programming fundamentals 9 th lecture szabolcs papp

Programming Fundamentals

9th lecture

Szabolcs Papp

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More Patterns of Algorithms (PoA)

Copy – calculation with function Multiple item selection Partitioning Intersection Union

PoAs – summary

Content

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Further PoAs

What is Pattern of Algorithm (PoA)?It is a general solution for a typical programming task.

Array single valueArray ArrayArray ArraysArrays Array

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7. Copy calculation with

function

Example tasks: Let's define the absolute values of

all elements in an array! Let's convert all letters of a text

to lowercase! Let's calculate the sum of two

vectors! Let's calculate sin(x) values for a

given x series! We know the ordinal number of N

months, let's give their names!

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function

What is in common? We have N something, and we have to assign N other things to these. The type of assigned values can differ from the type of original values, but the count (N) remains the same, as well as the order.

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function

Specification: Input: N:Integer,

X:Array[1..N:Type1]

Output: Y:Array[1..N:Type2] Precondition: N0 Post condition: i (1≤i≤N):

Y[i]=f(X[i])

Algorithm:i=1..N

Y[i]:=f(X[i])

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function

Specification (a frequent special case): Input: N:Integer,

X:Array[1..N:Type1]

Output: Y:Array[1..N:Type2] Precondition: N0 Post cond.: i (1≤i≤N):

Algorithm:i=1..N

T(X[i])

Y[i]:=f(X[i])

Y[i]:=X[i]

otherwise, iX

iX Tif, iXfY[i]

Y N

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function

Specification (other special case): Input: N:Integer,

X,Z:Array[1..N:Real] Output: Y:Array[1..N:Real] Precondition: N0 Post condition: i(1≤i≤N): Y[i]=X[i]

+Z[i]Algorithm:

i=1..N

Y[i]:=X[i]+Z[i]

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8. Multiple item selection

Example tasks: Let's define all excellent students

in a class! Let's calculate the divisors of an

integer! Let's list all the vowels from an

English word! Let's garble all the people who are

taller than 180cm, from a set of people!

Let's list those days of the year when it didn't freeze!

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What is in common?We have to list all elements from N "something", which have a common attribute T.

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X:Array[1..N:Type] Output: Cnt:Integer,

Y:Array[1..N:Integer] Precondition: N0 Post cond.: Cnt= and

i(1≤i≤Cnt): T(X[Y[i]])

and Y(1,2,…,N)

N

])i[X(T1i

1See PoA "Count"

In case of static In case of static array array

declarationdeclaration

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Algorithm:

Comment:Collecting ordinal numbers is more general than collecting elements. If we needed elements then we would have: Y[Cnt]:=X[i]. (And accordingly modified specification, too.)

Cnt:=0

i=1..N

T(X[i])

Cnt:=Cnt+1 Y[Cnt]:=i

Y N

See PoA "Count"

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8. Multiple item selection (not freezing days)


H:Array[1..N:Real] Output:

Cnt:Integer,NF:Array[1..N:Integer] Precondition: N0 Post cond.: Cnt= and

i(1≤i≤Cnt): H[NF[i]]>0 and

NF(1,2,…,N)

N

0H[i]1i

1

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8. Multiple item selection (not freezing days)

Algorithm:

Cnt:=0

i=1..N

H[i]>0

Cnt:=Cnt+1

NF[Cnt]:=i

Y N

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10. Partitioning

Example tasks: Let's define all excellent and non-

excellent students in a class! Let's garble all the people who are

taller than 180cm, and others who are not taller than 180cm, from a set of people!

Let's list all even and all odd numbers from a series!

Let's list those days of the year when it did freeze and didn't freeze!

Let's list all the vowels and consonants from an English word!

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10. Partitioning

What is in common? We have to list all elements from N "something", which have a common attribute T, and then also list those ones not having attribute T.

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10. Partitioning


X:Array[1..N:Type] Output: Cnt:Integer,

Y,Z:Array[1..N:Integer] Precondition: N0 Post cond.: Cnt= and

i(1≤i≤Cnt): T(X[Y[i]]) and i(1≤i≤N–Cnt): not T(X[Z[i]])

and Y(1,2,…,N) and Z(1,2,…,N)

N

])i[X(T1i

1

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10. Partitioning

Algorithm:

Comment:If we needed the partitioned elements, than here we could also have:=X[i] instead of :=i (And accordingly modified specification!)CntZ can be eliminated. How?

Cnt:=0

CntZ:=0

i=1..N

T(X[i])

Cnt:=Cnt+1

CntZ:=CntZ+1

Y[Cnt]:=I Z[CntZ]:=i

Y N

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10. Partitioning

Permutation(1,2,...,N):=set of all permutations of

1..N (set of sets)

Modification:There are exactly N number of elements in Y and Z together, so it's sufficient to have only one array. We will collect the T attributed elements at the beginning of the resulting array Y.

Solution: Input: N:Integer, X:Array[1..N:Type] Output: Cnt:Integer, Y:Array[1..N:Integer] Precondition: N0 Post cond.: Cnt= and

i(1≤i≤Cnt): T(X[Y[i]]) and i(Cnt+1≤i≤N): not T(X[Y[i]])

and YPermutation(1,2,…,N)

N

])i[X(T1i

1

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10. Partitioning

Algorithm:Cnt:=0

CntZ:=N+1

i=1..N

T(X[i])

Cnt:=Cnt+1

CntZ:=CntZ–1

Y[Cnt]:=I Y[CntZ]:=i

Y N

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11. Intersection

Example tasks: Let's list all common divisors of

two integers! We investigate birds in winter

and summer. Let's collect the birds which are not migratory!

Based on the free hours from the calendars of two persons, let's determine when they can meet.

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11. Intersection

What is in common?We have two sets as input (with same typed items), and we have to list all elements that are part of both sets!

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11. Intersection

One item included One item included only onceonly once

Specification: Input: N,M:Integer,

X:Array[1..N:Type] Y:Array[1..M:Type]

Output: Cnt:Integer, Z:Array[1..Cnt:Type]

Precondition: N0 and M0 andIsSet(X) and IsSet(Y)

Post cond.: Cnt= and

i(1≤i≤Cnt): (Z[i]X and Z[i]Y) and

IsSet(Z)

N

YX[i]1i

1

Could be alsoCould be also: : min(N,M)min(N,M)

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11. Intersection

Algorithm:

Comment:The solution is a multiple item selection and a decision.

Cnt:=0i=1..N

j:=1

j≤M and X[i]≠Y[j]

j:=j+1

j≤M

Cnt:=Cnt+1

Z[Cnt]:=X[i]

Y N

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11. Intersection

Variations for intersection:

We have two sets, we have to determine the number of common elements.

We have two sets, we have to determine if they have at least one common element.

We have two sets, we have to provide one common element.

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12. Union

Example tasks: We have two courses, let's list the

students visiting both courses! Let's collect all the birds we can

investigate in winter as well as in summer!

Based on the free hours from the calendars of two persons, let's determine when we can reach at least one of them!

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12. Union

What is in common?We have two sets as input (with same typed items), and we have to list all elements that are included at least in one of the sets.

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12. Union

Specification: Input: N,M:Integer,

X:Array[1..N:Type] Y:Array[1..M:Type]

Output: Cnt:Integer, Z:Array[1..Cnt:Type]

Precondition: N0 and M0 andIsSet(X) and IsSet(Y)

Post cond.: Cnt=N+ and

i(1≤i≤Cnt): (Z[i]X or Z[i]Y) and IsSet(Z)

M

XY[j]

1j1

Could be alsoCould be also: : N+MN+M

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12. Union

Algorithm:Z:=XCnt:=N

j=1..Mi:=1

i≤N and X[i]≠Y[j]

i:=i+1

i>N

Cnt:=Cnt+1

Z[Cnt]:=Y[j]

Y N

CopyCopy

DecisionDecision

Multiple item Multiple item selectionselection

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Patterns of Algorithms (PoAs)

Sequence Sequence7. Copy – calculation with function

8. More item selection9. Sort (will have later) Sequence Sequences10.Partitioning Sequences Sequence11.Intersection12.Union

Programming Fundamentals

End of 9th lecture

programming fundamentals 9 th lecture szabolcs papp

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