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Data Structures
Amihood Amir
‫ביולוגיה‬
‫מדעי‬
‫פיסיקה‬
‫המחשב‬
‫מדעי החברה‬
‫כימיה‬
‫מתמטיקה‬
‫מדעי הרוח‬
‫מדעי המדינה‬
‫מדעי ההתנהגות‬
‫המחשב‬
INPUT
‫מדעי‬
OUTPUT
‫המחשב‬
INPUT
‫מדעי‬
OUTPUT
‫מדעי‬
‫‪OUTPUT‬‬
‫מכונת כביסה‬
‫‪INPUT‬‬
‫מדעי‬
‫הטרירמה‬
‫מדעי המחשב‬
‫רכילות‬
‫אפליקציות‬
‫תאוריה‬
‫ניתן מה ניתן‬
‫מתודולוגיה מה‬
‫מה ניתן‬
‫מתודולוגיה‬
‫לחישוב‬
‫שפות תכנות‬
‫לחישוב?? לחישוב‬
‫מערכות הפעלה‬
‫חישוביות‬
‫יעיל?‬
‫בינה מלאכותית‬
‫ואיך?‬
‫עיבוד נתונים‬
‫מסדי נתונים‬
‫הנדסת תכנה‬
‫‪.‬‬
‫‪.‬‬
‫‪.‬‬
‫מורכבות‬
‫אלגוריתמים‬
‫מבני 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 Example
Let’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
pile
1
3
1
7
33
32
31
30
27
26
25
17
10
0
Total
taken
1
1
2
3
6
7
8
16
23
1
1
8
10
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
n
 2   1
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.
Now recurse…
If player A bring the outside gogoim to number
n
 2   1
then he wins. So recurse with the same
strategy. Make sure that in the previous move
the ouside gogoim were
n 
  2   1
    1
 2 


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 Game
Our data structures:
V = the current number of gogoim outside.
L = the stack of “winning” number of gogoim outside.
Initialization:
n
1


V
2
L empty.
A Winning Algorithm (2)
Find winning sizes:
While V > 2 do:
PUSH(L,V)
V
V 
 2   1
End While
You start.
If V=2 take 2 gogoim. If V=1 take 1.
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 Run
n = 440
440/2 219/2 109/2 54/2 26/2 12/2 5/2 -
1 = 219
1 = 109
1 = 54
1 = 26
1 = 12
1=5
1=2
Stack Implementation
Option 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:
18
9
20
17
Top
15
3
15
20
9
17
3
18
Stack Implementation (4)
Advantages:
Efficient for dynamic allocation of records.
Disadvantages:
More complicated to maintain.
Linked List
Top
Linked List Operations
Insertion: (if we know where to insert)
7
3
5
Linked List Operations
Deletion:
3
5
7
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
1
2
3
4
5
0
0
0
0
6
7
8
9
0 248 26
10
11
12
Top
613
1
7
3
17
4
248 11
Similar scheme for multidimensions.
26 12