CSI 1306

ALGORITHMS - PART 4LIST PROCESSING

ListsSometimes a problem deals with a list of values We represent such a list with a single name, and

use subscripts to indicate the individual members of the list

For example, if X is a list of size N, then the individual members are:

X1 , X2 , X3 ,………., XN

Lists

Always include the size of the list as part of the information for the algorithm – remember storage? How much room must the

computer set aside for our list?

– there are techniques for working with lists of unknown length, but they are beyond the scope of this course.

– either you will know the length of the list or you will be able to calculate it

Algorithm 4.1

Write an algorithm that returns the Kth element of a given list L, with N numbers

– NAME: FindK– GIVENS: L, N, K

• Change: None– RESULTS: Val– INTERMEDIATES: None– DEFINITION: Val := FindK(L, N, K)– -------------------------– METHOD:

Get L, NGet K

Let Val = LK

Give Val

Algorithm 4.2

Write an algorithm that replaces the Kth element`s value of a given list L (with N numbers) with the given value M– NAME: ChangeK– GIVENS: L, N, K, M

• Change: L– RESULTS: None– INTERMEDIATES: None– DEFINITION: ChangeK(L, N, K, M)– -------------------------– METHOD:

Get L, NGet KGet M

Let LK = M

Give L

Trace 4.1

Trace Algorithm 4.2 with a list L of 5 elements (1, 8, 3, 6, 2) where K is 2 and M is 20

METHOD:(1) Get L, N(2) Get K(3) Get M

(4) Let LK = M

(5) Give L

Ln L N K M 1 (1,8,3,6,2) 5 2 2 3 20 4 (1,20,3,6,2) 5 output (1,20,3,6,2)

Algorithm 4.3

Given a list, L, of N numbers, where N is odd, find the middle number in the list

– NAME: Middle– GIVENS: L, N

• Change: None– RESULTS: Mid– INTERMEDIATES: Loc– DEFINITION: Mid := Middle(L, N)– -------------------------– METHOD:

• Get L, N

• Let Loc = N div 2 + 1• Let Mid = Lloc

• Give Mid

Trace 4.2

Trace Algorithm 4.3 with a list L of 5 elements (1, 8, 3, 6, 2)

METHOD:(1) Get L, N(2) Let Loc = N div 2 + 1(3) Let Mid = Lloc

(4) Give Mid

LN L N Loc Mid

1 (1,8,3,6,2) 5

2 3

3 3

4 output 3

List Processing

Remember that we represent a list with a single name and use subscripts to indicate the individual members of the list

Processing a list usually involves a loop in which we use the list subscript variable as the loop control variable

Frequently, we also use Boolean operators in our loop tests– The next slide reviews use of these operators

List Processing

Assume that the Boolean variables X, Y and Z are set to the following values:X = True, Y = False, Z = True

The order of precedence for Boolean operators is Not, And, Or

What is the result of each of the following expressions:(X and Y) or Z = TRUE(True or Y) and False = FALSEX or True and not Y or Z = TRUEnot(X and not Z or False and (True and Y)) = TRUEY and Y or not Z = FALSE

Algorithm 4.4

Given a list X of N numbers, find the sum of the numbers in X

Analysis– Basic Algorithm

• SUM• Let Total = 0• Let Total = Total + ????

– Total = X1 + X2 + …….. + XN – Total = Total + XI

• Let I go from 1 to N in a loop; then we will add each X I

• Loop control variable is the list subscript variable

Algorithm 4.4

Given a list X of N numbers, find the sum of the numbers in X– Name:SUMLIST

– Given: X, N

• Change: None

– Result: Total

– Intermediate: I

– Definition

• Total := SUMLIST(X,N)

Method

Get X, N

Let Total = 0

Let I = 1

Loop When (I <= N)

Let Total = Total + XI

Let I = I + 1

Finish Loop

Give Total

Trace 4.3

Trace algorithm 4.4 with the list (3,7,5). N is 3.

(1) Get X, N

(2) Let Total = 0

(3) Let I = 1

(4) Loop When (I <= N)

(5) Let Total = Total + XI

(6) Let I = I + 1

(7) Finish Loop

(8) Give Total

LN X N I Total Test 1 (3, 7, 5) 3 2 0 3 1

4 (1 <= 3)5 36 2 4 (2 <= 3) 5 106 3

4 (3 <= 3)5 156 44 (4 <= 3)

8 Output 15

Algorithm 4.5

Given a list X of N numbers, determine whether the given value V occurs in the list X.

Analysis– Basic Algorithm

• SEARCH

• Let Found = False

• Does XI = V?

– Stop the index at this point

» FOUND at XI

– I to process the list

Algorithm 4.5

Given a list X of N numbers, determine whether the given value V occurs in the list X.Name: SEARCHX

Given: X, N, V

Change: None

Result: Found

Intermediate: I

Definition:

Found := SEARCHX(X,N,V)

Method Get X, N Get V Let Found = False Let I = 1 Loop If (XI = V) Let Found = True Else Let I = I + 1

Finish Loop When (I >N) or (Found)

Give Found

Trace 4.4

Trace algorithm 4.5 with the list (3,7,5) and with a value of 7 for V

(1) Get X, N(2) Get V(3) Let Found = False(4) Let I = 1(5) Loop(6) If (XI = V)(7) Let Found = True(8) Else(9) Let I = I + 1

(10) Finish Loop When (I >N) or Found

(11) Give Found

LN X N V I Found Test 1 (3,7,5) 3 2 7 3 False 4 1 6 (3 = 7) 9 2 10 (2>3) or (F)

6 (7 = 7) 7 True10 (2>3) or (T) 11 Output True

Algorithm 4.6

Given a list X of N positive numbers, find the maximum number and its position in X

Analysis– Basic Algorithm

• MAX

• Let Max = -1

• (XI > Max)?

– I for list processing• Must track not only the Max, but also the location of Max (the

value of I)

Algorithm 4.6

Given a list X of N positive numbers, find the maximum number and its position in X

– Name: MAXLIST

– Given: X, N

• Change:None

– Result: Max, Loc

– Intermediate: I

– Definition:

– (Loc,Max) := MAXLIST(X,N)

Method Get X, N Let Max = -1 Let I = 1

Loop When (I <= N) If (XI > Max) Let Max = XI

Let Loc = I Let I = I + 1 Finish Loop

Give MaxGive Loc

Trace 4.5 Trace algorithm 4.6 with the list (8,3,25,9)

(1) Get X, N(2) Let Max = -1(3) Let I = 1

(4) Loop When (I <= N)(5) If (XI > Max)(6) Let Max = XI

(7) Let Loc = I (8) Let I = I + 1(9) Finish Loop

(10) Give Max(11) Give Loc

LN X N I MAX LOC TEST

1,2,3 (8,3,25,9) 4 1 -1

4 (1 <= 4)

5 (8 > -1)

6 8

7 1

8 2

4 (2 <= 4)

5 (3 > 8)

8 3

4 (3 <= 4)

5 (25>8)

6 25

7 3

8 4

4 (4 <= 4)

5 (9>25)

8 5

4 (5 <= 4)

10 Output 25

11 Output 3

Algorithm 4.7

Given a list X of N numbers, find the number of times zero occurs in the list

Analysis– Basic Algorithm

• COUNT

• Let Count = 0

• XI = 0?

• Let Count = Count + 1 if a match

– I for list processing

Algorithm 4.7

Given a list X of N numbers, find the number of times zero occurs in the list

– Name: NUM0

– Given: X, N

• Change: None

– Result: Count

– Intermediate: I

– Definition:

– Count := NUM0(X,N)

Method

Get X, N

Let Count = 0

Let I = 1

Loop When (I <= N)

If (XI = 0)

Let Count = Count + 1

Let I = I + 1

Finish Loop When

Give Count

Algorithm 4.8

Given a list X of N numbers, see if X contains any two numbers which are the same

Analysis– Basic Algorithm

• Variant of SEARCH• Compare each XI with every other element of the list• Loop within a Loop

– For each XI

» Search for XI

• Search only until one match is found– Stop the Indexes at the locations

Algorithm 4.8

Given a list X of N numbers, see if X contains any two numbers which are the same– Name: DUPLICATE– Given: X, N

• Change: None– Result: Double– Intermediate: I, Test– Definition:– Double := DUPLICATE(X,N)

Method Get X, N Let Double = False Let Test = 1 Loop Let I = Test + 1 Loop If (XTest = XI) Let Double = True Else Let I = I + 1 Finish Loop When (I >N) or (Double) If (Not (Double)) Let Test = Test + 1 Finish Loop When (Test = N) or (Double)

Give Double

Trace 4.6 Trace algorithm 4.8 with the list

(2,8,7,8)(1) Get X, N(2) Let Double = False(3) Let Test = 1(4) Loop(5) Let I = Test +1(6) Loop(7) If (XTest = XI)(8) Let Double = True(9) Else(10) Let I = I + 1(11) Finish Loop When (I >N) or (Double)(12) If (Not(Double))(13) Let Test = Test + 1(14) Finish Loop When (Test = N) or (Double)

(15) Give Double

LN X N Double Test I TEST1,2,3 (2,8,7,8) 4 False 1 5 2 7 (2=8)10 311 (3>4)or(F) 7 (2=7)10 411 (4>4)or(F) 7 (2=8)10 511 (5>4)or(F)12 Not (F)13 214 (2=4)or(F) 5 3 7 (8=7)10 411 (4>4)or(F) 7 (8=8) 8 True11 (5>4)or(T)12 Not (T)14 (3=4)or(T)15 Output True

Grouped Lists

If we want to process multiple, related lists (as in a database), we must maintain the order relationship between elements in each list

E.g. Student grades– Name - Midterm - Final

– Ann - 100 - 30

– Bob - 75 - 28

– Cathy - 60 - 90

– Dave - 40 - 80

Grouped Lists

We have three lists– Name = (Ann, Bob, Cathy, Dave)

– Midterm = (100, 75, 60, 40)

– Final = (30, 28, 90, 80)

Since Ann is Name1, then her midterm mark would be Midterm1 and her final mark would be Final1

We can also create a new list from other lists.

Algorithm 4.9

Given two lists, Midterm and Final, of N numbers each, calculate a Final grade of 75% Final and 25% Midterm

– Name: FINDGRADE

– Given: M, F, N

• Change: None

– Result: G

– Intermediate: I

– Definition:

– G := FINDGRADE(M,F,N)

Method

Get M, F, N

Let I = 1

Loop When (I <= N)

Let GI = 0.75*FI + 0.25*MI

Let I = I + 1 Finish Loop

Give G

Trace 4.7

Trace algorithm 4.9 with the grouped list defined previously

Method

(1) Get M, F, N

(2) Let I = 1

(3) Loop When (I <= N)

(4) Let GI = 0.75*FI + 0.25*MI

(5) Let I = I + 1(6) Finish Loop

(7) Give G

F = (30, 28, 90, 80)

M = (100, 75, 60, 40)

LN I N G Test

1,2 1 4 (??,??,??,??)

3 (1 <= 4)

4 (48,??,??,??)

5 2

3 (2 <= 4)

4 (48,40,??,??)

5 3

3 (3 <= 4)

4 (48,40,83,??)

5 4

3 (4 <= 4)

4 (48,40,83,70)

5 5

3 (5 <= 4)

7 Output (48,40,83,70)

Algorithm 4.10

Given a set of lists, Name and Grade, determine who received the lowest grade.– Name: WORST

– Given: Name,G, N

• Change: None

– Result: LName

– Intermediate:

• Loc, Min, I

– Definition:

– Lname :=WORST(Name,G,N)

Method Get Name, G, N Let Min = 101 Let I = 1 Loop When (I <= N)

If (GI < Min)

Let Min = GI

Let Loc = I

Let I = I + 1 Finish Loop

Let LName = NameLoc

Give LName

Trace 4.8

Trace algorithm 4.10 with the grouped list defined previously

(1) Get Name, G, N(2) Let Min = 101(3) Let I = 1(4) Loop When (I <= N)(5) If (GI < Min)(6) Let Min = GI

(7) Let Loc = I

(8) Let I = I + 1(9) Finish Loop(10) Let LName = NameLoc

(11) Give LName

Name = (Ann, Bob, Cathy, Dave)G = (48,40,83,70)

LN N I LOC MIN Lname TEST1,2,3 4 1 101 4 (1 <= 4) 5 (48<101) 6 48 7 1 8 2

4 (2<=4) 5 (40<48) 6 40 7 2 8 3

4 (3<=4) 5 (83<40) 8 4

4 (4<=4) 5 (70<40) 8 5 4 (5<=4)

10 Bob 11 Output Bob

Additional Material

Flow Charts

Algorithm 4.1NAME: FindK

GIVENS: L, N, K

Change: None

RESULTS: Val

INTERMEDIATES: None

DEFINITION:

Val := FindK(L, N, K)

StartFINDK

Get L,NGet K

Let Val = LK

Give Val

FinishFINDK

Algorithm 4.2

NAME: ChangeK

GIVENS: L, N, K, M

Change: L

RESULTS: None

INTERMEDIATES: None

DEFINITION:

ChangeK(L, N, K, M)

StartCHANGEK

Get L,NGet KGet M

Let LK =M

Give L

FinishCHANGEK

Algorithm 4.3NAME: Middle

GIVENS: L, N

Change: None

RESULTS: Mid

INTERMEDIATES: Loc

DEFINITION:

Mid := Middle(L, N)

StartMIDDLE

Get L,N

Let Loc = N div 2 + 1Let Mid = LLoc

Give Mid

FinishMIDDLE

Algorithm 4.4Name:SUMLIST

Given: X, N

Change: None

Result: Total

Intermediate: I

Definition

Total := SUMLIST(X,N)

StartSUMLIST

Get X, N

Let Total = 0Let I = 1

(I <= N)

Let Total = Total + XI

Let I = I + 1

Give Total

FinishSUMLIST

Y

N

Algorithm 4.5

Name: SEARCHX

Given: X, N, V

Change: None

Result: Found

Intermediate: I

Definition:

Found := SEARCHX(X,N,V)

StartSEARCHX

Get X, NGet V

Let Found = FalseLet I = 1

If(XI = V)

Let Found = True Let I = I + 1

(I > N) or(Found)

Give Found

FinishSEARCHX

N

Y

N

Y

Algorithm 4.6

Name: MAXLIST

Given: X, N

Change:None

Result: Max, Loc

Intermediate: I

Definition:

(Loc,Max) := MAXLIST(X,N)

StartMAXLIST

Get X,N

Let Max = -1Let I = 1

(I <= N)

Let Max = XI Let Loc = I

Give MaxGive Loc

FinishMAXLIST

Y

N

If (XI > Max)

Let I = I + 1

Y

N

Algorithm 4.7

Name: NUM0

Given: X, N

Change: None

Result: Count

Intermediate: I

Definition:

Count := NUM0(X,N)

StartNUM0

Get X,N

Let Count = 0Let I = 1

(I <= N)

Let Count = Count + 1

Give Count

FinishNUM0

Y

N

If (XI = 0)

Let I = I + 1

Y

N

Algorithm 4.8

Name: DUPLICATE

Given: X, N

Change: None

Result: Double

Intermediate: I, Test

Definition:

Double := DUPLICATE(X,N)

StartDUPLICATE

Get X,N

Let Double = FalseLet Test = 1

Give Double

FinishDUPLICATE

Let Double = True Let I = I + 1

(I > N) or (Double)

If (XTest = XI )

Let I = Test + 1

If (Not (Double))

Let Test = Test + 1

(Test = N) or (Double)

N

Y

N

Y

Y

N

Y

N

Algorithm 4.9

Name: FINDGRADE

Given: M, F, N

Change: None

Result: G

Intermediate: I

Definition:

G := FINDGRADE(M,F,N)

StartFINDGRADE

Get M,F,N

Let I = 1

(I <= N)

Let GI = 75%FI + 25%MI

Let I = I + 1

Give G

FinishFINDGRADE

Y

N

Algorithm 4.10

Name: WORST

Given: Name,G, N

Change: None

Result: LName

Intermediate:

Loc, Min, I

Definition:

Lname :=WORST(Name,G,N)

StartWORST

Get Name, G, N

Let Min = 101Let I = 1

(I <= N)

Let Min = GI

Let Loc = I

Give LName

FinishWORST

Y

N

If (GI < Min)

Let I = I + 1

Y

N

Let LName = NameLoc

NSD

Algorithm 4.1NAME: FindK

GIVENS: L, N, K

Change: None

RESULTS: Val

INTERMEDIATES: None

DEFINITION:

Val := FindK(L, N, K)

Get L,NGet KLet Val = LK

Give Val

Algorithm 4.2

NAME: ChangeK

GIVENS: L, N, K, M

Change: L

RESULTS: None

INTERMEDIATES: None

DEFINITION:

ChangeK(L, N, K, M)

Get L, NGet K

Get M

Let LK = M

Give L

Algorithm 4.3NAME: Middle

GIVENS: L, N

Change: None

RESULTS: Mid

INTERMEDIATES: Loc

DEFINITION:

Mid := Middle(L, N)

Get L, NLet Loc = N div 2 + 1Let Mid = LLoc

Give Mid

Algorithm 4.4Name:SUMLIST

Given: X, N

Change: None

Result: Total

Intermediate: I

Definition

Total := SUMLIST(X,N)

While (I <=N)

Let Total = Total + XI

Let I = I + 1

Get X, NLet Total = 0

Let I = 1

Give Total

Algorithm 4.5

Name: SEARCHX

Given: X, N, V

Change: None

Result: Found

Intermediate: I

Definition:

Found := SEARCHX(X,N,V)

Y N

Let Found = True Let I = I + 1

Let I = 1

Let Found = False

Get VGet X, N

Give FoundUntil (I > N) or (Found)

If (XI = V)

Algorithm 4.6

Name: MAXLIST

Given: X, N

Change:None

Result: Max, Loc

Intermediate: I

Definition:

(Loc,Max) := MAXLIST(X,N)

Y NLet Max = XI

Let Loc = I

While (I <= N)

Give MaxGive Loc

Get X, NLet Max = -1

Let I = 1

If (XI > Max)

Do Nothing

Let I = I + 1

Algorithm 4.7

Name: NUM0

Given: X, N

Change: None

Result: Count

Intermediate: I

Definition:

Count := NUM0(X,N)

Y N

Let Count = Count + 1 Do Nothing

Give CountLet I = I + 1

If (XI = 0)

While (I <= N)

Let Count = 0Get X, N

Let I = 1

Algorithm 4.8

Name: DUPLICATE

Given: X, N

Change: None

Result: Double

Intermediate: I, Test

Definition:

Double := DUPLICATE(X,N)Y N

Let Double = True Let I = I + 1

NDo Nothing

Let I = Test + 1

Give Double

If (XTest = XI)

Until (I > N) or (Double)If (Not(Double))

YLet Test = Test + 1Until (Test = N) or (Double)

Get X, NLet Double = False

Let Test = 1

Algorithm 4.9

Name: FINDGRADE

Given: M, F, N

Change: None

Result: G

Intermediate: I

Definition:

G := FINDGRADE(M,F,N)

While (I <= N)

Let GI = 75%FI + 25% MI

Let I = I + 1

Get M, F, NLet I = 1

Give G

Algorithm 4.10

Name: WORST

Given: Name,G, N

Change: None

Result: LName

Intermediate:

Loc, Min, I

Definition:

Lname :=WORST(Name,G,N)

Y NLet Min = GI

Let Loc = I

Get Name, G, NLet Min = 101

Let I = 1

If (GI < Min)

While (I <= N)

Let LName = NameLoc

Give Lname

Do Nothing

Let I = I + 1

Homework

Write an algorithm to find the maximum value in a list of positive numbers. Each entry in the list is unique (ie. There are no duplicate numbers)

Write an algorithm that fills a list with the first 50 multiples of 7. (ie. 7, 14, 21…350)

Write an algorithm to populate a list with names. The list will continue to grow until the user indicates it is time to stop.

Rewrite Algorithm 4.6, such that the variable Max is not used at all:

– Name: MAXLIST

– Given: X, N

• Change:None

– Result: Loc

– Intermediate: I

– Definition:

– Loc := MAXLIST(X,N)