Efficient Compressed Tries Using Path Decompositions

1. Fast Compressed Tries through Path Decompositions Roberto Grossi Giuseppe Ottaviano* Università di Pisa * Part of the work done while at Microsoft Research Cambridge

2. Compacted tries Node label Branching character t h r three trial triangle trie triple triply ree i e a p l ε ε e y n l ε ε gle ε

3. Applications • String dictionaries • With prefix lookup, predecessor, … • Exploit prefix compression • Monotone perfect hash functions • “Hollow” or “Blind” tries [ALENEX 09] • Binary tree (no need store branching chars) • No need to store node labels, just lengths (skips)

4. Height vs. performance • Tries can be deep – no guarantee on height • Bad with pointer-based trees • ~1 cache miss per child operation • Worse with succinct tree encodings • Need to access several directories • Many cache misses per child operation • Large constants hidden in the O(1)

5. Path decomposition t triangle h r p h e l ree i e a p Recurse here withsuffix le l ε ε e y n l ε ε gle ε Query: triple

6. Centroid path decomposition • Decompose along the heavy paths • choose the edge that has most descendants • Height of the decomposed tree: O(log n) • Usually lower • Average height

7. Succinct encoding • [PODS 08] presents a succinct data structure for centroid path-decomposed tries • Not practical: need complex operations on succinct trees • We introduce a simpler and practical encoding • This encoding enables also simple compression of the labels

8. Succinct encoding • Node label written literally, interleaved with number of other branching characters at that point in array L • Corresponding branching characters in array B • Tree encoded with DFUDS in bitvectorBP • Variant of Range Min-Max tree [ALENEX 10] to support Find{Close,Open}, more space-efficient (Range Min tree) triangle L : t1ri2a1ngle BP: ( ((( ) B : h epl (spaces added for clarity) p h e l

9. Compression of L ...$...index.html$....html$....html$...index.html$ … 3 index … 5 .html … Dictionary ...$...35$...5$...5$...35$ • Dictionary codewords shared among labels • Codewords do not cross label boundaries ($) • Use vbyte to compress the codeword ids

10. Compression of L • Node labels (t1ri2a1ngle, l1e, …): • each label is suffix of a string in the set • interleaved with few “special characters” 1, 2, 3,… • Compressible if strings are compressible • Dictionary and parsing computed withmodified Re-Pair • Domain-specific compression can be used instead • Decompression overhead negligible

11. Experimental results (time) • Experiments show gains in time comparable to the gains in height • Confirm that bottleneck is traversal operations Code available at https://github.com/ot/path_decomposed_tries

12. Experimental results (space) • For strings with many common prefixes, even non-compressed trie is space-efficient • Labels compression considerably increases space-efficiency • Decompression time overhead: ~10% Code available at https://github.com/ot/path_decomposed_tries

13. Thanks for your attention! Questions?

14. References • [ALENEX 10]D. Arroyuelo, R. Cánovas, G. Navarro, and K. Sadakane. Succinct trees in practice. In ALENEX, pages 84–97, 2010. • [ALENEX 09] D. Belazzougui, P. Boldi, R. Pagh, and S. Vigna. Monotone minimal perfect hashing: searching a sorted table with O(1) accesses. In SODA, pages 785–794, 2009. • [PODS 08] P. Ferragina, R. Grossi, A. Gupta, R. Shah, and J. S. Vitter. On searching compressed string collections cache-obliviously. In PODS, pages 181–190, 2008.