Data structuresData structures

What does that mean?

In general there are two aspects:

• how data will be organized in computer memory

• what will be the operations that will be performed with them

Data structuresData structuresData organizationThe basic possibilities are to store data either in arrays:

or to link them with pointers:

Data structuresData structuresSome types of “linked objects”Linked lists:

Double-linked lists:

Data structuresData structuresSome types of “linked objects”Trees:

Data structuresData structuresImplementation of linked lists

Key Pointer 1

Data structuresData structuresImplementation of binary trees

Key Pointer 1

Pointer 2

Data structuresData structuresImplementation of general trees

Data structuresData structuresOperations with data structures

Dynamic Dictionaries

LookUp(Key)Insert(Key)Delete(Key)Make()

Priority Queues

Min()ExtractMin()DecreaseKey(Key)Insert(Key)Delete(Key)Make()

Other popular operations with data structures

- unify elements of 2 data structures into one (Union, Meld, )

Data structuresData structuresStacks

Operations

MakeStack()Push(Key,S)Pop(S)IsEmpty(S)

[Picture from J.Morris]

Data structuresData structuresStacks

Operations

MakeStack()Push(Key,S)Pop(S)IsEmpty(S)

LIFOLast - in - first - out

Data structuresData structures

struct Cell{int Key, pointer Next}struct Stack{pointer Head}

procedure MakeStack(): S new Stack S.Head 0return S

procedure Push(int Key, Stack S):C new Cell C.Next S.HeadC.Key KeyS.Head C

Stacks - MakeStack, Push

Data structuresData structures

procedure Pop(Stack S):C S.Head Key C.KeyS.Head C.Nextdelete Creturn Key

procedure IsEmpty(Stack S):if S.Head 0 then return 0else return 1

Stacks - Pop, IsEmpty

Data structuresData structuresQueues

Operations

MakeQueue()Enqueue(Key,Q)Dequeue(Q)IsEmpty(Q)

FIFOFirst - in - first - out

Data structuresData structures

struct Cell{int Key, pointer Next}struct Queue{pointer Head, pointer Tail}

procedure MakeQueue(): Q new Queue Q.Head 0Q.Tail 0return Q

Queues - MakeQueue

Data structuresData structures

procedure Enqueue(int Key, Queue Q):C new Cell C.Next 0C.Key Key if Q.Head = 0 then

Q.Head Celse Tail Q.Tail

Tail.Next CQ.Tail C

Queues - Enqueue

Data structuresData structures

procedure Dequeue(Queue Q): C Q.Head Key C.KeyQ.Head C.Nextif Q.Head = 0 then Q.Tail 0delete Creturn Key

procedure IsEmpty(Queue Q):if Q.Head 0 then return 0else return 1

Queues - Dequeue, IsEmpty

Data structuresData structuresHeaps• They are binary trees with all levels completed, except the lowest one which may have uncompleted section on the right side

• They satisfy so called Heap Property - for each subtree of heap the key for the root of subtree must not exceed the keys of its (left and right) children

Data structuresData structuresHeaps - Examples This may be Heap

Data structuresData structuresHeaps - Examples This may be Heap

Data structuresData structuresHeaps - ExamplesThis can not be Heap

Data structuresData structuresHeaps - ExamplesThis can not be Heap

Data structuresData structuresHeaps - Examples

2

45

12

143

1

13

This is Heap

Data structuresData structuresHeaps - Examples

2

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143

1

5

This is not Heap

Data structuresData structuresHeaps - Operations

Min()ExtractMin()DecreaseKey(Key)Insert(Key)Delete(Key)MakeHeap()

Heapify()InitialiseHeap()

Data structuresData structuresHeaps - Relation between size and heightTheoremFor heap with n elements the height h of the correspondingbinary tree is log n, i.e. h = (log n)

Data structuresData structuresHeaps - Implementation with an array

2

45

12

3

1

13

13 45 3 12 2 1

LC(j) = 2j – n

RC(j) = 2j – n – 1

P(j) = (j + n)/2

Data structuresData structuresHeaps - Implementation with an array

[Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structuresData structuresHeaps - Insert

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7

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13 1

T(n) = (h) = (log n)

Data structuresData structuresHeaps - Delete

3

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13 14

T(n) = (h) = (log n)

Data structuresData structuresHeaps - ExtractMin

3

45

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7 13 14

T(n) = (h) = (log n)

1