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Efficient Solutions for Dynamic Predecessor Problems via RAMBO

This paper presents a comprehensive solution to the dynamic predecessor problem utilizing the RAMBO model of computation. It defines critical operations including querying the largest element, inserting and deleting elements, and updating values efficiently. The approach leverages a cell probe model with AC0 CPU and offers solutions that utilize O(Nw) bits while achieving constant time complexity for operations. Additionally, it explores related literature, implementation strategies, and potential generalizations of the RAMBO model, providing valuable insights into efficient data structure management.

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Efficient Solutions for Dynamic Predecessor Problems via RAMBO

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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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