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Title: i206: Lecture 7: Analysis of Algorithms, continued


1
i206 Lecture 7Analysis of Algorithms, continued
  • Marti Hearst
  • Spring 2012

2
Algorithms
  • A clearly defined procedure for solving a
    problem.
  • A clearly described procedure is made up of
    simple clearly defined steps.
  • The procedure can be complex, but the steps must
    be simple and unambiguous.

3
Problem Solving Strategies
  • Try to work the problem backwards
  • Reverse-engineer
  • Once you know it can be done, it is much easier
    to do
  • What are some examples?
  • Stepwise Refinement
  • Break the problem into several sub-problems
  • Solve each sub-problem separately
  • Produces a modular structure
  • Look for a related problem that has been solved
    before
  • Software design patterns

4
Example of Stepwise RefinementSpiderman
  • Peter Parkers goal Make Mary Jane fall in love
    with him
  • To accomplish this goal
  • Get a cool car
  • To accomplish this goal
  • Get 3000
  • To accomplish this goal
  • Appear in a wrestling match
  • Each goal requires completing just one subgoal

5
Example of Stepwise RefinementStar Wars Episode
IV
  • Luke Skywalkers goal Make Princess Leia fall in
    love with him (they werent siblings yet)
  • To accomplish this goal
  • Rescue her from the death star
  • To accomplish this goal
  • Land on Death Star
  • Remove her from her Prison Cell
  • Disable the Tractor Beam
  • Get her onto the Millennium Falcon
  • To accomplish subgoal (2)
  • Obtain Storm Trooper uniforms
  • Have Wookie pose as arrested wild animal
  • Find Location of Cell
  • Have R2D2 communicate coordinates
  • To accomplish subgoal (3)
  • Have last living Jedi walk across catwalks
  • To accomplish subgoal (4)
  • Run down hall
  • Survive in garbage shoot
  • Fight garbage monster

6
Divide and Conquer Algorithms
  • Break the problem into smaller pieces
  • Recursively solve the sub-problems
  • Combine the results into a full solution
  • If structured properly, this can lead to a much
    faster solution.

7
Divide and Conquer Algorithms
  • Problem Cut up the licorice into 64 pieces!.

8
Divide and Conquer Algorithms
  • Problem Cut up the licorice into 64 pieces!
  • How do we do this in
  • O(n)
  • O(log n)
  • O(sqrt n)

9
Santas Socks
  • Problem
  • The elves have packed all 1024 boxes with skates
  • But Santas dirty smelly socks fell into one of
    them.
  • We have to find the box with the dirty socks!
    But its almost midnight!!!!

10
Santas Socks
  • http//www.youtube.com/watch?vwVPCT1VjySA

11
Analysis of Algorithms
  • Characterizing the running times of algorithms
    and data structure operations
  • Secondarily, characterizing the space usage of
    algorithms and data structures

12
Algorithm Complexity
  • Worst Case Complexity
  • the function defined by the maximum number of
    steps taken on any instance of size n
  • Best Case Complexity
  • the function defined by the minimum number of
    steps taken on any instance of size n
  • Average Case Complexity
  • the function defined by the average number of
    steps taken on any instance of size n

Adapted from http//www.cs.sunysb.edu/algorith/le
ctures-good/node1.html
13
Best, Worst, and Average Case Complexity
Number of steps
Worst Case Complexity
Average Case Complexity
Best Case Complexity
N (input size)
Adapted from http//www.cs.sunysb.edu/algorith/le
ctures-good/node1.html
14
Monotonicity
  • monotone (m n -t n )
  • A succession of sounds or words uttered in a
    single tone of voice
  • monotonic (m n -t n k) Mathematics.
    Designating sequences, the successive members of
    which either consistently increase or decrease
    but do not oscillate in relative value.
  • A function is monotonically increasing if
    whenever agtb then f(a)gtf(b).
  • A function is strictly increasing if whenever agtb
    then f(a)gtf(b).
  • Similar for decreasing
  • Nonmonotonic is the opposite

15
Monotonic and Nonmonotonic
  • https//en.wikipedia.org/wiki/Monotonic_function

16
The Seven Common Functions
  • Constant f(n) c
  • E.g., adding two numbers, assigning a value to
    some variable, comparing two numbers, and other
    basic operations
  • Useful free web tool Wolframs Alpha

17
The Seven Common Functions
  • Linear f(n) n
  • Do a single basic operation for each of n
    elements, e.g., reading n elements into computer
    memory, comparing a number x to each element of
    an array of size n

18
The Seven Common Functions
  • Quadratic f(n) n2
  • E.g., nested loops with inner and outer loops

Brookshear Figure 5.19
19
The Seven Common Functions
  • Cubic f(n) n3

20
The Seven Common Functions
  • More generally, all of the above are polynomial
    functions
  • f(n) a0 a1n a2n2 a3n3 annd
  • where the ais are constants, called the
    coefficients of the polynomial.

Brookshear Figure 5.19
21
The Seven Common Functions
  • Logarithmic
  • f(n) logbn
  • Intuition number of times we can divide n by b
    until we get a number less than or equal to 1
  • The base is omitted for the case of b2, which is
    the most common in computing
  • log n log2n

Brookshear Figure 5.20
22
The Seven Common Functions
  • n-log-n function f(n) n log n
  • Product of linear and logarithmic

23
Comparing Growth Rates
  • 1 lt log n lt n1/2 lt n lt n log n lt n2 lt n3 lt bn

http//www.cs.pomona.edu/marshall/courses/2002/sp
ring/cs50/BigO/
24
Polynomials
  • Only the dominant terms of a polynomial matter in
    the long run.  Lower-order terms fade to
    insignificance as the problem size increases.

http//www.cs.pomona.edu/marshall/courses/2002/sp
ring/cs50/BigO/
25
Definition of Big-Oh A running time is O(g(n))
if there exist constants n0 gt 0 and c gt 0 such
that for all problem sizes n gt n0, the running
time for a problem of size n is at most
c(g(n)). In other words, c(g(n)) is an upper
bound on the running time for sufficiently large
n.
c g(n)
Example the function f(n)8n2 is O(n) The
running time of algorithm arrayMax is O(n)
http//www.cs.dartmouth.edu/farid/teaching/cs15/c
s5/lectures/0519/0519.html
26
More formally
  • Let f(n) and g(n) be functions mapping
    nonnegative integers to real numbers.
  • f(n) is ?(g(n)) if there exist positive constants
    n0 and c such that for all ngtn0, f(n) lt cg(n)
  • Other ways to say this
  • f(n) is order g(n)
  • f(n) is big-Oh of g(n)
  • f(n) is Oh of g(n)
  • f(n) ? O(g(n)) (set notation)

27
Comparing Running Times
Adapted from Goodrich Tamassia
28
Analysis Example Phonebook
  • Given
  • A physical phone book
  • Organized in alphabetical order
  • A name you want to look up
  • An algorithm in which you search through the book
    sequentially, from first page to last
  • What is the order of
  • The best case running time?
  • The worst case running time?
  • The average case running time?
  • What is
  • A better algorithm?
  • The worst case running time for this algorithm?

29
Analysis Example (Phonebook)
  • This better algorithm is called Binary Search
  • What is its running time?
  • First you look in the middle of n elements
  • Then you look in the middle of n/2 ½n elements
  • Then you look in the middle of ½ ½n elements
  • Continue until there is only 1 element left
  • Say you did this m times ½ ½ ½ n
  • Then the number of repetitions is the smallest
    integer m such that

30
Analyzing Binary Search
  • In the worst case, the number of repetitions is
    the smallest integer m such that
  • We can rewrite this as follows

Multiply both sides by
Take the log of both sides
Since m is the worst case time, the algorithm is
O(logn)
31
Hint for Homework Problem 1
  • Hint 1
  • Write a program that computes the number of gifts
    for each value of N just to get a feeling for how
    it works.
  • Hint 2
  • Last time we showed how triangles of dots prove
  •     1 2 3 ... n      n(n 1)/2
  • Similarly, to add up the number of gifts, a
    closed form version of the total numbers is
  • (n/6)(n1)(n2)

32
Summary Analysis of Algorithms
  • A method for determining, in an abstract way, the
    asymptotic running time of an algorithm
  • Here asymptotic means as n gets very large
  • Useful for comparing algorithms
  • Useful also for determing tractability
  • Meaning, a way to determine if the problem is
    intractable (impossible) or not
  • Exponential time algorithms are usually
    intractable.
  • Well revisit these ideas throughout the rest of
    the course.
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