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Uniform algorithms for deterministic construction of efficient dictionaries Milan Ružić IT University of Copenhagen Faculty of Mathematics University of Belgrade ESA 2004 / ARCO 2005 presentation

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2 The dictionary problem How to store a set S U and answer inquires about membership: is x S ?. In the dynamic dictionary problem, S may change over time. Conditions: Compute on a unit-cost RAM with word length w and a standard instruction set, including multiplication and division. Finite universe U {0,1} w. Use space linear in n | S |.

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3 Randomized solutions Started with a static dictionary with O(n) expected construction time, using (nw) random bits [Fredman, Komolós, Szmerédi 82]. Reached a dynamic dictionary with: Constant search time. Constant update time with probability O(1 – n -c ). Use of only O(log n + log w) random bits. [Dietzfelbinger et al 92] However, what if: random bits are not easily available, or performance without a guarantee is unacceptable?

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4 Deterministic dictionaries with fast lookups ReferenceLookup time Construction time Compile-time precomputation Alon-Naor 94 O(w / log n)O(n w log 4 n) _ Andersson 96 O(log w loglog n)O(n)O(w O(1) ) Raman 96 O(1)O(n 2 w) _ Hagerup Miltersen Pagh 01 O(1)O(n log n) (2 (w) w) ? Our results O( t(n) ) O(n 1+1/t(n) + n t(n) log w) _ O(1)O(n w log 2 n) _

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5 The family of hash functions Viewing the problem in a continuous setting - H R. A sufficient condition for avoiding collisions :

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6 The set of good parameters The set of multipliers which generate less than m collisions on the set of s differences has the measure of at least We can calculate the measures with numbers of bounded precision. The set of good parameters contains sufficiently large intervals – that is, there are good multipliers which can be represented by a constant number of machine words.

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7 Finding a good function Problem: Given a set of s differences, deterministically find a multiplier a which produces less than m colliding differences. Not all differences need to be explicitly stored in memory. We use bit by bit construction – sometimes several consecutive bits are set at once. Choosing a bit is equivalent to choosing a half of a working interval. Key observation: sets with relatively small support intervals are insignificant to current choice.

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8 Three classes of differences The recurrence for measure estimates: 1 ( p+1 ) + 2 ( p+1 ) + E ( p+1 ) ( p ) + E ( p ) Several bits are chosen at once when D mid. O(w) term represents the total cost of finding the leftmost bits of keys.

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9 Reducing the construction time We employ multi-level hashing scheme. The number of levels can be set by adjusting the parameters m and s. The structure of the set of differences: In the case of O(1) lookup time we set n k n, m 4n and r n. Note on evaluation: When input consists of multi-word keys, full multiplication is usually not necessary.

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