Algorithm Discovery and Design

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Algorithm Discovery and Design
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Outline

• Representing Algorithms
• Examples of Algorithmic Problem Solving

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Representing Algorithm
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How to Express Algorithms Clear, Precise, and
Unambiguous

• Natural language
• Extremely verbose, with the resulting algorithms
  ending up as rambling, unstructured paragraphs
  that are hard to follow
• Too rich in interpretation and meaning
• Formal programming language like C, C, Java
• During the initial phase of algorithm design, we
  should be thinking and writing at a highly
  abstract level
• Using a programming language forces us to deal
  immediately with such low-level language issues
  as punctuation, grammar, and syntax
• Pseudo Code
• Compromise between natural language and formal
  programming language
• Translation of algorithms to programming language
  is relatively simple

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Use Natural Language to Express the Addition
Algorithm
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Use Java to Express the Addition Algorithm (Part)
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How to Write Pseudo Code

• Pseudo Code for
• Sequential operations
• Computation
• Input
• Output
• Conditional and iterative operations (control
  operations)

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Computation

• Set the value of variable to arithmetic
  expression
• A named variable is simply a named storage
  location that can hold a data value
• A variable is like a mailbox into which one can
  store a value and from which one can retrieve a
  value
• Examples
• Set the value of carry to 0
• Set the value of Area to (?r2)
• Add the two digits ai and bi to the current value
  of carry to get ci
• Set the value of ci to (ai bi carry)
• Add 1 to I, effectively moving one column to the
  left
• Set the value of I to (I1)
• Set cm to the value of carry
• Set the value of cm to carry

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Variable
carry
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Set the value of carry to 0
carry
5
0
I
4
415
Set the value of I to I1(or II1)
I
4
5
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Input/Output

• Input receive from the outside world data values
  that it may then use in later instructions
• Get values for variable, variable,
• Get a value for r, the radius of the circle
• Output send results to outside world for display
• Print the values of variable, variable,
• Print the value of area
• The output operation can be used to display a
  message rather than the desired results
  (something wrong during the computation)
• Print the message sorry, no answers were
  computed

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Input/Output
Input
ComputingAgent
Outside World
Output
Input Equipments keyboard, mouse Output
Equipments screen, printer
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Average Miles Per Gallon Algorithm (Version 1)
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Conditional Operations

• Conditional operations question-asking
• Allow an algorithm to ask a question and, on the
  basis of the answer to that question, to select
  the next operation to perform
• if a true/false condition is true then First
  set of algorithm operationselse (or
  otherwise) Second set of algorithm operations
• Example
• if (ci gt 10) then set the value of ci to (ci -
  10) set the value of carry to 1else set the
  value of carry to 0

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Average Miles Per Gallon Algorithm (Version 2)
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Iteration (Loop) Operations

• Iteration operations implement the repetition of
  a block of operations
• The real power of a computer comes from doing a
  calculation many, many times
• Repeat step i to step j until a true/false
  condition becomes true
• Step i operation
• Step i1 operation
• Step j operation

Termination Condition
Loop Body
This iterative primitive assumes the loop body
will always be done at least once. The first test
of termination is not done until after the
completion of the first iteration
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Example of A Loop

• Set the value of count to 1
• Repeat step 3 to 5 until (count gt 100)
• set square to (count count)
• print the value of count and square
• add 1 to count

?Loop??????Termination Condition?????????,???Loop?
???

• 1
• 4
• 9
• 100 10000

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Average Miles Per Gallon Algorithm (Version 3)
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Variation of Iterative Operations (I)

• Repeat until a true/false condition becomes
  true Operation OperationEnd of the loop

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Variation of Iterative Operations (II)

• While a true/false condition remains
  true Operation OperationEnd of the loop
• The following three iterative operations are
  nearly equal
• Repeat until the sum is greater than 100
• While the sum is not greater than 100 do
• While the sum is less than or equal to 100 do

Continuation Condition
Loop Body

• A while loop can execute 0, 1, or more times (the
  loop condition is checked immediately rather than
  at the end of the first iteration)
• A repeat loop must execute at least once

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(No Transcript)
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Examples of Algorithmic Problem Solving

• Trick 1?????,???,?????(?????)??????,??????

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Example 1 Looking, Looking, Looking (Sequential
Search)

• Purpose searching for a particular persons name
  in a non-alphabetized telephone book
• Name Telephone NumberN1 T1N2 T2 N1000
  0 T10000
• Input A NAME to be searched
• Output
• If NAME is found at position j, then output the
  telephone number of NAME, TJ
• If not, then output I am sorry but this name is
  not in the directory

10,000 (name, phone number) pair
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Example
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First Attempt at Designing a Sequential Search
Algorithm

• ??????????????(index)?
• Trick 2 ???Iteration?(1) ??????????(2)
  ???Iteration??? ????????,? ?????????? ??

No iteration is used
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Second Attempt at Designing a Sequential Search
Algorithm
What if a desired NAME does not appear anywhere
in the list
?????Iteration?????????,??????????
??Iteration?????????
A variable called i is used as an index (pointer)
to the list of all names.That is, Ni refers to
the ith name in the list
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Sequential Search Algorithm
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Notes about Looking, Looking, Looking

• How do you search a telephone book by hand?
• A telephone book is usually alphabetized (sorted)
• Binary Search may be better
• But the telephone book used here is
  non-alphabetized
• The selection of an algorithm to solve a problem
  is greatly influenced by the way the data are
  organized

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Example 2 Big, Bigger, Biggest

• Purpose search for the numerically largest value
  in the entire list
• Given a list of examinations, which student has
  the highest score
• Given a list of annual salaries, which person
  earns the most money
• Given a value n gt 2 and a list containing n
  unique numbers called A1, A2,, An, find and
  print out both the largest value in the list and
  the position in the list where that largest value
  occurred
• Example list 19, 41, 12, 63, 22 (n5)

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Find the Largest by Hand (I)
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• ???????,????????????
• ??????????,???????? (i.e. ??????????????)

The Pile
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19
1
The largest so far
The largest position
The Pile
12
41
2
The largest so far
The largest position
The Pile
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Find the Largest by Hand (II)
63
4
63
The largest so far
The largest position
The Pile
4
63
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The largest so far
The largest position
The Pile
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Find Largest Algorithm
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Example 3 Meeting Your Match (Pattern Match)

• You will be given some text composed of n
  characters that will be referred to as T1T2Tn.
  You will also be given a pattern of m characters,
  mltn, that will be represented as P1P2Pm. The
  algorithm must locate every occurrence of the
  pattern within the text. The output of the
  algorithm is the location in the text where each
  match occurred. For this problem, the location of
  a match is defined to be the index position in
  the text where the match begins
• Text to be or not to be, that is the
  questionPattern toOutput matches starting at
  position 1 and 14

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??????
Text We must band together and handle
adversityPattern and
We must band together and handle adversityand
We must band together and handle adversity and
We must band together and handle adversity and
We must band together and handle adversity and
We must band together and handle adversity
and
We must band together and handle adversity
and
We must band together and handle adversity
and
We must band together and handle adversity
and
We must band together and handle adversity
and
We must band together and handle adversity
and
We must band together and handle adversity
and Match!!!!
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Pattern Matching Algorithm

• The pattern-matching algorithm is composed of two
  parts
• The pattern is aligned under a specific position
  of the text, and the algorithm sees whether there
  is a match at that given position
• T1T2T3T4T5T6P1P2P3
• The pattern is slided ahead one character
  position
• T1T2T3T4T5T6 P1P2P3

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First Draft of the Pattern Match Algorithm

• Abstraction Use of high-level instructions
  during the initial design process
• Top-down design

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Go to the Primitives(I)

• Attempt to match every character in the pattern
  beginning at position K of the text
• Text T1T2TkTk1Tk2Tk(m-1)Pattern
  P1P2 P3 Pm
• Compare P1 to TkCompare P2 to Tk1Compare P3 to
  Tk2Compare Pm to Tk(m-1)

• If all the pairs are equal ? match
• Any one pair is not equal ? mismatch(No need to
  compare further)

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Go to the Primitives(II)

• Repeat until we have fallen off the end of the
  text
• Text T1T2 Tn-m1 Tn-2Tn-1Tn
  Pattern P1 Pm-2Pm-1Pm
• Repeat until k gt (n-m1)

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Pattern Matching Algorithm
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Pattern-Match Tracing
assignment is equal to?
Text We must band together and handle
adversity Pattern and
n 42 m3 ? (n-m1) 40 k1
Repeat until k gt (n-m1) 1 gt 40 i1
MismatchNo Repeat until either (i gt m) or
(MismatchYes) (1 gt 3), MismatchNo Pi
Tk(i-1) ? p1T1(1-1)? ? a W? ?
MismatchYes Exit the Repeat Loop Mismatch
No? Mismatch Yes ? Do not execute the two
Print Increment k by 1 ? set k to 2 Go back to
test the termination condition of the
outer-repeat loop and re-enter the loop
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Pattern-Match Tracing (Cont.)
Suppose now k 10 (i.e., we start comparing
the and in band Repeat until k gt (n-m1) 10 gt
40 i1 MismatchNo Repeat until either (i
gt m) or (MismatchYes) (1 gt 3), MismatchNo
Pi Tk(i-1) ? p1T10(1-1)? ? a a ?
Increment i by 1 (i2) Test the termination
condition of the inner-repeat and re-enter the
body Pi Tk(i-1) ? p2T10(2-1)? ? n
n ? Increment i by 1 (i3) Test the
termination condition of the inner-repeat and
re-enter the body Pi Tk(i-1) ?
p3T10(3-1)? ? d d ? Increment i by 1
(i4) Test the termination condition of the
inner-repeat and find (4 gt 3). Do notre-enter
the body Mismatch No? Mismatch No ?
Execute the two Print (print k10) Increment
k by 1 ? set k to 11 Go back to test the
termination condition of the outer-repeat and
re-enter the body