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When RAMBO goes Cuckoo

When RAMBO goes Cuckoo. Andrej Brodnik, University of Primorska John Iacono, Polytechnic University. Problem definition. The dynamic predecessor : Query(x) : Returns the largest element from S that is at most x Insert(x) : Adds x to S Delete(x) : Removes x from S

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When RAMBO goes Cuckoo

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  1. When RAMBO goes Cuckoo Andrej Brodnik, University of Primorska John Iacono, Polytechnic University

  2. Problem definition The dynamic predecessor: • Query(x): Returns the largest element from S that is at most x • Insert(x): Adds x to S • Delete(x): Removes x from S • Update(x, δ): Changes value of x When RAMBO goes Cuckoo

  3. Model of computation • Cell probe model with: • AC0 CPU with w-bit operands • Memory: • of M bits • control circuit of size wO(1) and of depth (log w/log log w) When RAMBO goes Cuckoo

  4. Result • Solution to the predecessor problem in O(Nw) bits and (1) time • Solution to the dynamic predecessor problem in O(Nw) bits and (1) time whp When RAMBO goes Cuckoo

  5. Literature • O(log log M) time and O(M) bits (vEB 1977) • O(log log M) time and O(N w) bits (Willard, 1983) • O(min(log w/ log log w, sqrt(log N/ log log N))) (Beame and Fich, 2002) • Unit-cost word-level RAM with multiplication • Communication game model • (1) time with (M) bit memory (Brodnik et al., 2005) • RAMBO When RAMBO goes Cuckoo

  6. Overall structure When RAMBO goes Cuckoo

  7. Split tagged tree When RAMBO goes Cuckoo

  8. RAMBO When RAMBO goes Cuckoo

  9. RAMBO – as implemented a d DEMUX 0 DEMUX 1 DEMUX 2 DEMUX 3 When RAMBO goes Cuckoo

  10. One bit plane – compressed Yggdrasil When RAMBO goes Cuckoo

  11. Compressed Yggdrasil - analysis • Circuit depth: (log w / log log w) • Size: (N w) bits • Time: • Extended circuit RAM: (log w / log log w) • Extended bit probe: (1) When RAMBO goes Cuckoo

  12. Conclusions • Neighbor problem solved in (1) time and (Nw) bits under cell probe model • Dynamic neighbor problem solved in (1) time whp and (Nw) bits under cell probe model • The solution is implementable (Leben at al. 1999) • Can it be generalized to cover CAMs!? • Can be worked out to (B) bits!? • What is the relation of RAMBO and RAM – is this the separating problem When RAMBO goes Cuckoo

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