1. Data 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

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

3. Data structures Some types of “linked objects” Linked lists: Double-linked lists:

4. Data structures Some types of “linked objects” Trees:

5. Data structures Implementation of linked lists KeyPointer 1

6. Data structures Implementation of binary trees KeyPointer 1Pointer 2

7. Data structures Implementation of general trees

8. Data structures 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,  )

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

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

11. Data structures struct Cell{int Key, pointer Next} struct Stack{pointer Head} procedure MakeStack(): S  new Stack S.Head  0 return S procedure Push(int Key, Stack S): C  new Cell C.Next  S.Head C.Key  Key S.Head  C Stacks - MakeStack, Push

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

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

14. Data structures struct Cell{int Key, pointer Next} struct Queue{pointer Head, pointer Tail} procedure MakeQueue(): Q  new Queue Q.Head  0 Q.Tail  0 return Q Queues - MakeQueue

15. Data structures procedure Enqueue(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 - Enqueue

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

17. Data structures 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

18. Data structures Heaps - Examples This may be Heap

19. Data structures Heaps - Examples This may be Heap

20. Data structures Heaps - Examples This can not be Heap

21. Data structures Heaps - Examples This can not be Heap

22. Data structures Heaps - Examples 2 45 12 14 3 1 13 This is Heap

23. Data structures Heaps - Examples 2 45 12 14 3 1 5 This is not Heap

24. Data structures Heaps - Operations Min() ExtractMin() DecreaseKey(Key) Insert(Key) Delete(Key) MakeHeap() Heapify() InitialiseHeap()

25. Data structures 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)

26. Data structures Heaps - Implementation with an array 2 45 12 3 1 13 4531221 LC(j) = 2j – n RC(j) = 2j – n – 1 P(j) =  (j + n)/2 

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

28. Data structures Heaps - Insert 3 45 12 7 2 131 T(n) =  (h) =  (log n)

29. Data structures Heaps - Delete 3 45 12 7 2 1314 T(n) =  (h) =  (log n)

30. Data structures Heaps - ExtractMin 3 45 12 7 1314 T(n) =  (h) =  (log n) 1