data structures amihood amir. מדעיהמחשב מדעי ביולוגיה פיסיקה כימיה...
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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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