Introduction to algorithmic problem solving

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• LECTURE 1
• Introduction to algorithmic problem solving

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Outline

• Problems solving
• What is an algorithm ?
• What properties an algorithm should have ?
• Algorithms description
• What types of data will be used ?
• What are the basic operations ?

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Problem solving

• Problem a set of questions concerning some
  entities representing the universe of the problem
• Problem statement description of the properties
  of the entities and of the relation between input
  data and problems solution
• Solving method procedure to construct the
  solution starting from the input data

Input data
Solving method
Result
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Problem solving

• Example
• Let a and b be two non-zero natural numbers.
  Find the natural number c having the following
  properties
• c divides a and b (c is a common divisor of a and
  b)
• c is greater than any other common divisor of a
  and b
• Problems universe natural numbers (a and b
  represent the input data, c represents the
  result)
• Problems statement (relations between the input
  data and the result) c is the greatest common
  divisor of a and b

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Problem solving

• Remark
• This is a problem which asks for computing a
  function (that which associates to a pair of
  natural numbers the greatest common divisor)
• Another kind of problems are those which ask to
  decide if the input data satisfy or not some
  properties.
• Example verify if a natural number is a
  prime number or not
• In both cases the solution can be obtained by
  using a computer only if it exists a method which
  after a finite number of operations produces the
  solution such a method is an algorithm

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Outline

• Problems solving
• What is an algorithm ?
• What properties an algorithm should have ?
• What types of data will be used ?
• What are the basic operations ?

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What is an algorithm ?

• Different definitions
• Algorithm something like a cooking recipe used
  to solve problems
• Algorithm step by step problem solving method
• Algorithm finite sequence of operations applied
  to some input data in order to obtain the problem
  solution

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What is the origin of the word ?

• al-Khowarizmi - Persian mathematician (9th
  century)
• algorism algorithm
• More on etymology
• algor coolness (in Latin)
• algos pain (in Greek)

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Examples

• Algorithms in day by day life
• using a phone
• pick up the phone
• dial the number
• talk .
• Algorithms in mathematics
• Euclids algorithm (it is considered to be the
  first algorithm)
• find the greatest common divisor of two numbers
• Eratostenes algorithm
• generate prime numbers in a range
• Horners algorithm
• compute the value of a polynomial

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Outline

• Problems solving
• What is an algorithm ?
• What properties an algorithm should have ?
• Algorithms description
• What types of data will be used ?
• What are the basic operations ?

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What properties an algorithm should have ?

• Generality
• Finiteness
• Non-ambiguity
• Efficiency

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Generality

• The algorithm applies to all instances of input
  data not only for particular instances
• Example
• Lets consider the problem of increasingly
  sorting a sequence of values.
• For instance
• (2,1,4,3,5)
  (1,2,3,4,5)
• input data
  result

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Generality (contd)
Description - compare the first two elements
if there are not in the desired order swap
them - compare the second and the third
element and do the same .. - continue until the
last two elements were compared

• Method

2 1 4 3 5
Step 1
1 2 4 3 5
Step 2
1 2 4 3 5
Step 3
1 2 3 4 5
Step 4
The sequence has been ordered
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Generality (contd)

• Is this algorithm a general one ? I mean, does it
  ensure the ordering of ANY sequence of values ?

• Answer NO
• Counterexample
• 3 2 1 4 5
• 2 3 1 4 5
• 2 1 3 4 5
• 2 1 3 4 5
• In this case the method doesnt work, thus it
  isnt a general sorting algorithm

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What properties an algorithm should have ?

• Generality
• Finiteness
• Non-ambiguity
• Efficiency

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Finiteness

• An algorithm have to terminate, i.e. to stop
  after a finite number of steps
• Example
• Step1 Assign 1 to x
• Step2 Increase x by 2
• Step3 If x10 then STOP
• else GO TO Step 2
• How does this algorithm work ?

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Finiteness (contd)

• How does this algorithm work and what does it
  produce?
• Step1 Assign 1 to x
• Step2 Increase x by 2
• Step3 If x10
• then STOP
• else Print x GO TO Step 2

x1
x3
x5
x7
x9
x11
The algorithm generates odd numbers but it never
stops !
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Finiteness (contd)

• The algorithm which generate all odd naturals
  smaller than 10
• Step1 Assign 1 to x
• Step2 Increase x by 2
• Step3 If xgt10
• then STOP
• else Print x GO TO Step 2

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What properties an algorithm should have ?

• Generality
• Finiteness
• Non-ambiguity
• Efficiency

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Non-ambiguity

• The operations in an algorithm must be rigorously
  specified
• At the execution of each step one have to know
  exactly which is the next step which will be
  executed
• Example
• Step 1 Set x to value 0
• Step 2 Either increment x with 1 or decrement x
  with 1
• Step 3 If x?-2,2 then go to Step 2 else Stop.
  
• As long as does not exist a criterion by which
  one can decide if
• x is incremented or decremented, the sequence
  above cannot be
• considered an algorithm.

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Non-ambiguity (contd)

• Lets modify the previous algorithm as follows
• Step 1 Set x to value 0
• Step 2 Flip a coin
• Step 3 If one obtains head
• then increment x with 1
• else decrement x with 1
• Step 3 If x?-2,2 then go to Step 2, else Stop.
  
• This time the algorithm can be executed but
  different executions can be different
• This is a so called random algorithm

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What properties an algorithm should have ?

• Generality
• Finiteness
• Non-ambiguity
• Efficiency

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Efficiency

• An algorithm should use a reasonable amount of
  computing resources memory and time
• Finiteness is not enough if we have to wait too
  much to obtain the result
• Example
• Lets consider a dictionary containing 50000
  words.
• Write an algorithm that given a word as input
  returns all anagrams of that word appearing in
  the dictionary.
• Example of an anagram ship -gt hips

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Efficiency

• First approach
• Step 1 generate all anagrams of the word
• Step 2 for each anagram search for its
  presence in the dictionary (using binary search)
• Lets consider that
• the dictionary contains n words
• the analyzed word contains m letters
• Rough estimate of the number of letters
  comparisons
• number of anagrams m!
• words comparisons for each anagram log2n
• letters comparisons for each word m
• m! mlog2n

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Efficiency

• Second approach
• Step 1 sort the letters of the initial word
• Step 2 for each word in the dictionary
  having m letters
• Sort the letters of this word
• Compare the sorted version of the word with the
  original word
• Rough estimate of the number of operations
• Sorting the initial word needs almost m2
  operations
• Sequentially searching the dictionary and sorting
  each word of length m needs at most nm2
  comparisons
• Comparing the sorted words requires at most nm
  comparisons
• n m2 nm m2

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Efficiency

• What approach is better ?
• First approach Second approach
• m! m log2n n m2 n
  m m2
• Example m12 (e.g. word algorithmics)
• n50000 (number of words in
  dictionary)
• 8 1010 8106
• one comparison 1ms10-3 s
• 24000 hours 2 hours

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Outline

• Problems solving
• What is an algorithm ?
• What properties an algorithm should have ?
• Algorithms description
• What types of data will be used ?
• What are the basic operations ?

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How can be described the algorithms ?

• The methods for solving problems are usually
  described in a mathematical language
• The mathematical language is not always adequate
  to describe algorithms because
• Operations which seems to be elementary when are
  described in a mathematical language are not
  elementary when they have to be coded in a
  programming language
• Example compute the value of a polynomial

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How can be described the algorithms ?

• There are two basic instruments to describe
  algorithms
• Flowcharts
• graphical description of the flow of processing
  steps
• they are not very often used
• however, sometimes are used to describe the
  overall structure of an application
• Pseudocode
• artificial language based on
• vocabulary (set of keywords)
• syntax (set of rules used to construct the
  languages phrases)
• not so restrictive as a programming language

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Why do we call it pseudocode ?

• Because
• It is similar to a programming language (code)
• It is not so rigorous as a programming language
  (pseudo)
• In pseudocode the phrases are
• Statements (used to describe processing steps)
• Declarations (used to specify the data)

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What types of data will be used ?

• Data container of information
• Characteristics
• name
• value
• constant (same value during the entire algorithm)
• variable (the value varies during the algorithm)
• type
• simple (numbers, characters, truth values )
• structured (array)

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What types of data will be used ?

• Arrays will be used to represent
• Sets (e.g. 3,7,43,4,7)
• the order of the elements doesnt matter
• Sequences (e.g. (3,7,4) is not (3,4,7))
• the order of the elements matters
• Matrices
• bidimensional arrays

3
7
4
3 7 4
Index 1 2 3
(1,1)
(1,2)
1
0
1
0
0
1
1
0
(2,1)
(2,2)
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How can we specify the data ?

• Simple data
• Integers INTEGER ltvariablegt
• Reals REAL ltvariablegt
• Boolean BOOLEAN ltvariablegt
• Characters CHAR ltvariablegt

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How can we specify the data ?

• Arrays
• One dimensional
• ltelements typegt ltnamegtn1..n2
• (ex REAL x1..n)
• Bi-dimensional
• ltelements typegt ltnamegtm1..m2, n1..n2
• (ex INTEGER A1..m,1..n)

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How can we specify the data ?

• Specifying elements
• One dimensional
• xi - i is the elements index
• Bi-dimensional
• Ai,j - i is the rows index, while j is the
  columns index

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How can we specify the data ?

• Specifying subarrays
• Subarray contiguous portion of an array
• One dimensional xi1..i2 (1lti1lti2ltn)
• Bi dimensional Ai1..i2, j1..j2
• (1lti1lti2ltm,
  1ltj1ltj2ltn)

1
n
1
j1
j2
i1
1
n
i2
i1
i2
m
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Outline

• Problems solving
• What is an algorithm ?
• What properties an algorithm should have ?
• Algorithms description
• What types of data will be used ?
• What are the basic operations ?

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What are the basic instructions ?

• Instruction (statement)
• action to be executed by
  the algorithm
• There are two main types of instructions
• Simple
• Assignment (assigns a value to a variable)
• Transfer (reads an input data writes a result)
• Control (specifies which is the next step to be
  executed)
• Structured .

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Assignment

• Aim give a value to a variable
• Description
• v ltexpressiongt
• Expression syntactic construction used to
  describe a computation
• It consists of
• Operands variables, constant values
• Operators arithmetical, relational, logical

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Operators

• Arithmetical
• (addition), - (subtraction), (multiplication),
  
• / (division), (power),
• DIV (integer quotient),
• MOD (remainder)
• Relational
• (equal), ! (different),
• lt (less than), lt (less than or equal),
• gt(greater than) gt (greater than or equal)
• Logical
• OR (disjunction), AND (conjunction), NOT
  (negation)

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Input/Output

• Aim
• get the input data
• output the results
• Description
• read v1,v2,
• write e1,e2,

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What are the instructions ?

• Structured
• Sequence of instructions
• Conditional statement
• Looping statement

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Conditional statement

• Aim allows choosing between two or many
  alternatives depending on the value of a/some
  condition(s)

• General variant
• if ltconditiongt then ltS1gt
• else ltS2gt
• endif
• Simplified variant
• if ltconditiongt then ltSgt
• endif

True
False
condition
ltS1gt
ltS2gt
indentation
True
False
condition
ltSgt
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Loop statements

• Aim allows repeating a processing step
• Example compute a sum
• S 12in
• A loop is characterized by
• The processing step which have to be repeated
• A stopping (or continuation) condition
• Depending on the moment of analyzing the stopping
  condition there are two main loop statements
• Preconditioned loops (WHILE loops)
• Postconditioned loops (REPEAT loops)

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WHILE loop

• First, the condition is analyzed
• If it is true then the statement is executed and
  the condition is analyzed again
• If the condition becomes false the control of
  execution passes to the next statement in the
  algorithm
• If the condition never becomes false then the
  loop is infinite
• If the condition is false from the beginning then
  the statement inside the loop is not at all
  executed

False
ltconditiongt
Next statement
True
ltstatementgt
while ltconditiongt do
ltstatementgt endwhile
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FOR loop

• Sometimes the number of repetitions of a
  processing step is known
• Then we can use a counting variable which varies
  from an initial value to a final value using a
  step value
• Repetitions v2-v11 if step1

vv1
False
v lt v2
Next statement
True
ltstatementgt
vvstep
vv1 while vltv2 do ltstatementgt vvstep endwh
ile
for vv1,v2,step do
ltstatementgt endfor
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REPEAT loop

• First, the statement is executed. Thus it is
  executed at least once
• Then the condition is analyzed and if it is false
  the statement is executed again
• When the condition becomes true the control
  passes to the next statement of the algorithm
• If the condition doesnt become true then the
  loop is infinite

ltstatementgt
ltconditiongt
True
Next statement
repeat ltstatementgt until ltconditiongt

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Next lecture will be on

• Simple examples
• What about not so simple examples subalgorithms