessed depends on the range of each attribute queried. However,it is bound by half of the bitmaps for each at-tribute.

4.3.1Traditional Bitmaps

For traditional bitmaps,the time required to execute a query over b bitmaps is:

t ex=(b?1)?t bo

where t ex is the estimated time to execute a query,b is the number of bitmaps that need to be accessed to answer the query,and t bo is the time to perform a bitwise operation. The time t bo will vary depending on the size of the bitmaps.

4.3.2Update Conscious Bitmaps

For update conscious bitmaps,query processing is identical as before with the addition that the?nal answer obtained from the bitwise operations is ANDed with the existence bitmap.The time to perform a query is:

t ex=b?t bo

And the speedup(slowdown)is:

speedup=(b?1)/b

5Experimental Results

5.1Experimental Setup

We performed experiments in order to measure the im-pact on bitmap update time,query execution time,and in-dex storage for update conscious bitmaps compared to tra-ditional bitmaps.

Four data sets were used in the experiments.Each data set contains2million rows and attributes of varying car-dinalities.The HEP data set is a set of real bitmap data generated from High Energy Physics experiments with12

Figure5.Index Update Time vs.Number of

Rows Inserted.

attributes.Each attribute has between2and12bins,for a total of122bitmaps.The UR data set is a synthetically gen-erated data set of uniformly distributed random data.The Z1and Z2data sets are synthetically generated data sets following the zipf distribution with parameters1and2re-spectively.

Experiments are performed using a Windows XP Pro-fessional Operating System on a Pentium42.26GHz pro-cessor with512MB of RAM.A Java implementation of traditional bitmaps and update conscious bitmaps was de-veloped to measure the relative differences between tech-niques and to compare to predicted results.Each bitmap is stored in a separate binary?le.The bitmap update time in-cludes time to read the relevant bitmap?les,update the last word(s)of the?le,and write the updated?le back to disk. Query execution time is the time to run point queries using the appropriate bitmap query execution technique.For HEP data,the query points are randomly selected from the last 100,000rows which were not used during bitmap creation. For synthetic data,the point queries correspond to randomly generated data points following the same distribution as the data set.

5.2Update Time

5.2.1Insertion Time versus Rows Appended

Figure5shows the index update time required as the num-ber of records appended is varied for the synthetic data sets. Each append is done individually,not in a batch fashion. In this experiment we append one attribute with cardinal-ity of50.For both the original bitmap representation and the UCBs,the update of the Existence Bitmap is included in this experiment.Both techniques exhibit linear perfor-mance with respect to the number of rows appended.The data sets,which vary in compressibility,have little effect on the append times.The ratio between the update

times

Figure6.Index Update Time vs.Number of

Rows

Inserted.

Figure7.Index Update Time vs.Cardinality

of the Attribute Indexed.

for the original technique and the UCBs is close to the pre-dicted https://www.wendangku.net/doc/4011198011.html,ing our experimental setup,on the average, one insertion is done in7ms for UCB as opposed to212ms for traditional bitmaps.

Figure6shows index update time for the real HEP data set.This graph shows the linear performance of insertions and includes the time to append all12attributes of an in-serted record.Appends to the real data show the same pat-tern of append time as the synthetic data.To insert one new row,the average time is50ms using UCB as opposed to470 ms using traditional bitmaps.The speedup is9.46(123/13).

5.2.2Insertion Time versus Attribute Cardinality Figure7shows the record append time for the synthetic data as attribute cardinality varies.For each stated cardinality, 1,000rows of a single attribute are appended.The UCBs shows constant update time with respect to attribute cardi-nality while the original bitmaps exhibit update times that linearly increase with respect to attribute cardinality.

Figure8shows the relationship between append time and attribute cardinality for the real HEP data set.It shows

Figure 8.Index Update Time vs.Cardinality of the Attribute

Indexed.

Figure 9.Index Update Time vs.Number of Attributes Indexed.

similar performance to the synthetic data sets.5.2.3

Insertion Time versus Number of Attributes

Figure 9shows how insertion time is affected by the num-ber of attributes indexed.Insertion time measures the time needed to append 1,000rows with a varying number of at-tributes.The attributes involved are equally taken from the UR,Z1,and Z2data sets and all have cardinality 10.As ex-pected,the performance for both the original and UCBs is linear with respect to the number of indexed attributes and the ratio between original bitmap and UCB append time is as predicted.The real HEP data performance is similar and is not shown here due to space considerations.

5.3Query Execution Time

Figure 10shows how the average query execution time compares using original bitmaps and UCB.Times are pro-vided for performing 100point queries using the indicated bitmap type as the number of attributes involved in

the

Figure 10.Point Query Execution Time vs.Number of Attributes

Queried.

Figure 11.Point Query Execution Time for best case EB and worst case EB.

query vary.Times are provided for both the original bitmaps and UCB with and without an extra AND operation with a compressed existence bitmap.UCB and original bitmap query execution time is nearly identical when both use ex-istence bitmap and when both do not.

5.4Worst Case Existence Bitmap

As deletes occur over time,the existence bitmap be-comes less compressible (the 0’s associated with deleted records interrupt runs of 1’s).Since query execution time over bitmaps is dependent on the compressed length of those bitmaps,queries that involve the existence bitmap can become more time consuming as the existence bitmap in-creases in size.Figure 11demonstrates this effect and the magnitude of the effect.We compare the results using a best case (all 1’s existence bitmap)to a worst case existence bitmap (alternating 1’s and 0’s,no WAH-compression pos-sible).The graph displays results for the Z2dataset using

cardinality10attributes.The results show that a worst case existence bitmap adds less than20ms in query execution time for this experiment.Other data sets provide similar results.

5.5Compressed Size

Considering the case when we only add one pad-word, there is no overhead in terms of storage space when the last word of the compressed bitmap is zero,since this word would be subsumed by the pad-word.If the last word of the compressed bitmap has a one,then we are increasing the compressed size by1word.In general,if we add p pad-words,the overhead would be between|B|?(p?1)?w and|B|?p?w bits,where|B|is the number of bitmaps for the given table.For example,if we add one32-bit pad-word to each bitmap and the number of bitmaps for the ta-ble is1,000bitmaps,in the worse case we are increasing the size of the bitmap indices by4KB.This is clearly a small overhead when compared with the overall bene?ts achieved using this technique.

6Conclusion

In this paper,we have introduced an update conscious bitmap index structure.In order to insert a record,only those bitmaps associated with attribute-value pairs that match the new tuple need to be updated.This means that for each attribute appends are performed in constant time regardless of the attribute cardinality.This contrasts to traditional bitmaps which would require all bitmaps to be updated in order to maintain the index up-to-date.The speedup of bitmap index insertion using an update con-scious bitmap index is on the order of the average indexed cardinality.In addition,query performance using UCB is as ef?cient as traditional bitmaps when the updates are in the form of appends.The storage overhead is minimal since the compressed size of each bitmap is increased by at most one word.Even in the case when there have been so many deletes and modi?cations of existing records that the Exis-tence Bitmap becomes incompressible,there is only a mi-nor impact on query execution time.Furthermore,there is nothing in the update conscious bitmap architecture that prevents periodic rebuilds of the bitmap index to mitigate this effect.

Update conscious bitmaps are particularly a good choice for domains where new data is added to the data set at a rate which allows for updates to take place,and it is desirable to keep the index up-to-date in real time.This approach ex-pands the applicable realm of bitmap indices to include a wide range of dynamic data domains,such as scienti?c ap-plications where data is appended as experiments are con-ducted.To the best of our knowledge,this is the?rst work that addresses the update of bitmap indices and proposes a solution to handle appends of new data ef?ciently. References

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