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

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t three trial triangle trie triple triply iree εε l εεεgle hr e p a l n Node label Branching character Compacted tries ye

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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)

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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)

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Path decomposition t iree εε l εεεgle hr e p a l n ye he p l Query: triple Recurse here with suffix le

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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 Web QueriesURLsSynthetic Compacted trie11.018.1504.4 Centroid trie5.26.22.8 Hollow trie50.867.31005.3 Centroid hollow trie8.09.22.8

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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

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Succinct encoding he p l L : t1ri2a1ngle BP: ( ((( ) B : h epl (spaces added for clarity) 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 bitvector BP – Variant of Range Min-Max tree [ALENEX 10] to support Find{Close,Open}, more space-efficient (Range Min tree)

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

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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 with modified Re-Pair – Domain-specific compression can be used instead Decompression overhead negligible

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Experimental results (time) Experiments show gains in time comparable to the gains in height Confirm that bottleneck is traversal operations Web QueriesURLsSynthetic Trie3.57.0119.8 Centroid trie2.44.35.1 Hollow trie [ALENEX 09]16.622.4462.7 Hollow trie7.213.9137.1 Centroid hollow trie2.84.411.1 (microseconds, lower is better) Code available at https://github.com/ot/path_decomposed_tries

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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% Web QueriesURLsSynthetic Hu-Tucker Front Coding40.9%24.4%19.1% Centroid trie55.6%22.4%17.9% Centroid trie + compression31.5%13.6%0.4% (compression ratio, lower is better) Code available at https://github.com/ot/path_decomposed_tries

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Thanks for your attention! Questions?

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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.

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