Dictionaries and tolerant retrieval

1
Dictionaries and tolerant retrieval
2
Type/token distinction

• Token An instance of a word or term occurring
  in a document.
• Type An equivalence class of tokens.
• In June, the dog likes to chase the cat in the
  barn.
• How many tokens? How many types?
• 12 tokens, 9 types

3
Inverted index

• For each term t, we store a list of all documents
  that contain t.

4
Dictionaries

• The dictionary is the data structure for storing
  the term vocabulary.
• Term vocabulary the data
• Dictionary the data structure for storing the
  term vocabulary

5
Dictionary as array of ?xed-width entries

• For each term, we need to store a couple of
  items
• document frequency
• pointer to postings list
• . . .
• Assume for the time being that we can store this
  information in a ?xed-length entry.
• Assume that we store these entries in an array.

6
Dictionary as array of ?xed-width entries

• How do we look up an element in this array at
  query time?

7
Data structures for looking up term

• Two main classes of data structures hashes and
  trees
• Some IR systems use hashes, some use trees.
• Criteria for when to use hashes vs. trees
• Is there a ?xed number of terms or will it keep
  growing?
• Do we support prefix search?

8
Hashes

• Each vocabulary term is hashed into an integer.
• Try to avoid collisions
• At query time, do the following hash query term,
  resolve collisions, locate entry in ?xed-width
  array
• Pros
• Lookup in a hash is faster than lookup in a tree.
• Cons
• no way to ?nd minor variants (resume vs.
  resume)
• no pre?x search (all terms starting with automat)
• need to rehash everything periodically if
  vocabulary keeps growing

9
Trees

• Trees solve the pre?x problem (?nd all terms
  starting with automat).
• Simplest tree binary search tree
• Search is slightly slower than in hashes
• O(logM), where M is the size of the vocabulary.
• O(logM) only holds for balanced trees!
• Rebalancing binary trees is expensive.
• B-trees mitigate the rebalancing problem.
• B means Balance
• B-tree de?nition every internal node has a
  number of children in the interval a, b where
  a, b are appropriate positive integers, e.g., 2,
  4.
• Note that we need a standard ordering for
  characters in order to be able to use trees.

10
Binary search tree
11
Wildcard queries

• mon ?nd all docs containing any term beginning
  with mon
• Easy with B-tree dictionary retrieve all terms
  t in the range
• mon t lt moo
• mon ?nd all docs containing any term ending
  with mon
• Maintain an additional tree for terms backwards
• Then retrieve all terms t in the range nom t lt
  non

12
Query Processing

• At this point, we have an enumeration of all
  terms in the dictionary that match the wildcard
  query.
• We still have to look up the postings for each
  enumerated term.
• E.g., consider the query gen and universit
• This may result in the execution of many Boolean
  and queries.

13
How to handle in the middle of a term

• Example mnchen
• We could look up m and nchen in the B-tree and
  intersect the two term sets.
• Expensive!
• Alternative permuterm index
• Basic idea Rotate every wildcard query, so that
  the occurs at the end.

14
Permuterm index

• For term hello add hello, elloh, llohe,
  lohel, and ohell to the B-tree where is a
  special symbol
• Queries

15
Permuterm ? term mapping
16
Permuterm index

• For hello, weve stored hello, elloh, llohe,
  lohel, and ohell
• Queries
• For X, look up X
• For X, look up X
• For X, look up X
• For X, look up X
• For XY, look up YX
• Example For helo, look up ohel
• How do we handle XYZ?
• Its really a tree and should be called permuterm
  tree.
• But permuterm index is more common name.

17
Processing a lookup in the permuterm index

• Rotate query wildcard to the right
• Use B-tree lookup as before
• Problem Permuterm quadruples the size of the
  dictionary compared to a regular B-tree.
  (empirical number)

18
k-gram indexes

• More space-e?cient than permuterm index
• Enumerate all character k-grams (sequence of k
  characters) occurring in a term
• 2-grams are called bigrams.
• Example from April is the cruelest month we
  get the bigrams
• a ap pr ri il l i is s t th he e c cr ru
  ue el le es st t m mo on nt h
• is a special word boundary symbol.
• Maintain an inverted index from bigrams to the
  terms that contain the bigram

19
Bigram indexes

• Note that we now have two di?erent types of
  inverted indexes
• The term-document inverted index for ?nding
  documents based on a query consisting of terms
• The k-gram index for ?nding terms based on a
  query consisting of k-grams

20
Processing wildcarded terms in a bigram index

• Query mon can now be run as
• m and mo and on
• Gets us all terms with the pre?x mon . . .
• . . . but also many false positives like moon.
• We must post?lter these terms against query.
• Surviving terms are then looked up in the
  term-document inverted index.
• k-gram indexes are fast and space e?cient
  (compared to permuterm indexes).

21
Processing wildcard queries in the term-document
index

• Very expensive
• Does Google allow wildcard queries?
• Why?
• Users hate to type.

22
B -Tree Index Files

• A B -Tree is a rooted tree satisfying the
  following properties
• All paths from root to leaf are of the same
  length
• Each node that is not a root or a leaf has
  between ?n/ 2? and n children
• A leaf node has between ?(n-1)/2? and n-1 values
• Special casesif the root is not a leaf it has at
  least 2 children if the root is a leaf (that is
  there are no other nodes in the tree) it can have
  between 0 and(n-1) values.

23
B -Tree Node Structure

• Typical node
• Ki are the search-key values
• Pi are pointers to children (for nonleaf nodes)
  or pointers to records or buckets of records (for
  leaf nodes).
• The search-keys in a node are ordered
  K1 lt K2 lt K3 lt ... lt Kn

24
Leaf Nodes in B -Trees

• Properties of a leaf node
• For i 1, 2, . . . , n i points to a file record
  with search-key value Ki
• If Li , Lj are leaf nodes and i lt j , Li s
  search-key values are less than Lj s search-key
  values
• Pn points to next leaf node in search-key order

25
Non-Leaf Nodes in B -Trees

• Non leaf nodes form a multi-level sparse index on
  the leaf nodes. For a non-leaf node with m
  pointers
• All the search-keys in the subtree to which P1
  points are less than K1
• For 2 ? i ? n-1 all the search-key in the subtree
  to which Pi points have values greater than or
  equal to Ki-1 and less than Ki
• All the search-keys in the subtree to which Pm
  points are greater than or equal to Km-1

26
Example of a B-tree
27
Observations about B -Trees

• Insertions and deletions to the main file can be
  handled efficiently, as the index can be
  restructured in logarithmic time (as we shall
  see).
• Logm/2N

28
Queries on B-Trees

• Find all records with a search-key value of k.
• Start with the root node
• Examine the node for the smallest search-key
  value gt k .
• If such a value exists, assume it is Ki . Then
  follow Pi to the child node
• Otherwise k ?Km , where there are m pointers in
  the node. Then follow Pm to the child node.
• If the node reached by following the pointer
  above is not a leaf node, repeat the above
  procedure on the node, and follow the
  corresponding pointer.
• Eventually reach a leaf node. If key Ki k ,
  follow pointer Pi to the desired record or
  bucket. Else no record with search-key value k
  exists.

29
Queries on B-Trees (Cont.)

• In processing a query, a path is traversed in the
  tree from the root to some leaf node.
• If there are K search-key values in the file, the
  path is no longer than ?log ?n/ 2? (K )?.
• A node is generally the same size as a disk
  block, typically 4 kilobytes, and n is typically
  around 100 (40 bytes per index entry).
• With 1 million search key values and n 100, at
  most log50 (1, 000, 000) 4 nodes are accessed
  in a lookup.
• Contrast this with a balanced binary tree with 1
  million search key values --- around 20 nodes are
  accessed in a lookup

30
Updates on B-Trees Insertion

• Find the leaf node in which the search-key value
  would appear
• If the search-key value is already there in the
  leaf node, record is added to file and if
  necessary pointer is inserted into bucket.
• If the search-key value is not there, then add
  the record to the main file and create bucket if
  necessary. Then
• if there is room in the leaf node, insert
  (search-key value, record/bucket pointer) pair
  into leaf node at appropriate position.
• if there is no room in the leaf node, split it
  and insert (search-key value, record/bucket
  pointer) pair as discussed in the next slide.

31
Updates on B-Trees Insertion (Cont.)

• Splitting a node
• take the n (search-key value, pointer) pairs
  (including the one being inserted) in sorted
  order. Place the first ?n/ 2? in the original
  node, and the rest in a new node.
• let the new node be p, and let k be the least key
  value in p. Insert (k, p) in the parent of the
  node being split. If the parent is full, split it
  and propagate the split further up.
• The splitting of nodes proceeds upwards till a
  node that is not full is found. In the worst case
  the root node may be split increasing the height
  of the tree by 1.

32
Updates on B-Trees Insertion (Cont.)
33
Updates on B-Trees Deletion

• Find the record to be deleted, and remove it from
  the main file and from the bucket (if present)
• Remove (search-key value, pointer) from the leaf
  node if there is no bucket or if the bucket has
  become empty
• If the node has too few entries due to the
  removal, and the entries in the node and a
  sibling fit into a single node, then
• Insert all the search-key values in the two nodes
  into a single node (the one on the left), and
  delete the other node.
• Delete the pair (Ki - 1 , Pi ), where P i is the
  pointer to the deleted node, from its parent,
  recursively using the above procedure.

34
Updates on B-Trees Deletion

• Otherwise, if the node has too few entries due to
  the removal, and the entries in the node and a
  sibling can not fit into a single node, then
• Redistribute the pointers between the node and a
  sibling such that both have more than the minimum
  number of entries.
• Update the corresponding search-key value in the
  parent of the node.
• The node deletions may cascade upwards till a
  node which has ?n/ 2? or more pointers is found.
  If the root node has only one pointer after
  deletion, it is deleted and the sole child
  becomes the root.

35
Examples of B-Tree Deletion

• The removal of the leaf node containing
  Downtown'' did not result in its parent having
  too little pointers. So the cascaded deletions
  stopped with the deleted leaf node's parent.

36
Examples of B-Tree Deletion (Cont.)

• The deleted Perryridge nodes parent became too
  small, but its sibling did not have space to
  accept one more pointer. So redistribution is
  performed. Observe that the root nodes
  search-key value changes as a result

37
B-Tree Index Files

• Similar to B-Tree, but B-Tree allows search-key
  values to appear only once eliminates redundant
  storage of search keys.
• Search keys in nonleaf nodes appear nowhere else
  in the B-Tree an additional pointer field for
  each search key in a nonleaf node must be
  included.
• Generalized B-Tree leaf node
• Nonleaf node - pointers Bj are the bucket or file
  record pointers.

38
B-Tree Index Files(Cont.)

• Advantages of B-Tree indices
• May use less tree nodes than a corresponding
  B-Tree.
• Sometimes possible to find search-key value
  before reaching leaf node.
• Disadvantages of B-Tree indices
• Only small fraction of all search-key values are
  found early
• Non-leaf nodes are larger, so fan-out is reduced.
  Thus B-Trees typically have greater depth than
  corresponding B-Tree
• Insertion and deletion more complicated than in
  B-Trees
• Implementation is harder than B-Trees.
• Typically, advantages of B-Trees do not outweigh
  disadvantages.

39
Static Hashing

• A bucket is a unit of storage containing one or
  more records (a bucket is typically a disk
  block). In a hash file organization we obtain the
  bucket of a record directly from its search-key
  value using a hash function.
• Hash function h is a function from the set of all
  search-key values K to the set of all bucket
  addresses B.
• Hash function is used to locate records for
  access, insertion as well as deletion.
• Records with different search-key values may be
  mapped to the same bucket thus entire bucket has
  to be searched sequentially to locate a record.

40
Hash Functions

• Worst hash function maps all search-key values to
  the same bucket this makes access time
  proportional to the number of search-key values
  in the file.
• An ideal hash function is uniform, i.e. each
  bucket is assigned the same number of search-key
  values from the set of all possible values.
• Ideal hash function is random, so each bucket
  will have the same number of records assigned to
  it irrespective of the actual distribution of
  search-key values in the file.

41
Handling of Bucket Overflows

• Bucket overflow can occur because of
• Insufficient buckets
• Skew in distribution of records. This can occur
  due to two reasons
• multiple records have same search-key value
• chosen hash function produces non-uniform
  distribution of key values
• Although the probability of bucket overflow can
  be reduced, it cannot be eliminated it is
  handled by using overflow buckets.
• Overflow chaining -- the overflow buckets of a
  given bucket are chained together in a linked
  list
• Above scheme is called closed hashing. An
  alternative, called open hashing, is not suitable
  for database applications.

42
Hash Indices

• Hashing can be used not only for file
  organization, but also for index-structure
  creation. A hash index organizes the search keys,
  with their associated record pointers, into a
  hash file structure.
• Hash indices are always secondary indices --- if
  the file itself is organized using hashing, a
  separate primary hash index on it using the same
  search-key is unnecessary. However, we use the
  term hash index to refer to both secondary index
  structures and hash organized files.

43
Example of Hash Index
44
Deficiencies of Static Hashing

• In static hashing, function h maps search-key
  values to a fixed set B of bucket addresses.
• Databases grow with time. If initial number of
  buckets is too small, performance will degrade
  due to too much overflows.
• If file size at some point in the future is
  anticipated and number of buckets allocated
  accordingly, significant amount of space will be
  wasted initially.
• If database shrinks, again space will be wasted.
• One option is periodic re-organization of the
  file with a new hash function, but it is very
  expensive.
• These problems can be avoided by using techniques
  that allow the number of buckets to be modified
  dynamically.

45
Comparison of Ordered Indexing and Hashing

• Issues to consider
• Cost of periodic re-organization
• Relative frequency of insertions and deletions
• Is it desirable to optimize average access time
  at the expense of worst-case access time?
• Expected type of queries
• Hashing is generally better at retrieving records
  having a specified value of the key.
• If range queries are common, ordered indices are
  to be preferred