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The Wavelet Trie : Maintaining an Indexed Sequence of Strings in Compressed Space

The Wavelet Trie : Maintaining an Indexed Sequence of Strings in Compressed Space. Roberto Grossi Giuseppe Ottaviano * Università di Pisa * Part of the work done while at Microsoft Research Cambridge. Disclaimer. THIS HAS NOTHING TO DO WITH WAVELETS!. Indexed S tring S equences.

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The Wavelet Trie : Maintaining an Indexed Sequence of Strings in Compressed Space

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  1. The Wavelet Trie: Maintaining anIndexed Sequence of Strings in Compressed Space Roberto GrossiGiuseppe Ottaviano*Università di Pisa* Part of the work done while at Microsoft Research Cambridge

  2. Disclaimer THIS HAS NOTHING TO DO WITH WAVELETS!

  3. Indexed String Sequences (foo, bar, foobar, foo, bar, bar, foo) 0 1 2 3 4 5 6 • Queries • Access(i): access the i-th element • Access(2) = foobar • Rank(s, pos): count occurrences of s before pos • Rank(bar, 5) = 2 • Select(s, i): find the i-th occurrence of a s • Select(foo, 2) = 6

  4. Prefix operations (foo, bar, foobar, foo, bar, bar, foo) 0 1 2 3 4 5 6 • Queries • RankPrefix(p, pos): count strings prefixed by p before pos • RankPrefix(foo, 5) = 3 • SelectPrefix(p, i): find the i-th string prefixed by p • SelectPrefix(foo, 2) = 3

  5. Example: storing relations • Write the columns as string sequences • Store them separately • Reduce relational operations to sequence queries What does Sheldon like? Who likes pages from domain wikipedia.org? Other operations: range counting, …

  6. Dynamic sequences We want to support the following operations: • Insert(s, pos): insert the string s immediately before position pos • Append(s): append the string s at end of the sequence (special case of Insert) • Delete(pos): delete the string at position pos If data structure only supports Append, we call it append-only, otherwise dynamic (or fully dynamic)

  7. Requirements • Store the sequence in as little space as possible • Close to the information-theoretic lower bound • But still be able to support all the described operations (query and update) efficiently • Aim for worst-case polylog operations

  8. Some notation (foo, bar, foobar, foo, bar, bar, foo) 0 1 2 3 4 5 6 • Sequence S, |S| = n • In the example n = 7 • String set Sset is unordered set of distinct strings appearing in S • In the example, {foo, bar, foobar}, |Sset| = 3 • Also called alphabet • Sequence symbols can also be integers, characters, … • As long as they are binarized to strings

  9. Wavelet Trees • Introduced in 2003 to represent Compressed Suffix Arrays • Support Access/Rank/Select on sequences on a finite alphabet (of integers) • Reduces to operations on bitvectors by recursively partitioning the alphabet • String sequences can be reduced to integer sequences

  10. Wavelet Trees • S = (a, b, r, a, c, a, d, a, b, r, a), Sset={a, b, c, d, r} abracadabra00101010010 {a, b} {c, d, r} abaaaba0100010 rcdr1011 {d, r} rdr101 a b c d r

  11. Wavelet Trees • Space equal to entropy of the sequence • Plus negligible terms • Supports Access/Rank/Select in O(log |Sset|) • Later extended to support Insert/Delete… • … but tree structure is fixed a priori • String set Sset is cannot be changed! • Unrealistic restriction in many database applications

  12. The Wavelet Trie • The Wavelet Trie is a Wavelet Tree on sequences of binary strings (Sset⊂ {0, 1}*) • Supports Access/Rank(Prefix)/Select(Prefix) • Fully dynamic… • … or append only (with better bounds) • The string set need not be known in advance

  13. Wavelet Trie: Construction Sequence of binary strings Branching bit: β α: 010β: 1001011 010111 0100110 0100001 0101010 0100110 010111 010110 010111 0100110 0100001 0101010 0100110 010111 010110 0 1 110 001 110 11 010 11 10 Common prefix: α

  14. Wavelet Trie: Construction α: 010β: 1001011 010111 0100110 0100001 0101010 0100110 010111 010110 α:εβ: 101 α:εβ: 1011 α: 01 α: 10 α: 10 α:εβ: 110 α: ε α: ε

  15. Wavelet Trie: Access α: 010β:1001011 α: 010β: 1001011 010111 0100110 0100001 0101010 0100110 010111 010110 0123456 α:εβ:101 α:εβ: 101 α:εβ: 1011 α:εβ:1011 α: 01 α: 10 α: 10 α:εβ: 110 Access(5) = 1 010 1 1 α: ε α: ε Rank is similar

  16. Wavelet Trie: Select α: 010β: 1001011 α: 010β:1001011 010111 0100110 0100001 0101010 0100110 010111 010110 0123456 α:εβ: 101 α:εβ:101 α:εβ: 1011 α:εβ:1011 α: 01 α: 01 α: 10 α: 10 α: 10 α: 10 α:εβ: 110 α:εβ: 110 4 Select(0100110, 1) = α: ε α: ε α: ε α: ε

  17. Wavelet Trie: Append α: 010β: 1001011 0 010111 0100110 0100001 0101010 0100110 010111 010110 α:εβ: 101 α:εβ: 1011 1 α: 01 α: 10 α:εβ: 11 α: 10 α:εβ: 110 0 010010 Insert/Delete are similar α: ε α: ε α: 0 α: ε

  18. Space analysis • Information-theoretic lower bound • LB(S) = LT(Sset) + nH0(S) • LT is the information-theoretic lower bound for storing a set of strings • Static WT: LB(S) + o(ĥn) • Append-only WT: LB(S) + PT(Sset)+ o(ĥn) • PT(Sset): space taken by the Patricia Trie • Fully dynamic WT: LB(S) + PT(Sset)+ O(nH0(S))

  19. Operations time complexity • Need new dynamic bitvectors to support initialization (create a bitvector0n or 1n) • Static and Append-only Wavelet Trie • All supported operations in O(|s| + hs) • hs is number of nodes traversed by string s • Fully dynamic Wavelet Trie • All supported operations in O(|s| + hs log n) • Deletion may take O(|ŝ| + hs log n) where ŝ is longest string in the trie

  20. Thanks for your attention! Questions?

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