Efficiency of Algorithms

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Efficiency of Algorithms

• Csci 107
• Lecture 5

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• Last time
• Algorithms for
• Find all occurences of target
• Find number of occurences of target
• Find number of values larger than target
• Find largest /smallest, sum, average
• Pattern matching
• Today
• Pattern matching algorithm
• Efficiency of algorithms
• Data cleanup algorithms
• Reading start on Chapter 3, textbook

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Pattern Matching

• Problem Suppose we have a gene (text) T
  TCAGGCTAATCGTAGG and a probe (pattern) P TA.
  Design an algorithm that searches T to find the
  position of every instance of P that appears in
  T.
• E.g., for this text, the algorithm should return
  the answer
• There is a match at position 7
• There is a match at position 13
• Algorithm
• What is the idea?
• Check if pattern matches starting at position 1,
  then check if it matches starting at position
  2,and so on
• How to check if pattern matches text starting at
  position k?
• Check that every character of pattern matches
  corresponding character of text

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Pattern Matching

• Input
• Gene (text) of n characters T1, T2, , Tn
• Probe (pattern) of m (m lt n) characters P1, P2,
  Pm
• Output
• Location (index) of every occurrence of pattern
  within text
• Algorithm idea
• Get input (text and pattern)
• Set starting location k to 1
• Repeat until reach end of text
• Attempt to match every character in the pattern
  beginning at pos k in text
• If there was a match, print k
• Add 1 to k
• Stop

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Comparing Algorithms

• Algorithm
• Design
• Correctness
• Efficiency
• Also, clarity, elegance, ease of understanding
• There are many ways to solve a problem
• Conceptually
• Also different ways to write pseudocode for the
  same conceptual idea
• How to compare algorithms?

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Efficiency of Algorithms

• Efficiency Amount of resources used by an
  algorithm
• Space (number of variables)
• Time (number of instructions)
• When designing an algorithm must be aware of its
  use of resources
• If there is a choice, pick the more efficient
  algorithm!

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Efficiency of Algorithms

• Does efficiency matter?
• Computers are so fast these days
• Yes, efficiency matters a lot!
• There are problems (actually a lot of them) for
  which all known algorithms are so inneficient
  that they are impractical
• Remember the shortest-path-through-all-cities
  problem from Lab1

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Efficiency of Algorithms

• How to measure time efficiency?
• Running time let it run and see how long it
  takes
• On what machine?
• On what inputs?
• Time efficiency depends on input
• Example the sequential search algorithm
• In the best case, how fast can the algorithm
  halt?
• In the worst case, how fast can the algorithm
  halt?

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Time Efficiency

• We want a measure of time efficiency which is
  independent of machine, speed etc
• Look at an algorithm pseudocode and estimate its
  running time
• Look at 2 algorithm pseudocodes and compare them
• (Time) Efficiency of an algorithm
• the number of pseudocode instructions (steps)
  executed
• Is this accurate?
• Not all instructions take the same amount of
  time
• But..Good approximation of running time in most
  cases

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(Time) Efficiency of an algorithm
worst case efficiency is the maximum number of
steps that an algorithm can take for any input
data values. best case efficiency is the
minimum number of steps that an algorithm can
take for any input data values. average case
efficiency -the efficiency averaged on all
possible inputs - must assume a distribution of
the input - we normally assume uniform
distribution (all keys are equally probable)
If the input has size n, efficiency will be a
function of n
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Analysis of Sequential Search

• Time efficiency
• Best-case 1 comparison
• target is found immediately
• Worst-case 3n 5 comparisons
• Target is not found
• Average-case 3n/24 comparisons
• Target is found in the middle
• Space efficiency
• How much space is used in addition to the input?

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Worst Case Efficiency for Sequential Search

• Get the value of target, n, and the list of n
  values 1
• Set index to 1 1
• Set found to false 1
• Repeat steps 5-8 until found true or index gt
  n n
• 5 if the value of listindex target then n
• Output the index 0
• Set found to true 0
• 8 else Increment the index by 1 n
• 9 if not found then 1
• 10 Print a message that target was not found 0
• Stop 1
• Total 3n5

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Order of Magnitude

• Worst-case of sequential search
• 3n5 comparisons
• Are these constants accurate? Can we ignore them?
  
• Simplification
• ignore the constants, look only at the order of
  magnitude
• n, 0.5n, 2n, 4n, 3n5, 2n100, 0.1n3 .are all
  linear
• we say that their order of magnitude is n
• 3n5 is order of magnitude n 3n5 ?(n)
• 2n 100 is order of magnitude n 2n100?(n)
• 0.1n3 is order of magnitude n 0.1n3?(n)
• .

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Data Cleanup Algorithms
What are they? A systematic strategy for
removing errors from data. Why are they
important? Errors occur in all real computing
situations. How are they related to the search
algorithm? To remove errors from a series of
values, each value must be examined to determine
if it is an error. E.g., suppose we have a list d
of data values, from which we want to remove all
the zeroes (they mark errors), and pack the good
values to the left. Legit is the number of good
values remaining when we are done.
5 3 4 0 6 2 4 0
d1 d2 d3 d4 d5 d6 d7
d8
Legit
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Data Cleanup Copy-Over algorithm

• Idea Scan the list from left to right and copy
  non-zero values to a new list
• Copy-Over Algorithm (Fig 3.2)
• Variables n, A1, , An, newposition, left,
  B1,,Bn
• Get values for n and the list of n values A1, A2,
  , An
• Set left to 1
• Set newposition to 1
• While left lt n do
• If Aleft is non-zero
• Copy A left into B newposition
• (Copy it into position newposition in new list
• Increase left by 1
• Increase newposition by 1
• Else increase left by 1
• Stop

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Data Cleanup The Shuffle-Left Algorithm

• Idea
• go over the list from left to right. Every time
  we see a zero, shift all subsequent elements one
  position to the left.
• Keep track of nb of legitimate (non-zero) entries
  
• How does this work?
• How many loops do we need?

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Shuffle-Left Algorithm (Fig 3.1)

• Variables n, A1,,An, legit, left, right
• Get values for n and the list of n values A1, A2,
  , An
• Set legit to n
• Set left to 1
• Set right to 2
• Repeat steps 6-14 until left gt legit
• 6 if Aleftt ? 0
• 7 Increase left by 1
• 8 Increase right by 1
• 9 else
• 10 Reduce legit by 1
• Repeat 12-13 until right gt n
• Copy Aight into Aright-1
• Increase right by 1
• 14 Set right to left 1
• 15 Stop

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Exercising the Shuffle-Left Algorithm
5 3 4 0 6 2 4 0
d1 d2 d3 d4 d5 d6 d7
d8
legit
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Data Cleanup The Converging-Pointers Algorithm

• Idea
• One finger moving left to right, one moving right
  to left
• Move left finger over non-zero values
• If encounter a zero value then
• Copy element at right finger into this position
• Shift right finger to the left

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Converging Pointers Algorithm (Fig 3.3)

• Variables n, A1,, An, legit, left, right
• Get values for n and the list of n values A1,
  A2,,An
• Set legit to n
• Set left to 1
• Set right to n
• Repeat steps 6-10 until left right
• If the value of Aleft?0 then increase left by 1
• Else
• Reduce legit by 1
• Copy the value of Aright to Aleft
• 10 Reduce right by 1
• if Aleft0 then reduce legit by 1.
• Stop

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Exercising the Converging Pointers Algorithm
5 3 4 0 6 2 4 0
d1 d2 d3 d4 d5 d6 d7
d8
legit