Data Structures and Operations: Arrays, Linked Lists, Stacks, Queues, Heaps

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 organization The basic possibilities are to store data either in arrays: or to link them with pointers:

Some types of “linked objects” Linked lists: Double-linked lists:

Some types of “linked objects” Trees:

Implementation of linked lists

Implementation of binary trees

Implementation of general trees

Operations 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, )

Stacks Operations MakeStack() Push(Key,S) Pop(S) IsEmpty(S) [Picture from J.Morris]

Stacks Operations MakeStack() Push(Key,S) Pop(S) IsEmpty(S) LIFO Last - in - first - out

Stacks - MakeStack, Push struct Cell{intKey, pointerNext} struct Stack{pointerHead} procedure MakeStack(): S newStack S.Head  0 return S procedurePush(int Key, Stack S): C newCell C.Next  S.Head C.Key  Key S.Head  C

Stacks - Pop, IsEmpty procedurePop(Stack S): C S.Head Key  C.Key S.Head  C.Next delete C return Key procedureIsEmpty(Stack S): if S.Head  0then return 0 else return 1

Queues FIFO First - in - first - out Operations MakeQueue() Enqueue(Key,Q) Dequeue(Q) IsEmpty(Q)

Queues - MakeQueue struct Cell{intKey, pointerNext} struct Queue{pointerHead, pointerTail} procedure MakeQueue(): Q newQueue Q.Head  0 Q.Tail  0 return Q

Queues - Enqueue procedureEnqueue(int Key, Queue Q): C new Cell C.Next  0 C.Key  Key if Q.Head= 0 then Q.Head  C else Tail  Q.Tail Tail.Next  C Q.Tail  C

Queues - Dequeue, IsEmpty procedureDequeue(Queue Q): C Q.Head Key  C.Key Q.Head  C.Next if Q.Head= 0 then Q.Tail  0 delete C return Key procedureIsEmpty(Queue Q): if Q.Head  0then return 0 else return 1

Heaps • 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

Heaps - Examples This may be Heap

Heaps - Examples This can not be Heap

Heaps - Examples This is Heap 1 2 12 3 45 13 14

Heaps - Examples This is not Heap 1 2 12 3 45 5 14

Heaps - Operations Min() ExtractMin() DecreaseKey(Key) Insert(Key) Delete(Key) MakeHeap() Heapify() InitialiseHeap()

Heaps - Relation between size and height Theorem For heap with n elements the height h of the corresponding binary tree is log n, i.e. h = (log n)

Heaps - Implementation with an array LC(j) = 2j–n – 1 1 RC(j) = 2j–n–2 2 12 P(j) = 1 + (j+ n)/2 3 45 13

Heaps - Implementation with an array [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Heaps - Insert 2 3 12 7 45 13 1 T(n) = (h) = (log n)

Heaps - Delete 2 3 12 7 45 13 14 T(n) = (h) = (log n)

Heaps - ExtractMin 1 3 12 7 45 13 14 T(n) = (h) = (log n)

Data Structures and Operations: Arrays, Linked Lists, Stacks, Queues, Heaps