Algorithms

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Algorithms
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Algorithms

• How to explain to a dumb mechanical/computing
  device how to solve a problem?
• How to solve a problem using basic set of
  primitive instructions.
• Complexity of solving problems.

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Goals

• To develop framework for instructing computers to
  perform tasks.
• To introduce notion of algorithm as means of
  specifying how to solve a problem.
• To introduce and appreciate approaches for
  defining and solving very complex tasks in terms
  of simpler tasks.

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• Computer Science can be considered as the study
  of algorithms including
• Their formal properties
• Their hardware and software realisations
• Their applications

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Real Life Examples

• Cook Recipe for preparing a dish
• Paper Toy
• Directions How to go to Nyugati railway station

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Real Life Examples

• Imprecise Instructions
• Job can often be done even if the instructions
  are not followed precisely
• Modifications may be done by the person following
  the instructions
• Computers need to be told in precise manner what
  to do. Instructions cannot be vague or ambiguous.

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Algorithm Definition

• An algorithm for solving a problem is a finite
  sequence of unambiguous, executable
  steps/instructions, which if followed would
  ultimately terminate and give the solution of the
  problem.
• Finite set of steps
• Unambiguous
• Executable
• Terminate

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Operations of Algorithms

• sequence of steps/instructions
• Iteration repeating task
• Selection the continue of task depend on a
  condition

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Main description of Alg.

• Flow Charts
• PseudoCode

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PseudoCode

• Somewhere between human language such as
  English and Computer languages.
• Precise enough to describe what is meant
  without being tedious.

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Flow Charts

• Flow Charts Defined
• A Flow charts is a pictorial representation
  showing all the step of a process

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Creating a Flow Chart

• First, familiarize the participants with the flow
  chart symbols.
• Draw the process flow chart and fill it out in
  detail about each element.
• Analyze the flow chart. Determine which steps
  add value and which dont in the process of
  simplifying the work.

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Flow chart techniques

• Basic symbols
• Terminator
• Process
• Decision
• Predefined process
• On-page connector
• Connector

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Easy example
I have an idea
I raise my hand
Did my teacher call my name?
no
Wait
yes
Share my idea out loud
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Example Adding two numbers

• Input Two positive m digit decimal numbers
• am, am-1, ., a1
• bm, bm-1, ., b1
• Output cm1, cm, ., c1

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• Step 1 Set the value of carry to 0
• Step 2 Set the value of i to 1.
• Step 3 Repeat steps 4 through 6 until the value
  of i is gt m.
• Step 4 Add ai and bi to the current value of
  carry, to get xi.
• Step 5 If xi lt 10, then let cixi and reset
  carry to 0.
• Else (i.e. xi ? 10)
• let cixi -10 and reset carry to 1.
• Step 6 Increase the value of i by 1.
• Step 7 Set cm1 to the value of carry.
• Step 8 Print the final answer cm1, cm, ., c1
• Step 9 Stop.

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Iteration

• For Instruction
• For x ? a to b Do
• ..
• ..
• Endfor
• Execute the instructions .., first using xa,
  then using xa1, , and finally using xb.

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Example Divisibility

• A positive integer n is divisible by m if for
  some positive integer i?n, m ? i n

Is n divisible by m?
Yes
No
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• Divisible(m,n)
• 1. For i?1 To n Do
• If m?in, Then Output True and Stop
• Else do nothing (continue with next
  i).
• EndFor
• 2. Output False and Stop
• End

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• To communicate the algorithm to computer
• need a way to represent the algorithm
• Syntax --- representation
• Semantics --- meaning
• complex task --- Programming Languages
• Need more English Like

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Subprogram

• Problems already shown to be doable, can be used
  as primitive (already known) instruction
• Simplifies the process of writing
  algorithms/programs
• Modularity, Abstraction, division of labour
• Complex tasks can be divided and each part solved
  separately and combined later
• Subroutines, procedures, modules, etc.

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Example Prime Numbers

• A number n is a prime iff it has no positive
  factors except for 1 and n.
• In other words, n is not divisible by any
  positive number m such that 1 lt m lt n.

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• Prime(n)
• 1. For m?2 To n-1 Do
• If Divisible(m,n) Then Output False
• and Stop
• Else do nothing (continue with next
  m).
• EndFor
• 2. Output True and Stop
• End

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Recursion

• As a means of solving bigger problems using
  smaller subproblems/subparts which may be of same
  kind.
• Solve simplest cases directly
• Solve bigger problems using smaller (simpler)
  subproblems.
• Abstraction.

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Example Fibonacci Numbers

• F1 1
• F2 1
• F3 2
• F4 3
• ..
• Fn Fn-2Fn-1

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• Fibonacci(n)
• If n1 or n2, Then Fibonacci(n) 1
• Else Fibonacci(n)Fibonacci(n-1)Fibonacci(n-2)
• End

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Example Tower of Hanoi

• Three Pegs A, B and C
• Peg A initially has n disks, all of different
  size, stacked one over another
• Larger disks are below smaller disks
• Problem is to move the n disks to Peg C
• Can move only one disk at a time
• Smaller disk should be above larger disk
• Can use Peg B as intermediate

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Example Tower of Hanoi

• Example for 3 disks (disk 3 is largest, disk 1 is
  smallest)
• (1) Move disk 1 from Peg A to Peg C.
• (2) Move disk 2 from Peg A to Peg B.
• (3) Move disk 1 from Peg C to Peg B.
• (4) Move disk 3 from Peg A to Peg C.
• (5) Move disk 1 from Peg B to Peg A.
• (6) Move disk 2 from Peg B to Peg C.
• (7) Move disk 1 from Peg A to Peg C.

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• Hanoi(n,A,B,C)
• ( Move top n disks from A to C via B )
• 1. If n1, Then move top disk on Peg A to Peg C.
• 2. If ngt1, Then,
• (a) Hanoi (n-1,A,C,B)
• ( That is move top n-1 disks on Peg A to Peg
  B using Peg C as intermediate Peg. )
• (b) Move the current top disk on Peg A to Peg C.
• (c) Hanoi(n-1,B,A,C)
• ( That is move top n-1 disks on Peg B to Peg C
  using
• Peg A as intermediate Peg. )
• End

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Searching

• 13, 28, 19, 74, 76, 14, 12, 18, 22, 55

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Searching

• List of numbers A1, A2, A3, ., An
• A number x
• Question Is x in the list?
• Sequential Search(x, A1, A2, A3, ., An )
• For i?1 To n Do
• If xAi then output Yes and Stop.
• Else do nothing (continue with next i).
• EndFor
• Output No and Stop
• End

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Binary Search

• List is sorted.
• That is A1? A2 ? A3 ? . ? An
• Then we can do better.

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Binary Search

• 1, 4, 9, 11, 14, 43, 78

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Binary Search

• Input Sorted List A1? A2 ? A3 ? . ? An
• A number x.
• Question Is x in the list.
• 1. First 1
• Last n
• 2. While First ? Last
• mid ? (firstlast)/2?
• If xAmid Then output Yes and Stop
• Else If xlt Amid Then Lastmid-1
• Else If x gt AmidThen Firstmid1
• EndWhile
• 3. Output False and Stop
• End

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Time Complexity

• Sequential Search
• Worst Case n comparisons
• Best Case 1 comparison
• Average Case n/2 comparisons
• Binary Search
• Worst Case approximately log2 n comparisons
• Best Case 1 comparisons
• Average Case approximately log2 n-1
  comparisons

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Characteristics of Algorithm

• Correctness
• Complexity --- time, space (memory), any other
  measure
• Ease of understanding
• Ease of coding
• Ease of maintainence

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Logo?

• Full-featured computer programming language
  derived from LISP
• A language for learning
• A tool to teach the process of learning and
  thinking
• An environment where students assume the role of
  teacher

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Sequences

• forward 50 right 90 forward 50right
  90forward 50right 90forward 50right 90

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Iteration

• repeat 4 forward 50 right 90

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Procedures

• to squarerepeat 4 forward 50 right 90end

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Procedures

• You could draw a flag
• forward 60squareback 60

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Further Iteration

• You could make a circle of squaresrepeat
  12square right 30

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Procedures

• to house square forward 50 right
  90 triangleend
• to trianglerepeat 3 forward 50 left 120end

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Passing Values into Procedures

• to square sizerepeat 4 forward size right
  90end
• square 10
• square 20 etc

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Recursion

• to design clearscreen right 30 polyspi 5
  120end
• to polyspi size angleif size gt 205
  stop forward size right angle polyspi
  size5 angle.12end