Data Structures

Amihood Amir

המחשבמדעי מדעי

ביולוגיה

פיסיקה

כימיה

מתמטיקה

מדעי הרוח

מדעי החברה

מדעי המדינה

מדעי ההתנהגות

המחשבמדעי

INPUT OUTPUT

המחשבמדעי

INPUT OUTPUT

כביסה מדעי מכונת

INPUT OUTPUT

הטרירמהמדעי

המחשבמדעי

אפליקציותרכילות תאוריהאפליקציותרכילות

מתודולוגיהשפות תכנות

מערכות הפעלהבינה מלאכותית

עיבוד נתוניםמסדי נתונים

הנדסת תכנה...

מה ניתןמתודולוגיהלחישוב?

חישוביות

מה ניתןלחישוב?

מה ניתן לחישוב

יעיל?ואיך?

מורכבותאלגוריתמיםמבני נתונים

What Does Data Structures Mean?

EXAMPLE:

Input: Text T=T[1], … , T[n] of words.

Query: Find occurrences of word P.

Time: O(n|P|).

Can we do better?

Concordance (index) :

Construct a table C of pairs:

<word, index>

Sort the table lexicographically by word.

EXAMPLE: Let

T= boa, aba, xavier, abracadabra, wonderful

C= <aba,2>

<abracadabra,4>

<boa,1>

<wonderful,5>

<xavier, 3>

Do binary search on C:

Time: O((log n)|P|).

What Happened?

We constructed an external structure

(in the example, a table)

That enabled answering our question faster.

In this course we will see some basic such structures and the type of problems they enable to solve more efficiently.

Another ExampleLet’s play a game:

Players: Two people.Input: A pile of gogoim.Moves: Take gogoim out of the pile. First player takes out 1or 2 gogoim. Then alternately, can not exceed the number already taken out.Winner: The player who takes out the last

gogoim.

Example

333231302726251710 0

1

1

1

8

10

pileTotal taken

123678

1623

1

3

1

7

I WIN!!!!

What is the strategy?

Note that if there are n elements in the pile, and player A manages to bring the gogoim taken out of the pile to the number

Then player B still can not win, but no matter what B does, he will bring the number of gogoim to more than half so A will then be able to win in the next move.

12

n

Now recurse…

If player A bring the outside gogoim to number

then he wins. So recurse with the same strategy. Make sure that in the previous move the ouside gogoim were

12

n

12

12

n

The LIFO

The data structure we need is the LIFO –

Last In First Out –

Or stack (מחסנית).

Using the LIFO

Let L be a stack, V a variable.

Basic Operations:

Push(L,V) -- pushes V into stack L.

Pop(L,V) -- pops the top value out of stack L and puts it in variable V

A Winning Algorithm for our GameOur data structures:

V = the current number of gogoim outside.

L = the stack of “winning” number of gogoim outside.

Initialization:

V

L empty.

12

n

A Winning Algorithm (2)Find winning sizes:

While V > 2 do:

PUSH(L,V)

V

End While

You start.

If V=2 take 2 gogoim. If V=1 take 1.

12

V

A Winning Algorithm (3)The Game:Variables: S = number of gogoim taken out so far.Initialize: S 0.

The Game moves:While pile not empty do:

POP(L,V)If this is first pop then take out V gogoim.

S V else take out V-S gogoim. S V.Opponent makes his move.

End While

A Sample Runn = 440

440/2 - 1 = 219

219/2 - 1 = 109

109/2 - 1 = 54

54/2 - 1 = 26

26/2 - 1 = 12

12/2 - 1 = 5

5/2 - 1 = 2

Stack ImplementationOption 1:

Array S[1],…,S[n].First element in S[1].Pointer top points to last element.

Advantages:

Simple.

Disadvantages:

Need to keep a long array for stack.

When it fills, need to copy everything to a longer array.

Stack Implementation (2)Option 2:

Linked list of records.

A record has a number of data fields and a pointer to the next record.

Stack Implementation (3)In our case: Stack =

May be implemented in memory as:1520917318

Top

15

20

9

17

3

18

Stack Implementation (4)

Advantages:

Efficient for dynamic allocation of records.

Disadvantages:

More complicated to maintain.

Linked ListTop

Linked List Operations

Insertion: (if we know where to insert)

3 7

5

Linked List Operations

Deletion:

3 75

Take care: properly maintain the free space

Doubly Linked Lists

Insertion and Deletion Time: constant.

Can we do Binary Search on Linked List?

For example to have a dynamic concordance.

No No constant time random access on linked

lists

Searching a linked list of length n:

O(n) time.

Other Data Structures for which Linked Lists are Suitable

FIFO – First In First Out – Queue

FIFO Implementation

Head End

FIFO Operations

Front Rear

1) ENQUEUE (x,Q) : INSERT (x,END (Q),Q)

2) DEQUEUE (x,Q) : x (FIRST (Q))

Remove(FIRST(Q),Q)

Double Ended Queue

Can get in or out at head or end of queue.

Linked Lists used for Compression

Sparse Arrays:

613 0 7 17 0 0 0 0 0 0 248 26 1 2 3 4 5 6 7 8 9 10 11 12

Similar scheme for multidimensions.

613 1 7 3 17 4 248 11 26 12

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