Elementary Data Structures: Stacks and Linked Lists, Study notes of Data Structures and Algorithms

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Introduction toData Structures and Algorithms Chapter:

Elementary Data Structures(1)

Data Structures

and Algorithms

Overview on simple data structuresfor representing dynamic sets of data records ^ Main operations on these data structures are^ ^

Insertion

and^ deletion

of an element

^ searching

for an element ^ finding the

minimum

or^ maximum

element

^ finding the

successor

or the^

predecessor

of an element

^ And similar operations …  These data structures are often implementedusing^ dynamically allocated objects

and^ pointers

Elementary Data Structures

Data Structures

and Algorithms

Stack ^ A^ stack

implements the LIFO (last-in, first-out) policy ^ like a stack of plates, where you can either placean extra plate at the top or remove the topmost plate  For a stack, ^ the^ insert^ operation is called

Push

^ and the

delete

operation is called

Pop

Elementary Data Structures

Data Structures

and Algorithms

Where are Stacks used? ^ A^ call stack

that is used for the proper execution of a computer program with subroutine or function calls  Analysis of

context free languages

(e.g. properly nested brackets)

^ Properly nested: (()(()())), Wrongly nested: (()((())  Reversed Polish notation of terms ^ Compute 2 + 3*

⇨^ 2 Push 3 Push 5 * +

Elementary Data Structures

Data Structures

and Algorithms

Typical Implementation of a Stack ^ A typical implementation of a stack of size nis based on an

array^ S[1…n]

Elementary Data Structures^ ⇨^ so it can hold at most n elements^ ^ top(S) is the index of the most recentlyinserted element^ ^ The stack consists of elementsS[1 … top(S)], where^ ^ S[1] is the element at the bottom of the stack,^ ^ and S[top(S)] is the element at the top.^ ^ The unused elements S[top(S)+1 … n]are not in the stack

S

Top

Push^

Pop^4321

Data Structures

and Algorithms

Stack ^ If top(S) = 0 the stack is empty

⇨ no element can be popped

^ If top(S) = n the stack is full

⇨ no further element can be pushed

Elementary Data Structures

Data Structures

and Algorithms

Elementary Data Structures

S:^

S:^

S:^

S:

7 6 5 4 3 2 1

7 6 5 4 3 2 1

7 6 5 4 3 2 1

7 6 5 4 3 2 1

Top[S]=

Top=

Top=

Top=

3 3

3 23 3 23

(^53) 17 push(S,17)

pop (S)^

17 pop (S)^

3 pop (S)^

23 push(S,5)

pop (S)^

5 pop (S)^

3 pop (S)

Error:underflow

Data Structures

and Algorithms

Elementary Data Structures^ Pseudo Code for Stack Operations^ ^ Number of elements

NumElements

(S)

return

top[S]

Data Structures

and Algorithms

Elementary Data Structures^ Pseudo Code for Stack Operations^ ^ Pushing and Popping

Push(S,x)if^ Stack_Full(S)then^ error "overflow" else^ top[S]

:=^ top[S]+ S[top[S]]

:=^ x

Pop(S)if^ Stack_Empty(S)then^ error "underflow" else^ top[S]

:=^ top[S]- return

S[top[S]+1]

This pseudo code containserror handling functionality

Data Structures

and Algorithms

Pseudo Code for Stack Operations ^ (Asymptotic) Runtime^ ^

NumElements

: number of operations independent of size n of stack ⇨ constant ⇨ O(1)  Stack_Empty

and^ Stack_Full

:

number of operations independent of size n of stack ⇨ constant ⇨ O(1)  Push^ and

Pop : number of operations independent of size n of stack ⇨ constant

⇨ O(1)

Elementary Data Structures

Data Structures

and Algorithms

Elementary Data Structures^ Where are Queues used?^ ^ In multi-tasking systems (communication, synchronization)^ ^ In communication systems (store-and-forward networks)^ ^ In servicing systems (queue in front of the servicing unit)^ ^ Queuing networks (performance evaluation of computer andcommunication networks)

Data Structures

and Algorithms

Typical Implementation of a Queue ^ A typical implementation of a queue consisting of at most n-1elements is based on an

array^ Q[1 … n]

^ Its attribute

head(Q)

points to the head of the queue.

^ Its attribute

tail(Q)

points to the position

Elementary Data Structures where a new element will be inserted into the queue(i.e. one position behind the last element of the queue).^ ^ The elements in the queue are in positionshead(Q), head(Q)+1, …, tail(Q)-1, where we wrap around the arrayboundary in the sense that Q[1] immediately follows Q[n]

Data Structures

and Algorithms

Elementary Data Structures^ Q

1 2

3 4

5 6

7 8

9 10 head(Q)^

tail(Q)

Example (2) ^ Insert one more element (5.) ^ And again: Insert one more element (6.)

1.^ 2.^

3.^ 4.

Q

1 2

3 4

5 6

7 8

9 10 head(Q)

tail(Q)

1.^ 2.^

3.^ 4.^

Q

1 2

3 4

5 6

7 8

9 10 head(Q) tail(Q)

1.^ 2.^

3.^ 4.^

Data Structures

and Algorithms

Elementary Data StructuresTypical Implementation of a Queue^ ^ Number of elements in queue^ ^

If tail > head:

NumElements(Q) = tail - head ^ If tail < head:

NumElements(Q) = tail – head + n ^ If tail = head:

NumElements(Q) = 0 ^ Initially: head[Q] = tail[Q] = 1  Position of elements in queue ^ The x. element of a queue Q (

≤^ x^ ≤^ NumElements(Q)

is mapped to array positionhead(Q) + (x - 1)

if x

≤^ n – head +1 (no wrap around)

head(Q) + (x - 1) - n

if x > n – head +1 (wrap around)