Efficiency of Algorithms

1
Efficiency of Algorithms

• Csci 107
• Lecture 7

2

• Last time
• Data cleanup algorithms and analysis
• ?(1), ?(n), ?(n2)
• Today
• Binary search and analysis
• ?(lg n)
• Sorting
• Selection sort

3
Searching

• Problem find a target in a list of values
• Sequential search
• Best-case ?(1) comparison
• target is found immediately
• Worst-case ?(n) comparisons
• Target is not found
• Average-case ?(n) comparisons
• Target is found in the middle
• Can we do better?
• Nounless we have the input list in sorted order

4
Searching a sorted list

• Problem find a target in a sorted list
• How can we exploit that the list is sorted, and
  come up with an algorithm faster than sequential
  search in the worst case?
• How do we search in a phone book?
• Can we come up with an algorithm?
• Check the middle value
• If smaller than target, go right
• Otherwise go left

5
Binary search

• Get values for list, A1, A2, .An, n , target
• Set start 1, set end n
• Set found NO
• Repeat until ??
• Set m middle value between start and end
• If target m then
• Print target found at position m
• Set found YES
• Else if target lt Am then end m-1
• Else start m1
• If found NO then print Not found
• End

6
Efficiency of binary search

• What is the best case?
• What is the worst case?
• Initially the size of the list in n
• After the first iteration through the repeat
  loop, if not found, then either start m or end
  m gt size of the list on which we search is
  n/2
• Every time in the repeat loop the size of the
  list is halved n, n/2, n/4,.
• How many times can a number be halved before it
  reaches 1?

7
Orders of magnitude

• Order of magnitude ?( lg n)
• Worst-case efficiency of binary search ?( lg n)
• Comparing order of magnitudes
• ?(1) ltlt ?(lg n) ltlt ?(n) ltlt ?(n2)

8
Sorting

• Problem sort a list of items into alphabetical
  or numerical order
• Why sorting?
• Sorting is ubiquitous (very common)!!
• Examples
• Registrar Sort students by name or by id or by
  department
• Post Office Sort mail by address
• Bank Sort transactions by time or customer name
  or accound number
• For simplicity, assume input is a list of n
  numbers
• Ideas for sorting?

9
Selection Sort

• Idea grow a sorted subsection of the list from
  the back to the front
• 5 7 2 1 6 4 8 3
• 5 7 2 1 6 4 3 8
• 5 2 1 6 4 3 7 8
• 5 2 1 3 4 6 7 8
• 1 2 3 4 5 6 7 8

10
Selection Sort

• Pseudocode (at a high level of abstraction)
• Get values for n and the list of n items
• Set marker for the unsorted section at the end of
  the list
• Repeat until unsorted section is empty
• Select the largest number in the unsorted section
  of the list
• Exchange this number with the last number in
  unsorted section of list
• Move the marker of the unsorted section forward
  one position
• End

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Selection Sort

• Level of abstraction
• It is easier to start thinking of a problem at a
  high level of abstraction
• Algorithms as building blocks
• We can build an algorithm from parts consisting
  of previous algorithms
• Selection sort
• Select largest number in the unsorted section of
  the list
• We have seen an algorithm to do this last time
• Exchange 2 values

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Selection Sort Analysis

• Iteration 1
• Find largest value in a list of n numbers n-1
  comparisons
• Exchange values and move marker
• Iteration 2
• Find largest value in a list of n-1 numbers n-2
  comparisons
• Exchange values and move marker
• Iteration 3
• Find largest value in a list of n-2 numbers n-3
  comparisons
• Exchange values and move marker
• Iteration n
• Find largest value in a list of 1 numbers 0
  comparisons
• Exchange values and move marker
• Total (n-1) (n-2) . 2 1

13
Selection Sort

• Total work (nb of comparisons)
• (n-1) (n-2) . 2 1
• This sum is equal to .5n2 -.5n (proved by Gauss)
• gt order of magnitude is ?( ? )
• Questions
• best-case, worst-case ?
• we ignored constants, and counted only
  comparisons.. Does this make a difference?
• Space efficiency
• extra space ?

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Selection Sort

• In conclusion Selection sort
• Space efficiency
• No extra space used (except for a few variables)
• Time efficiency
• There is no best-case and worst-case
• the amount of work is the same ?(n2)
  irrespective of the input
• Other sorting algorithms? Can we find more
  efficient sorting algorithms?

15
This week

• Tuesday Lab 4
• Exam 1 (Wednesday or Monday?)
• Material Algorithms
• Chapter 1, 2 3 from textbook
• For next time
• Binary search
• Review chapter 1,2 3
• Practice exam