Download Practical Session 3

Practical Session #3 - ADTs: Array, Queue, Stack,
Linked List
Basic Data Structures and Abstract Data Types
ADT
Abstract Data Type
A collection of data-storing entities with operations to create, access,
change, etc.
Vector(Array)
A linear sequence of elements that supports access to its elements by
their indexes.
A vector ADT supports the following basic methods:
 elementAt(i) – returns the element at index i
 replaceAt(i, element) – replace and return the element that in
index i with new given element.
 insertAt(i, element) – insert new element at index i
 removeAt(i) – remove the element in index i
 size() – returns the size of the vector
 isEmpty() - returns true if the vector is empty
FIFO - First In First Out
ADT that supports the following operations:
Queue



Enqueue - insert new element at the tail of the queue
Dequeue - remove element from the head of the queue
isEmpty - returns true if the queue is empty
LIFO - Last In First Out
ADT that supports the following operations:
Stack



Push - add element to the head of the stack
Pop - remove element from the head of the stack
isEmpty - returns true if the stack is empty
Linked List
A data structure with linear access to the elements in it.
Each element has a pointer to the next element in the list
Doubly-Linked
List
A data structure with linear access to the elements in it.
Each element has 2 pointers, one to the next element in the list and
the other to the previous element
Question 1
S is a set of at most n elements, where n is given.
Each element has a unique key in the range [0...n-1]. (i.e. no 2 different elements have
the same key)
Find a data structure of size O(n), that supports the following operations in the given
times:
Init(n)
Operation
Time
Initialize S (S is an empty set at the beginning)
O(n)
isElement(k) Check if S has an element whose key equals to k
O(1)
Insert(x)
Add element x to S
O(1)
Remove(k)
Remove element whose key equals to k
O(1)
isEmpty()
Check if S is empty
O(1)
hasAll
Check if S contains all the elements whose keys are
O(1)
in the range [0...n-1]
Solution: (Direct Addressing) :
1. Array A[1..n] of pointers to elements.
if A[k] == null, the element whose key is k is not in S.
2. count = number of elements in S
Init (n)
for (i=0 ; i<n ; i++)
A[i] = null;
count = 0;
isElement (k)
return (A[k] != null);
Insert (x)
A[x.key] = x;
count++;
Remove (k)
A[k] = null;
count--;
isEmpty ()
return (count == 0);
hasAll ()
return (count == n);
Question 2
Suggest 2 ways to implement a Queue using 2 Stacks.
Solution:
The queue/stack is abstract, meaning the implementation isn't known, only the interface.
In order to implement a queue we need to implement the following methods:



Enqueue(x)
Dequeue()
isEmpty()
We will use stack A in order to hold the queue's elements and stack B as a temporary
stack. We will use the Stack ADT operations: Push(x), Pop() and isEmpty().
Enqueue (x) {
moveElements(A,B)
A.Push(x)
moveElements(B,A)
}
Dequeue ( ) {
element = A.Pop()
return element
}
isEmpty ( ) {
return A.isEmpty()
}
moveElements(X,Y) {
while (! X.isEmpty()){
temp = X.Pop()
Y.Push(temp)
}
}
Another way to implement a queue using 2 stacks:
Enqueue (x) {
moveElements(A,B)
A.Push(x)
}
Dequeue ( ) {
moveElements(B,A)
element = A.Pop()
return element
}
isEmpty ( ) {
return ( A.isEmpty() && B.isEmpty())
}
In a very similar way you can implement stack using 2 queues.
Question 3:
L1 and L2 are two linked lists which have a common part of size k.
The length of L1, L2 until the 1st common node is n, m respectively.
Suggest a way to find the 1st common node in O(n+m+k).
(k,n,m are variables, not given)
Solution :
1. find the length of L1 and L2
2. In the longer list, find the (L1.length-L2.legth)th node.
3. Traverse both lists step by step together. In the shorter list start from the first
node. In the longer list start from the node you found in 2. Stop when the “next”
pointer of both lists points to the same node.
time complexity:
finding the length of L1: O(n+k)
finding the length of L2: O(m+k)
finding the common link: O(m+n)
total: O(n+m+k)
Question 4:
L is a singly linked list. Starting from the first element, suggest way to traverse forward
and backward from one element to its next or previous element at O(1). You can use only
O(1) extra memory.
Solution:
Converting the list to a doubly-linked list can do the trick, but it is not allowed (will cost
O(n) extra memory).
We will use two pointers:
 current – the current link we want to traverse from.
 prev – the previous link in the traversal.
As we traverse forward, all the links before current will be linked backward.
As we traverse backward, all the links after current will be linked forward as usual.
init()
current=head
prev=null
moveForward()
temp = prev
prev=current
curent=current.next
prev.next=temp
moveBackward()
temp=current
current=prev
prev=prev.next
current.next=temp
Question 5:
A is a Boolean (1 or 0 values) matrix of size n*n.
Find a data structure that supports the following operations in the given time:
Operation
Time
init(n)
Initialize A with the value 1
O(n2)
flip(i,j)
A[i,j]=!A[i,j]
O(1)
hasRowOf1 Return true iff A has a row that contains only 1-s O(1)
hasRowOf0 Return true iff A has a row that contains only 0-s O(1)
Solution:
We will use the uses the following data structures:
 A[n][n] – the matrix
 Sum[n] – An array containing sums of the rows in A. Sum[i] = the sum of the row
i in A.
 count1 = how many cells in Sum contain n
 count0 = how many cells in Sum contain 0
init()
fill A with 1’s
fill Sum with n.
count1=n
count0=0
flip(i,j)
if (A[i][j]==0) {
if (Sum[i]=0)
count0- Sum[i]++
if (Sum[i]=n)
count1++
A[i][j]=1
}
else {
if (Sum[i]=n)
count1- Sum[i]-if (Sum[i]=0)
count0++
A[i][j]=0
}
hasRowOf1()
return count1>0
hasRowOf0()
return count0>0
Question 6:
Suggest a way to implement an array of integers so that assignment of some value
INIT_VAL to all cells of an array will take O(1) time.
Remember that reading A[i] or writing to A[i] takes O(1) time.
Solution:
In order to implement an initialization of an array A of size n in O(1) time, we will
create a new ADT called smartArray.
The smartArray ADT uses the following data structures:





A - the array of values
top – counts the number of defined elements in A
INIT_VAL– the value to initiate the array with
C – Array of defined and undefined indexes
C[0..top-1] – indexes of defined elements in A
B – array of pointers to the defined indexes in C
B[k] = the index of k in the array C
The smartArray ADT supports the following methods:
new: [Arr = new int[n]]
NewSmartArray(n)
A = new int[n]
B = new int[n]
C = new int[n]
Initialization: [ for (int i=0 ; i < n ; i++) Arr[i] = INIT_VAL]
init(val)
INIT_VAL = val
top = 0
The following procedure returns true if a given index ‘i’ is not initialized.
not_defined(i)
return !((B[i] < top) and C[B[i]] = i))
Assumptions:
 The value of an undefined element B[i] is an unexpected non-negative
number.
Thus, for an undefined index, the comparison B[i]<top can be either true or
false.
 The and expression is evaluated from left to right.
Thus, C[B[i]]=i is evaluated only if B[i] < top.
Read(i): [val ← Arr[i]]
Read(i)
if (not_defined(i))
return INIT_VAL
else
return A[i]
Write(i,val): [Arr[i] ← val]
Write(i,val)
if (not_defined(i))
C[top] = i
B[i] = top
top++
A[i] ← val
Example:
x = new smartArray[n]
x.init(0)
x.write(2,20)
x.write(4,40)
x.write(5,50)
a = x.read(4) >> the value of a is 40
b = x.read(3) >> the value of b is 0
5 50
4 40
3
2 20
1
0
5 2
4 1
3
2 0
1
0
5
4
3
2 5
1 4
0 2
A
B
C
not set
top
set
‫שאלה מס' ‪7‬‬
‫נגדיר "מחסנית מינימום" כמבנה נתונים התומך בפעולות הבאות‪:‬‬
‫‪ – Create() ‬אתחול מבנה הנתונים‪.‬‬
‫‪ – Insert(x) ‬הכנסת איבר ‪ x‬למבנה‪.‬‬
‫‪ – RemoveLast() ‬הוצאת האיבר שהוכנס אחרון‪.‬‬
‫‪ – Min() ‬החזרת האיבר הקטן ביותר במבנה (ללא הוצאתו)‪.‬‬
‫‪ – ChangeMin(k) ‬שינוי ערך האיבר הקטן ביותר במבנה ל – ‪.k‬‬
‫הנחה‪ :‬כל האיברים שונים זה מזה‪.‬‬
‫הציעו מימוש ל"מחסנית מינימום"‪ ,‬כאשר סיבוכיות הזמן הנדרשת לארבע הפעולות הראשונות היא )‪,O(1‬‬
‫ולפעולה ‪ ,O(t) ChangeMin‬כש – ‪ t‬הינו מספר האיברים במבנה שהוכנסו אחרי האיבר המינימלי (הכוונה‬
‫לאיבר המינימלי לפני שהפעולה ‪ ChangeMin‬בוצעה)‪.‬‬
‫פתרון‪:‬‬