Lower Bounds, Alternative Models, etc'

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Lower Bounds, Alternative Models, etc.

• Ke Yi
• April 8, 2008

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External Memory Data Structures

• n N/B

(n, log nT/B) (nBe, logBnT/B)
(n, log n T/B)?
B
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Previous Results Internal Memory

• Computation model Pointer machine
• Most upper bound structures also fall into this
  model
• Range searching (T is the output size)
• O(N) space, O(NeT) time (BM 80)
• O(N logN / loglogN) space, O(logNT) time
  Chazelle 88
• Tight for O(logcNT) query structures, Chazelle
  90
• Point enclosure Chazelle 86
• Persistent interval tree
• ?(N) space, ?(logNT) time
• Optimal in both space and time

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External Memory Models

• External pointer machine
• Natural generalization of the internal pointer
  machine
• Each node contains B data objects
• Out-degree 2 ?B
• Little success
• Bounding-volume hierarchy (Non-replicating index
  structure)
• Tree structure
• Each object is stored only once
• Lower bound known for R-trees, kdB-trees
• Indexability model Hellerstein et al. PODS 97,
  Arge, Samoladas, and Yi, ESA04

D
Block I/O
M
P
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External Memory Models

• Indexability model
• No structure at all!
• Only models layout of data
• Each block contains B data objects
• Can magically find the smallest set ? of blocks
  whose union contains all results
• Cost is defined to be ?

Indexability model
External pointer machine
Bounding volume hierarchy
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Indexability Model in Details

• N data objects laid out in disk blocks, possibly
  with redundancy
• Each block holds at most B objects
• Cost of a query q minimum blocks needed to
  retrieve all answers
• Can find those blocks without cost
• Redundancy r and access overhead A0, A1
• r Average copies in the index
• Size is rn blocks
• A0, A1 Any query can be covered by
  
• Lower bound expressed as a tradeoff between r and
  A0, A1

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Trade-off Results

• Range searching
• Point enclosure
• Dual bounds in external memory!

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Alternative ModelsCache-Oblivious Model
?
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Memory Hierarchy

• Modern machines have complicated memory hierarchy
• Levels get larger and slower further away from
  CPU
• Block sizes and memory sizes are different!
• There are a few attempts to model the hierarchy
  but not successful
• They are too complicated!

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The Cache-Oblivious Model FLPR99

• Assume any (constant) number of levels in the
  hierarchy with different M and B
• Theorem If the algorithm works with any values
  of M and B in the two-level model, then the
  algorithm uses the optimal (up to a constant)
  number of memory transfers on any level in the
  memory hierarchy
• We still analyze the algorithm in the standard
  two-level model!

R A M
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Example Cache-Oblivious Search Tree
Question How to layout the binary tree in memory
such that any root-to-leaf path visits O(logBN)
blocks?
How can we make it work on any block size B?
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Van Emde Boas Layout

• Consider the first level of recursionwhere tree
  size lt B
• Tree size between and B
• So has height T(log B)

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Alternative ModelsDynamic Memory Model Barve
and Vitter, FOCS 98

• Allocated memory to algorithm changes over time

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Memory Allocated Changes Over Time

• Sorting
• One I/O allowed in each time unit
• Memory ( blocks) changes by 0, 1 or -1 in each
  time unit
• Resources consumed by one I/O when the memory
  has m blocks is defined to be log m
• A sorting algorithm is measured in terms of
  resource consumption
• Gave an merge-sort algorithm with resource
  consumption O(n log n), this is also tight