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Faster persistent data structures through hashing Johan Tibell johan.tibell@gmail.com 2011-02-15
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Motivating problem: Twitter data analys “I’m computing a communication graph from Twitter data and then scan it daily to allocate social capital to nodes behaving in a good karmic manner. The graph is culled from 100 million tweets and has about 3 million nodes.” We need a data structure that is fast when used with string keys, and doesn’t use too much memory.
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Persistent maps in Haskell Data.Map is the most commonly used map type. It’s implemented using size balanced trees. Keys can be of any type, as long as values of the type can be ordered.
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Real world performance of Data.Map Good in theory: no more than O(log n) comparisons. Not great in practice: up to O(log n) comparisons! Many common types are expensive to compare e.g String, ByteString, and Text. Given a string of length k, we need O(k ∗ log n) comparisons to look up an entry.
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Hash tables Hash tables perform well with string keys: O(k) amortized lookup time for strings of length k. However, we want persistent maps, not mutable hash tables.
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Milan Straka’s idea: Patricia trees as sparse arrays We can use hashing without using hash tables! A Patricia tree implements a persistent, sparse array. Patricia trees are as twice fast as size balanced trees, but only work with Int keys. Use hashing to derive an Int from an arbitrary key.
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Implementation tricks Patricia trees implement a sparse, persistent array of size 232 (or 264 ). Hashing using this many buckets makes collisions rare: for 224 entries we expect about 32,000 single collisions. Linked lists are a perfectly adequate collision resolution strategy.
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First attempt at an implementation −− D e f i n e d i n t h e c o n t a i n e r s p a c k a g e . data IntMap a = Nil | Tip {−# UNPACK #−} ! Key a | Bin {−# UNPACK #−} ! P r e f i x {−# UNPACK #−} ! Mask ! ( IntMap a ) ! ( IntMap a ) type P r e f i x = I n t type Mask = Int type Key = Int newtype HashMap k v = HashMap ( IntMap [ ( k , v ) ] )
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A more memory eﬃcient implementation data HashMap k v = Nil | Tip {−# UNPACK #−} ! Hash {−# UNPACK #−} ! ( FL . F u l l L i s t k v ) | Bin {−# UNPACK #−} ! Prefix {−# UNPACK #−} ! Mask ! ( HashMap k v ) ! ( HashMap k v ) type P r e f i x = I n t type Mask = Int type Hash = Int data F u l l L i s t k v = FL ! k ! v ! ( L i s t k v ) data L i s t k v = N i l | Cons ! k ! v ! ( L i s t k v )
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Reducing the memory footprint List k v uses 2 fewer words per key/value pair than [( k, v )] . FullList can be unpacked into the Tip constructor as it’s a product type, saving 2 more words. Always unpack word sized types, like Int, unless you really need them to be lazy.
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Benchmarks Keys: 212 random 8-byte ByteStrings Map HashMap insert 1.00 0.43 lookup 1.00 0.28 delete performs like insert.
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Can we do better? We still need to perform O(min(W , log n)) Int comparisons, where W is the number of bits in a word. The memory overhead per key/value pair is still quite high.
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Borrowing from our neighbours Clojure uses a hash-array mapped trie (HAMT) data structure to implement persistent maps. Described in the paper “Ideal Hash Trees” by Bagwell (2001). Originally a mutable data structure implemented in C++. Clojure’s persistent version was created by Rich Hickey.
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Hash-array mapped tries in Clojure Shallow tree with high branching factor. Each node, except the leaf nodes, contains an array of up to 32 elements. 5 bits of the hash are used to index the array at each level. A clever trick, using bit population count, is used to represent spare arrays.
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The Haskell deﬁnition of a HAMT data HashMap k v = Empty | BitmapIndexed {−# UNPACK #−} ! Bitmap {−# UNPACK #−} ! ( Array ( HashMap k v ) ) | L e a f {−# UNPACK #−} ! ( L e a f k v ) | F u l l {−# UNPACK #−} ! ( Array ( HashMap k v ) ) | C o l l i s i o n {−# UNPACK #−} ! Hash {−# UNPACK #−} ! ( Array ( L e a f k v ) ) type Bitmap = Word data Array a = Array ! ( Array# a ) {−# UNPACK #−} ! I n t
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Making it fast The initial implementation by Edward Z. Yang: correct but didn’t perform well. Improved performance by replacing use of Data.Vector by a specialized array type, paying careful attention to strictness, and using GHC’s new INLINABLE pragma.
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Where is all the time spent? Most time in insert is spent copying small arrays. Array copying is implemented using indexArray# and writeArray#, which results in poor performance. When cloning an array, we are force to ﬁrst ﬁll the new array with dummy elements, followed by copying over the elements from the old array.
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A better array copy Daniel Peebles have implemented a set of new primops for copying array in GHC. The ﬁrst implementation showed a 20% performance improvement for insert. Copying arrays is still slow so there might be room for big improvements still.
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Other possible performance improvements Even faster array copying using SSE instructions, inline memory allocation, and CMM inlining. Use dedicated bit population count instruction on architectures where it’s available. Clojure uses a clever trick to unpack keys and values directly into the arrays; keys are stored at even positions and values at odd positions. GHC 7.2 will use 1 less word for Array.
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Optimize common cases In many cases maps are created in one go from a sequence of key/value pairs. We can optimize for this case by repeatedly mutating the HAMT and freeze it when we’re done. Keys: 212 random 8-byte ByteStrings fromList/pure 1.00 fromList/mutating 0.50
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Abstracting over collection types We will soon have two map types worth using (one ordered and one unordered). We want to write function that work with both types, without a O(n) conversion cost. Use type families to abstract over diﬀerent concrete implementation.
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Summary Hashing allows us to create more eﬃcient data structures. There are new interesting data structures out there that have, or could have, an eﬃcient persistent implementation.
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