Estimating the Efficiency

1
Chapter 3

• Estimating the Efficiency
• and
• Randomness

2
Outline

• Software-development Life Cycle
• Skip section 3.2 3.4.2
• Big-O notation (3.4.3 3.4.5)
• Growth Rates
• clock() function
• time() function
• Random Number Generator (3.4.8)
• rand() function
• srand() function

3
Software-development Life Cycle

• Problem Analysis
• Determine what is to be done
• System Tests
• Sample input values are created and the
  corresponding output produced by hand
• Program Design
• What classes will be needed to solve the problem
  and how those classes are related
• Program Implementation
• Determine the correctness and efficiency of each
  of the classs methods

4
Mathematical Definition f(n) O(g(n))

• We say that f(n)O(g(n)) if there are positive
  constants C and K such that f(n) ? Cg(n) when n
  ? K.
• Here, we are concerned with n being positive real
  numbers (usually positive integers)
• Examples
• f(n) n O(n) with g(n) n, C 1, K 1
• f(n) 10n O(n) with g(n)n, C10, K 1
• Note that f(n) 10n O(n2) with g(n) n2, C
  1, K 10
• Also, f(n) 10n O(n2) with g(n) n2, C 2, K
  5
• Hence, g(n), C and K are not unique

5
Big-O notation

• g(n) is an upper bound for f(n)
• We say that f(n) is no larger than g(n)
• We say f(n) is of order of g(n)
• We say f(n) is Big-O of g(n), i.e., f(n)
  O(g(n))
• We say the growth rate of f(n) is less than or
  equal to the growth rate of g(n)

6
Graphic Representation of f(n) O(g(n))
K
7
Any Algorithm that is O(n2) is also O(n3)
K
K
8
Read Section 3.4.4 and 3.4.5

• log (n) The number of times to divide n by 2
  until n lt 1 is log2n
• worstTime(n) Analysis
• Examples 1 through 5 Text pages 77 81

9
Common Functions
10
Growth Rates
worstTime(n)
O(2n)
O(n2)
O(n log n)
O(n)
O(log n)
O(1)
n
11
Example of Run Times
Assume 1 operation per microsecond (0.000001)
12
Examples
Give the asymptotic notation (O-notation) of the
following functions and rearrange them in
ascending order by rate of growth. (Hint
)
13
clock() function

• clock() This function returns the number of
  clock ticks (the CPU time taken) the program has
  taken. To convert to the number of seconds,
  divide by CLOCKS_PER_SEC, a constant that is
  defined in time.h
• To determine the amount of time a section of code
  takes to run,
• include lttime.hgt//include ltstdlib.hgt
  //alternativefloat timertimer clock()
  //clock() returns the time in terms of ticks
• // Perform tasktimer clock() - timer
  //Gives the elapsed time in tickstimer /
  CLOCKS_PER_SEC //Converts ticks to
  secondstimer / 60 //Converts
  seconds to minutes

14
time() function

• To determine the amount of time (in seconds) a
  section of code takes to run, use the following
  code
• includelttime.hgt// include ltstdlib.hgt //
  alternative long timer time(NULL)
• //Perform tasktimer
  time(NULL) - timer //Gives the elapsed time in
  seconds

15
Random Number Generator
16
rand()

• Method is in the header file
• stdlib.h or time.h
• The header file has a compiler-dependent constant
  RAND_MAX
• The following program will generate a random
  number between 0 and RAND_MAX (0x7FFFFFFF,
  2,147,483,647)

• include ltiostreamgt
• include ltstdlib.hgt
• using namespace std
• int main ( )
• cout ltlt rand()
• return 0

17
rand()

• To find a random number between 0 and some int
  high, use the remainder portion of divide.
• The following program will generate a random
  number between 1 and 100

• include ltiostreamgt
• include ltstdlib.hgt
• using namespace std
• int main ( )
• int high 100
• cout ltlt rand() high 1 ltlt endl
• return 0

18
rand()

• Unfortunately, every execution of the program
  results in the same random number.
• The function depends on an initial value, called
  a seed.
• There is a function which allows the initial
  value to be changed.
• srand(seed)
• where seed is an integer.
• The following program will generate a random
  number between 1 and 100 depending on the
  inputted integer.

• include ltiostreamgt
• include ltstdlib.hgt
• using namespace std
• int main ( )
• int high100
• int seed
• cin gtgt seed
• srand(seed)
• cout ltlt rand() high 1 ltlt endl
• return 0

19
time()

• include ltiostreamgt
• include ltstdlib.hgt
• // include lttime.hgt
• // alternative
• using namespace std
• int main ( )
• int high 100
• long seed time(NULL)
• srand(seed)
• cout ltlt rand() high 1
• return 0

• A better method would be to use something that is
  constantly changing, time
• could use the header file time.h instead of
  stdlib.h
• Contains a function
• time()
• returns a long value representing the time
  elapsed in seconds since 01/01/1970.
• The following program will generate a different
  random number between 1 and 100 each time it is
  run (unless you run it twice in a second).

Text P85 (lab 5) generates a sequence of
pseudo-random numbers