Uniform algorithms for deterministic construction of efficient dictionaries

1
Uniform algorithms for deterministic construction
of efficient dictionaries

• Milan Ružic
• IT University of Copenhagen
• Faculty of Mathematics
• University of Belgrade
• ESA 2004 / ARCO 2005 presentation

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 ? 0,1w .
• Use space linear in n ? S .

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?

4
Deterministic dictionaries with fast lookups
Reference Lookup time Construction time Compile-time precomputation
Alon-Naor 94 O(w / log n) O(n w log4 n) _
Andersson 96 O(log w loglog n) O(n) O(wO(1))
Raman 96 O(1) O(n2 w) _
Hagerup Miltersen Pagh 01 O(1) O(n log n) ?(2?(w) w) ?
Our results O( t(n) ) O(n11/t(n) n t(n) log w) _
Our results O(1) O(n w log2 n) _
5
The family of hash functions

• Viewing the problem in a continuous setting - HR
  .
• A sufficient condition for avoiding collisions

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.

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.

8
Three classes of differences

• The recurrence for measure estimates
• ?1(p1) ?2(p1) E(p1) ? ?(p) E(p)
• Several bits are chosen at once when Dmid ? ? .
• O(w) term represents the total cost of finding
  the leftmost bits of keys.

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 nk? n?, m
  ? 4n? and r ? n.
• Note on evaluation When input consists of
  multi-word keys, full multiplication is usually
  not necessary.