List ADTs

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Algorithm Analysis
Capt Balazs Michelson 366
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Algorithm Analysis

• In other words Analysis of Algorithms
• Analysis examination of a complex, its elements,
  and their relations
• Algorithm step-by-step procedure for solving a
  problem or accomplishing some end
• Both from Mirriam-Webster dictionary

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Algorithm

• Well defined computational procedure
• May take some value or set of values as input
• Produces some value or set of values as output
• Sequence of computational steps to transform
  input to output
• Tool for solving a well-specified (computational)
  problem

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Algorithm Analysis

• Were going to look at
• How to analyze measure efficiency of algorithms
• How to compare two algorithms (both solving same
  problem) in terms of their efficiency

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Better Algorithms

• Robust able to deal with unexpected or
  erroneous inputs
• Time how fast it is
• Space how much memory or bandwidth it takes
• Ease how easy it is to implement, understand,
  or modify

Where well focus
Hard to measure
Easy to measure
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Algorithm

• Well defined problem

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• We have two algorithms, we can only use one
• Choose the better one
• What does it mean to be better?

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Time Analysis of alg1 alg2

• We use System.currentTimeMillis() to figure out
  how long the algorithms take

k
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alg1 alg2

• The execution times are not constant
• Both grow as k increases.
• For given algorithm, the running time is a
  function of input (in this case k)
• Shape of graph is important
• Rate at which it grows as k increases
• Growth rate of an algorithms run-time function
  is very important

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Time Analysis

• If we run algorithms on different computer we
  will get different times
• Processor speed, operating system, memory, etc.
• So run time wont really tell us an algorithms
  speed We need a better way

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So, How to Analyze Time?

• Time to execute a basic step varies from
  machine to machine
• We cant use execution time
• What can we use to tell how fast an algorithm is
  in terms that transcend individual machines

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Basic Steps

• Basic Step primitive operation or a set of
  operations that typically take same amount of
  time
• assigning values, method calls, arithmetic,
  comparison, array index, following a
  pointer/reference, returning from method, etc.
• We have some flexibility here
• One person's basic step is another person's
  multiple steps
• On a basic level, taking the not of a boolean is
  one step because it involves one bit, but adding
  two 32-bit numbers is 32 (or more!) steps
• The important thing is Be consistent in what a
  step is

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Counting Steps
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• The number of steps is a function of the input
  size (n)

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Function of Input

• Not only is number of steps dependant on size of
  input, but also the input itself.
• Ex Search for an element in an array.
• if element is at the front, then 1step
• if element not present or at the end, then n
  steps
• Same algorithm, same input size, different input
  yields different behaviors
• We can talk about best case, worst case and
  average case running times.

click
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Function of Input

• Can compare two methods
• f(n) 12n 22
• g(n) n2 7

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Linear vs. Quadriatic

• g(n) is smaller than f(n) for small values of n
  (up to about n 14), but eventually g(n)
  overtakes f(n)
• Comparing linear function to quadratic quadratic
  function will eventually overtake (dominate)
  linear function regardless of coefficients and
  constants
• We dont care much about small values of n, were
  interested in what happens when n gets extremely
  large

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f_alg1
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f_alg2
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Comparison

• alg_1 takes k2 6k 19 steps. While alg_2
  takes 5k 13 steps
• We are interested in long-term growth rate of the
  functions
• we can ignore constants and lower order terms
• alg_2 is said to have a linear growth rate
• alg_1 is said to grow like k2

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Common growth rates
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Algorithm Analysis

• For a given algorithm, the running time is a
  function of the input.
• The growth rate of an algorithms run-time
  function is very important.
• It doesnt matter what the constant coefficients
  are for a given function (i.e., linear vs.
  quadratic). There will always be a point where
  the function with the faster growth rate will
  dominate the function with the slower growth
  rate.
• Since we are interested in the long-term growth
  rate of the functions we can ignore the constants
  and lower order terms
• Were typically interested in worst case bounds.