Graph Theory in Computer Science

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Graph Theory in Computer Science

• Greg Stoll
• November 22, 2008

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Today's Topics

• Graph colorings
• Huffman encoding

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Graph colorings
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Graph colorings

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Graph colorings

• What's the fewest number of colors that are
  needed to color a given graph?

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Graph colorings

• A clique of size n is a graph with n vertices
  where all vertices have edges between them.

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Register allocation

• Modern computers do operations (, -, , etc.) on
  registers. Fixed number of these (x86
  architecture has 8, x64 has 16). When compiling
  code, need to assign variables to registers to
  use them.

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Register allocation

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Register allocation
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Register allocation

• In fact, this is always true for interval graphs
  the number of registers needed (i.e. the number
  of colors to color the graph) is equal to the
  size of the largest clique!

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Register allocation

• Unfortunately, even finding the largest clique in
  a graph is NP-hard, but a greedy algorithm does
  pretty well.

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Register allocation

• Another problem what if there aren't enough
  registers?

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Worksheet, part 1

• Work on questions 1-6 on the worksheet!

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Cookies!

• Mmmm....

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Sending a message

• On a computer, all you have are bits (0's and
  1's). How can you send a text message to another
  computer?

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ASCII encoding

• ASCII is a standard way to turn characters into
  bits.
• A 65 01000001
• B 66 01000010
• ...
• Z 90 01011010

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Modified ASCII

• What if we just want to send letter and a little
  punctuation? We can use a smaller code
• a 0 00000
• b 1 00001
• ...
• ltspacegt 26 11010
• 5 bits means 25 32 characters

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Modified ASCII

• Our modified ASCII uses 5 bits per character.
• Sample message hello mom, how are you? is 22
  characters 5 bits/character 110 bits.
• How could we reduce the average number of bits
  per character?

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Shrinking the message

• Not all letters are created equal. e is much
  more common than x in most English text.
• We could take advantage of this if we made the
  code for e smaller than the code for x.

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Shrinking the message

• o and are the most commonly used letters in
  our sample message. Let
• o 0
• 1
• m 00
• ....
• Will this work?

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Prefix-free code

• A prefix-free code is one in which no code word
  is a prefix of another.
• o 0
• 1
• m 00
• is not prefix free, since o0 is a prefix of
  m00.

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Binary tree

• A binary tree is a common data structure where
  each node has up to two children.

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Binary tree

• What if we assign a character to each leaf?

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Huffman encoding

• We want a way to build up a binary tree where the
  least common characters are lower. (i.e. their
  code words are longer) We can do this by
  starting at the bottom and always combining the
  least common two characters, and continuing until
  we've formed a binary tree.

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Huffman encoding

• Example hello mom, how are you?

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Huffman encoding

• Using Huffman encoding lets us send the message
  in 78 bits, as opposed to 110. With longer
  message, you can get even greater savings.

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Worksheet, part 2

• Work on questions 7-12 on the worksheet!