A Logarithmic Randomly Accessible Data Structure

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A Logarithmic Randomly Accessible Data Structure
Tom Morgan TJHSST Computer Systems Lab 2006-2007
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Purpose

• To create a data structure with
• O(log(N)) insertion
• O(log(N)) deletion
• O(log(N)) random access
• To implement the data structure and empirically
  test the general case run time
• To prove the data structure's worst case run time

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Background

• Dynamic Arrays
• O(N) insertion
• O(N) deletion
• O(1) random access
• Red-Black Trees
• O(log(N)) insertion
• O(log(N)) deletion
• not randomly accessible
• Splay Trees
• amortized O(log(N)) insertion
• amortized O(log(N)) deletion
• not randomly accessible

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The Basic Algorithm

• Similar to binary search tree
• Places values only at the leaves
• Each node keeps track of the number of leaves
  below it
• O(log(N)) on all operations in a random case, but
  not a sequential one
• Balancing is underway

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An Example Tree
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Balancing Techniques

• When performing any operation, if not balanced
• Shift one element from bigger side to smaller
  side
• Shift half of bigger side to smaller side
• Find best cutoff point along nearest side to
  smaller side (if right is bigger, the left part
  of it) and shift over corresponding sub-tree

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Red Black Tree

• Each node is either red or black
• The path from the root to each leaf contains the
  same number of black nodes
• A red node may not have a red parent
• Absurd amount of cases

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Splay Tree

• Standard binary search tree but for splay
  function
• Splay function is called everytime a node is
  accessed
• Splay function rotates accessed node up to the
  root position

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Testing

• Data structure implemented in Java as a class
• Test various operations and their runtime using a
  separate testing program in Java, which uses the
  class
• Tester allows for both random and ordered
  insertion and deletion
• Balanced tree compared to TreeSet and ArrayList

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Results Pre-Red Black Tree

• A tree that allows for insertion, deletion and
  random access
• All operations are logarithmic on the random case
• Three separate balancing techniques which vary in
  terms of effectiveness of balancing and time
  taken by the balancing
• None of the balancing techniques are perfect as
  of now

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Results Red Black Tree

• Started making randomly accessible but quickly
  encountered errors
• Insertion works fine but deletion appears to
  throw off properties
• Have added numerous testing and analysis
  functions to no avail
• Work suspended in favor of Splay Tree

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Results Splay Tree

• The Splay Tree is fully functional
• Has complete random access functionality
• Works with Java's templates
• Made compliant with Collection and AbstractList
• Added depth-first search iterator, allowing O(N)
  access of all elements in sequence