COMP 9020

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COMP 9020

• Dr. Rex Kwok
• Mondays 3-6pm
• OMB 31

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Motivation

• Proliferation of programming languages
• Many languages have come and gone
• Many languages will be developed in the future
• Logic and mathematics as a means of reasoning and
  analysis

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Course Overview

• Reasoning with logic
• Means of representing beliefs/knowledge
• Derivation of consequences
• Reasoning with sets and relations
• Complexity and order of growth

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Course Overview

• Counting and probability
• Common data structures
• Trees
• Graphs

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Assessment

• Two assignments worth 20 each
• Applications of theory
• Sorting algorithms?
• Bioinformatics, evolutionary genetics?
• A final exam worth 60

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Resources

• http//www.cse.unsw.edu.au/cs9020/
• lecture material
• practice questions and solutions

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Textbooks

• The lecture material comes from Discrete
  Mathematics by K.A. Ross and C.R.B. Wright.
• Other references are Logic and Discrete
  Mathematics A Computer Science Perspective by
  W.K. Grassmann and J. Tremblay.
• and Foundations of Computer Science C Edition by
  A.V. Aho and J.D. Ullman.

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Example The Priority Queue

• Data
• key which is used to identify or rank an object
• e.g. student number, a persons height,
  intelligence etc.
• Operations
• addKey adds a new piece of data
• findMin is an operation which returns the object
  with the smallest key

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An unordered array

• addKey is implemented by simply adding the new
  piece of data to the end of the list.

addKey(27)
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An unordered array

• Implement findMin in the following way
• set the leftmost element as the current candidate
  for the smallest key
• traverse the array from left to right updating
  the candidate as required

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An unordered array

• In every case, the search has to continue to the
  end of the array
• If there are n keys, findMin requires n-1
  comparison operations are necessary
• addKey does not require any comparison operations

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A heap

• A heap is a complete binary tree with the
  property that all for every node v (except the
  root), the key at v is greater than the key of
  the parent of v

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A heap
4
11
24
34
16
51
97
71
36
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A heap
4
11
24
34
16
51
97
71
36
1
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A heap
4
11
24
34
16
1
97
71
36
51
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A heap
4
11
1
34
16
24
97
71
36
51
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A heap
1
11
4
34
16
24
97
71
36
51
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Consequences of the heap property

• The smallest key is always held at the root node
• findMin does not require any comparison
  operations
• With n keys the height of the heap is log2n
• addKey in the worst case requires log2n
  comparisons

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Venn Diagrams
B
A
B
A
A ? B
A ? B
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Venn Diagrams
B
B
A
A
A \ B
A B
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Venn Diagrams
A
Ac