ICS 241

1
ICS 241

• Discrete Mathematics II
• William Albritton, Information and Computer
  Sciences Department at University of Hawaii at
  Manoa
• For use with Kenneth H. Rosens Discrete
  Mathematics Its Applications (5th Edition)
• Based on slides originally created by
• Dr. Michael P. Frank, Department of Computer
  Information Science Engineering at University
  of Florida

2
Section 9.2 Applications of Trees

• Binary search trees
• A simple data structure for sorted lists
• Decision trees
• Minimum comparisons in sorting algorithms
• Prefix codes
• Huffman coding
• Game trees

3
Binary Search Trees

• A representation for sorted sets of items.
• Supports the following operations in T(log n)
  average-case time
• Searching for an existing item.
• Inserting a new item, if not already present.
• Supports printing out all items in T(n) time.
• Note that inserting into a plain sequence ai
  would instead take T(n) worst-case time.

4
Binary Search Tree Format

• Items are stored at individual tree nodes.
• We arrange for the tree to always obey this
  invariant
• For every item x,
• Every node in xs left subtree is less than x.
• Every node in xs right subtree is greater than x.

Example
7
3
12
1
5
9
15
0
2
8
11
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Recursive Binary Tree Insert

• procedure insert(T binary tree, x item)v
  rootTif v null then begin rootT x
  return Done endelse if v x return Already
  presentelse if x lt v then return
  insert(leftSubtreeT, x)else must be x gt
  v return insert(rightSubtreeT, x)

6
Class Exercise

• Exercise 1., 3. (p. 656)
• Each pair of students should use only one sheet
  of paper while solving the class exercises

7
Decision Trees

• A decision tree represents a decision-making
  process.
• Each possible decision point or situation is
  represented by a node.
• Each possible choice that could be made at that
  decision point is represented by an edge to a
  child node.
• In the extended decision trees used in decision
  analysis, we also include nodes that represent
  random events and their outcomes.

8
Coin-Weighing Problem

• Imagine you have 8 coins, oneof which is a
  lighter counterfeit, and a free-beam balance.
• No scale of weight markings is required for this
  problem!
• How many weighings are needed to guarantee that
  the counterfeit coin will be found?

?
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As a Decision-Tree Problem

• In each situation, we pick two disjoint and
  equal-size subsets of coins to put on the scale.

A given sequence ofweighings thus yieldsa
decision tree withbranching factor 3.
The balance thendecides whether to tip left,
tip right, or stay balanced.
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Applying the Tree Height Theorem

• The decision tree must have at least 8 leaf
  nodes, since there are 8 possible outcomes.
• In terms of which coin is the counterfeit one.
• Recall the tree-height theorem, h?logm??.
• Thus the decision tree must have heighth
  ?log38? ?1.893? 2.
• Lets see if we solve the problem with only 2
  weighings

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General Solution Strategy

• The problem is an example of searching for 1
  unique particular item, from among a list of n
  otherwise identical items.
• Somewhat analogous to the adage of searching for
  a needle in haystack.
• Armed with our balance, we can attack the problem
  using a divide-and-conquer strategy, like whats
  done in binary search.
• We want to narrow down the set of possible
  locations where the desired item (coin) could be
  found down from n to just 1, in a logarithmic
  fashion.
• Each weighing has 3 possible outcomes.
• Thus, we should use it to partition the search
  space into 3 pieces that are as close to
  equal-sized as possible.
• This strategy will lead to the minimum possible
  worst-case number of weighings required.

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General Balance Strategy

• On each step, put ?n/3? of the n coins to be
  searched on each side of the scale.
• If the scale tips to the left, then
• The lightweight fake is in the right set of ?n/3?
  n/3 coins.
• If the scale tips to the right, then
• The lightweight fake is in the left set of ?n/3?
  n/3 coins.
• If the scale stays balanced, then
• The fake is in the remaining set of n - 2?n/3?
  n/3 coins that were not weighed!
• Except if n mod 3 1 then we can do a little
  better by weighing ?n/3? of the coins on each
  side.

You can prove that this strategy always leads to
a balanced 3-ary tree.
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Coin Balancing Decision Tree

• Heres what the tree looks like in our case

123 vs 456
left 123
balanced78
right 456
4 vs. 5
1 vs. 2
7 vs. 8
L1
L4
L7
R2
B3
R5
B6
R8
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Prefix Codes

• A way to encode letters (or any data) using bit
  strings, so that the bit string for a letter
  never occurs as the first part of a bit string
  for another letter
• Can be represented as a binary tree, where the
  characters are the leaves, and the bits
  correspond to the left and right child
• Example e (0), a(10), t(11), so that 101110
  ate
• If use 8-bit ASCII code, then 3824 bits

15
Class Exercise

• Exercise 21. (p. 657)
• Each pair of students should use only one sheet
  of paper while solving the class exercises

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Huffman Coding

• A way to compress data, using the frequency of
  symbols in a string to produce a prefix code that
  encodes the string using the fewest possible bits
• Tree is built from the bottom-up, connecting the
  symbols and nodes with the lowest frequencies
  first
• Parent frequency of left child freq. of right
  child
• Left child frequency gt right child frequency

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Huffman Coding Example

• Letters respective frequencies
• A0.08, B0.10, C0.12, D0.15, E0.20, F0.35
• Huffman coding gives following encodings
• A(111), B(110), C(011), D(010),E(10), F(00)
• Average number of bits used to encode a letter
• 30.08 3 0.10 30.12 30.15 20.20
  20.35 2.45

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Class Exercise

• Exercise 23. (p. 657)
• Each pair of students should use only one sheet
  of paper while solving the class exercises

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Game Trees

• Game of Nim
• Two players remove 1 or more stones from one of
  the piles of stones
• Lose by removing the last stone
• Represent game with a tree that shows all
  possible moves in the game
• Root is start
• Even level nodes are boxes (1st players move)
• Odd level nodes are circles(2nd players move)

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Game of Nim

• Determine who will win by minmax strategy
• Leaves are given numbers that correspond to who
  wins the game
• 1 if 1st player wins
• -1 if 2nd player wins
• For each node, take the maximum (for 1st player)
  or minimum value of children (for 2nd player)
• 1st player tries to maximize payoffs 2nd player
  tries to minimize payoffs

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Game of Nim Example

• Start with 3 piles of stones 2, 2, 1 stones
• Start with square node (1st players move)
• 3 children are circular nodes (2nd players move)
  
• Leaves are 1/-1 for 1st/2nd player wins
• Take min or max values of children to determine
  who will win the game

22
Class Exercise

• Exercise 33. (p. 659)
• Each pair of students should use only one sheet
  of paper while solving the class exercises