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 fixed-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 fixed-length entry. • Assume that we store these entries in an array.

6. Dictionary as array of fixed-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 fixed 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 fixed-width array • Pros • Lookup in a hash is faster than lookup in a tree. • Cons • no way to find minor variants (resume vs. r´esum´e) • no prefix search (all terms starting with automat) • need to rehash everything periodically if vocabulary keeps growing

9. Trees • Trees solve the prefix problem (find 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 definition: 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*: find all docs containing any term beginning with mon • Easy with B+-tree dictionary: retrieve all terms t in the range: • mon ≤ t < moo • *mon: find 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 < 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: m*nchen • 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$, ello$h, llo$he, lo$hel, and o$hell to the B-tree where $ is a special symbol • Queries

15. Permuterm → term mapping

16. Permuterm index • For hello, we’ve stored: hello$, ello$h, llo$he, lo$hel, and o$hell • Queries • For X, look up X$ • For X*, look up X*$ • For *X, look up X$* • For *X*, look up X* • For X*Y, look up Y$X* • Example: For hel*o, look up o$hel* • How do we handle X*Y*Z? • It’s 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-efficient 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 different types of inverted indexes • The term-document inverted index for finding documents based on a query consisting of terms • The k-gram index for finding 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 prefix mon . . . • . . . but also many “false positives” like moon. • We must postfilter these terms against query. • Surviving terms are then looked up in the term-document inverted index. • k-gram indexes are fast and space efficient (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 cases:if 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 < K2 < K3 < ... < 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 < 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 > 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“ node‘s parent became too small, but its sibling did not have space to accept one more pointer. So redistribution is performed. Observe that the root node’s 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 openhashing, 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