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VB codes use an adaptive number of bytesdepending on the size of the gap. Bit-level codes adapt thelength of the code on the finer grained bit level.The simplest bit-level code is unary code . The unarycode of is a string of 1s followed by a 0 (see thefirst two columns of Table 5.5 ).Obviously,this is not a very efficient code, but it will come in handyin a moment.How efficient can a code be in principle? Assuming the gaps with are all equallylikely, the optimal encoding uses bits for each . Sosome gaps ( in this case) cannot be encoded withfewer than bits. Our goal is to get as close tothis lower bound as possible.A method that is within a factor of optimalis encoding .codes implement variable-length encoding by splitting therepresentation of agap into a pair of length and offset.Offset is in binary, but with the leading 1removed. For example, for 13 (binary 1101) offset is 101. Length encodes the length of offsetin unary code. For 13,the length of offset is 3 bits, which is 1110 inunary. The code of 13 is therefore 1110101, theconcatenation of length 1110 and offset 101.The right hand column of Table 5.5 gives additionalexamples of codes. A code is decoded by first reading the unary codeup to the 0 that terminates it, for example, the four bits 1110when decoding 1110101. Now we know how long theoffset is: 3 bits. The offset 101 can then be read correctly andthe 1 that was chopped off in encoding is prepended: 101 1101 = 13.The length of offset is bits and the length of length is bits, so the length of the entirecode is bits. codes arealways of odd length and they are within a factor of2 of what we claimed to be the optimal encoding length .We derived this optimumfrom the assumptionthat the gapsbetween and are equiprobable.But this need not be the case. In general, we do not know theprobability distribution over gaps a priori.Figure 5.9:Entropy as a function of for a sample spacewith two outcomes and .The characteristic of a discrete probabilitydistribution thatdetermines its coding properties (including whether a code isoptimal) is its entropy , which isdefined as follows:(4)

It can be shownthat the lower bound for the expected length of acode is if certain conditions hold (see the references). It canfurther be shown that for , encoding is within a factor of 3 of this optimal encoding,approaching 2 for large :(5)

In addition to universality, codes have two other properties that are useful for indexcompression. First, they are prefix free , namely, no code is the prefix of another. This means thatthere is always a unique decoding of a sequence of codes - and we do not need delimiters between them,which would decrease the efficiency of the code. The secondproperty is that codes are parameter free . For many other efficient codes, wehave to fit the parameters of a model (e.g., the binomial distribution) tothe distribution of gaps in the index. This complicates theimplementation of compression and decompression. For instance, theparameters need to be stored and retrieved. And in dynamic indexing, the distribution ofgaps can change, so that the original parameters are no longerappropriate. These problems are avoided with aparameter-free code.How much compression of the inverted index do codesachieve? To answer this question we use Zipf's law, the termdistribution model introduced in Section 5.1.2 .According to Zipf's law, the collection frequency is proportional to theinverse of the rank , that is, there is a constant such that: (6)

Figure 5.10:Stratification of terms forestimating the size of a encoded inverted index.Now we have derived termstatistics that characterize the distribution of terms inthe collection and, by extension, the distribution of gaps inthe postings lists.From these statistics, we cancalculatethe space requirements for an inverted index compressed with encoding. We first stratify the vocabulary into blocks of size. On average, term occurs times perdocument. So the average number of occurrences per documentis for terms in thefirst block, corresponding to a total number of gaps perterm. The average is for terms in thesecond block, corresponding to gaps per term, and for terms in thethird block, corresponding to gaps per term, and so on. (Wetake the lower bound because it simplifies subsequent calculations.As we will see, the final estimate is toopessimistic, even with this assumption.)We will make the somewhat unrealistic assumption that allgaps for a given term have the same sizeas shown in Figure 5.10.Assuming such a uniform distribution of gaps,we then have gaps of size 1 in block 1, gaps of size 2 inblock 2, and so on.Encoding the gaps of size with codes, the number ofbits needed for the postings listof a term in the th block (corresponding to one row inthe figure) is:

Table 5.6 summarizesthe compression techniques covered in this chapter.Theterm incidence matrix (Figure 1.1 , page 1.1 )for Reuters-RCV1 has size bits or 40 GB.The numbers were the collection (3600 MB and 960 MB) are forthe encoding of RCV1 of CD, which uses one byte percharacter, not Unicode. codes achieve great compression ratios - about15% better than variable byte codes for Reuters-RCV1. But theyare expensive to decode. This is because manybit-level operations - shifts and masks - are necessary todecode a sequence of codes as the boundaries betweencodes will usually be somewhere in the middle of a machineword. As a result, query processing is more expensive for codes than for variable byte codes.Whether we choose variable byte or encoding depends on the characteristics of an application, for example,on the relativeweights we give to conserving disk space versus maximizing queryresponse time.The compression ratio for the index inTable 5.6 is about 25%: 400 MB(uncompressed, each posting stored as a 32-bit word) versus 101 MB() and 116 MB (VB). This shows that both and VB codes meetthe objectives we stated in the beginning of the chapter.Index compression substantially improves time and spaceefficiency of indexes by reducing the amount of disk space needed,increasing the amount of information that can be kept inthe cache, and speeding up data transfers from disk to memory. Exercises.Compute variable byte codes for the numbers inTables 5.3 5.5 .Compute variable byte and codes forthepostings list 777, 17743, 294068, 31251336. Use gaps insteadof docIDs where possible. Write binary codes in 8-bit blocks. Consider the postings list with a corresponding list of gaps.Assume that the length of the postings list is stored separately, so the system knows when a postings list is complete. Using variable byte encoding:(i) What is the largest gap you can encode in 1 byte?(ii) What is the largest gap you can encode in 2 bytes?(iii) How many bytes will the above postings list require under this encoding? (Count only space for encoding the sequence of numbers.) A little trick is to notice that a gap cannot be of length 0and that the stuff left to encode after shifting cannot be0. Based on these observations:(i) Suggest a modification to variable byte encoding thatallows you to encode slightly larger gaps in the same amountof space.(ii) What is the largest gap you can encode in 1 byte?(iii) What is the largest gap you can encode in 2 bytes? (iv) How many bytes will the postings list in Exercise 5.3.2 require under this encoding? (Count only space for encoding the sequence of numbers.) From the following sequence of -coded gaps,reconstruct first the gap sequence and then the postingssequence: 1110001110101011111101101111011. codes are relativelyinefficient for large numbers (e.g., 1025 inTable 5.5 ) as they encode the length of theoffset in inefficient unary code. codes differ from codes in that they encode the firstpart of the code (length) in code instead ofunary code. The encoding of offset is thesame. For example, the code of 7 is 10,0,11 (again,we add commas for readability). 10,0 is the codefor length (2 in this case) and the encoding of offset(11) is unchanged. (i) Compute the codes for the othernumbers in Table 5.5 . For what range of numbersis the code shorter than the code?(ii) code beats variable byte code inTable 5.6 because the index contains stop words and thusmany small gaps. Show that variable byte code is morecompact if larger gaps dominate. (iii) Compare thecompression ratios of code and variable byte codefor a distribution of gaps dominated by large gaps.Go through the above calculation of index size andexplicitly state all the approximations that were made toarrive at Equation 11.For a collection of your choosing, determine the numberof documents and terms and the average length of adocument. (i) How large is the inverted index predicted to be byEquation 11? (ii) Implement an indexer thatcreates a -compressed inverted index for thecollection. How large is the actual index? (iii) Implement anindexer that uses variable byte encoding. How large is thevariable byte encoded index?

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