Sets of Digital Data

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Sets of Digital Data

• CSCI 2720
• Fall 2005
• Kraemer

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Digital Data

• In earlier work with BSTs and various balanced
  trees, we compared keys for order or equality
• Here, we take advantage of structure of key
• Use it as an index, or
• Decompose string key into characters, or
• Treat key as numerical quantity on which we can
  perform operations

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Assumptions

• We will construct and manipulate sets that
• Are drawn from a universe U of size N
• U u0, uN-1
• A relatively simple procedure exists by which we
  can compute, for an element u ? U, the index i
  such that u ui.
• Easy if U is set of integers
• Also easy if U is set of characters with
  character codes in a contiguous interval

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Bit Vector

• Used to represent a subset S ? U
• A table of N bits, Bits0.. N-1
• Bitsi 1 if ui ? S
• Bitsi 0 if ui ? S
• Example todays attendance

0 1 2 3 4 5 6 --
student number
1 1 0 1 0 1 1
1 present 0 absent
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Bit Vectors

• Assume
• determining element index takes constant time
• accessing position in table takes constant time
• May actually take several ops, and depend
  somewhat on N(size of universe), but not on size
  of set represented
• Then
• Insert, Delete, Member are constant time ops

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Bit Vectors

• A subset of a set of size N always takes N bits
  to represent, independent of size of subset
• Makes sense if
• N is not too large
• need to represent sets of size comparable to N

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Storage Efficiency

• Bit Vector vs. Binary Trees
• Binary Tree, set of size n
• Requires n(2p K) bits
• K gt lg N, size of field to represent key value
• p number of bits in a pointer
• Bit Vector, takes N bits
• If n ? N, then bit vector more efficient
• If p K 32, then tree becomes more space
  efficient when n/N ? 1
• Actually, when n(2p K) N, which is when n/N
  1/96

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When to use Bit Vectors?

• When universe is relatively small
• When sets are large in relation to size of
  universe

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Advantages of Bit Vectors

• O(1) implementation of Insert, Delete, Member
• Union and Intersection easy
• Implement via Boolean and and or operations
• May actually take less than one op/element, as
  operations are performed on full machine word
• If machine word 32, then one machine operation
  handles 32 potential elements of set

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Disadvantages of Bit Vectors

• On some computers access to individual bits can
  require shifting and masking operations
  (expensive)
• Result is that Member may be much more expensive
  than Union
• Initialization takes ?(N) -- zero all the bits in
  the vector
• But can use constant time initialization
  algorithm
• But that makes storage requirement go to 2p 1
  bits per element
• So, in practice, just use machine ops to set to
  zero, which are efficient

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Tries and Digital Search Trees

• If the key can be decomposed into characters,
  then the characters of the key can be used as
  indices
• Tries are based on this idea
• trie is the middle symbol of retrieval, a pun
  on tree, but pronounced try

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Tries

• Assume k possible character values
• A trie is a (k1)-ary tree
• each node a table of k1 pointers
• One pointer for each possible character
• One for the end of string character, ?

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Trie Example
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Tries

• Path for key of m characters is length m, with
  pointer at ?
• Dont need to store key itself .. It is the path
  followed.
• Info field might be pointed to by ? element

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Tries Analysis

• Let
• n be the number of keys stored in a trie
• l be the length(in characters) of the longest key
• s be the number of nodes in the trie
• k be the size of the alphabet
• Pro
• Access time is O(l), independent of k, n and s
• Con
• Size -- requires (k1) s p bits
• Most pointers are null, so lots of wasted space

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Strategies for reducing storage requirements of
tries

• Implement a k-ary trie with m nodes as a 2-D, m
  by k table

A B C D E M . P . T
. ?
0 1 2 3 4 5
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Table approach

• Number the nodes in the diagram of slide 13 from
  1 to m
• The table entry corresponding to jth child of ith
  node is the index of the child node
• How does that save space? Just as many nodes and
  elements as on slide 13
• need only ceil(lg(m)) bits to represent,
  smaller than a pointer

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Patricia TreeAnother strategy for reducing
space in a trie

• Patricia tree
• Practical Algorithm to Retrieve Information Coded
  in Alphanumeric
• Eliminate nodes with only one nonempty child
• Can now skip right from T to ? in TURING in our
  example
• Skip from MA . To E or ? in the MENDEL ,
  MENDELEEV chain
• But need to store with each node the index of the
  character on which it discriminates
• And need to store the key itself at the leaf

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Patricia tree
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de la Briandais trees

• Another strategy to save space vs. standard tries
• Use a linked list instead of a table at the node
  level
• Each pointer labeled with the character it
  indexes
• longer search time than tries depends on size of
  character set
• saves significant amounts of memory

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de la Briandais
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Another strategy

• Use tries at the first few levels
• Use ordinary BSTs or de la Briandais at the lower
  levels
• reasoning
• speed advantage at the top, but not too much
  extra memory required
• save space at lower levels

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Digital Search Trees

• Treat keys as bit strings
• (strings over the alphabet 0,1)
• Binary tree search directed left on 0, right on
  1
• Each node contains not only two pointers, but
  also contains a key that matches that string
  prefix
• Compare for equality before searching left or
  right
• If frequencies are known, store higher frequency
  keys nearer root
• Can be grown dynamically
• Expected Search time O(log n)

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Digital Search Tree