8. Fundamental Data Structures

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1 Fundamental Data Structures Abstract data types stack, queue, implementation variants for linked lists, [Ottman/Widmayer, Kap , Cormen et al, Kap ]

2 Abstract Data Types 173 We recall A stack is an abstract data type (ADR) with operations

3 Abstract Data Types 173 We recall A stack is an abstract data type (ADR) with operations push(x, S): Puts element x on the stack S.

4 Abstract Data Types 173 We recall A stack is an abstract data type (ADR) with operations push(x, S): Puts element x on the stack S. pop(s): Removes and returns top most element of S or null

5 173 Abstract Data Types We recall A stack is an abstract data type (ADR) with operations push(x, S): Puts element x on the stack S. pop(s): Removes and returns top most element of S or null top(s): Returns top most element of S or null.

6 173 Abstract Data Types We recall A stack is an abstract data type (ADR) with operations push(x, S): Puts element x on the stack S. pop(s): Removes and returns top most element of S or null top(s): Returns top most element of S or null. isempty(s): Returns true if stack is empty, false otherwise.

7 173 Abstract Data Types We recall A stack is an abstract data type (ADR) with operations push(x, S): Puts element x on the stack S. pop(s): Removes and returns top most element of S or null top(s): Returns top most element of S or null. isempty(s): Returns true if stack is empty, false otherwise. emptystack(): Returns an empty stack.

8 Implementation Push 174 top x n x n 1 x 1 null push(x, S):

9 Implementation Push 174 top x n x n 1 x 1 null x push(x, S): 1 Create new list element with x and pointer to the value of top.

10 174 Implementation Push top x n x n 1 x 1 null x push(x, S): 1 Create new list element with x and pointer to the value of top. 2 Assign the node with x to top.

11 Implementation Pop 175 top x n x n 1 x 1 null pop(s):

12 Implementation Pop 175 top x n x n 1 x 1 null pop(s): 1 If top=null, then return null

13 175 Implementation Pop top x n x n 1 x 1 null r pop(s): 1 If top=null, then return null 2 otherwise memorize pointer p of top in r.

14 175 Implementation Pop top x n x n 1 x 1 null r pop(s): 1 If top=null, then return null 2 otherwise memorize pointer p of top in r. 3 Set top to p.next and return r

15 Analysis 176 Each of the operations push, pop, top and isempty on a stack can be executed in O(1) steps.

16 Queue (fifo) 177 A queue is an ADT with the following operations

17 Queue (fifo) 177 A queue is an ADT with the following operations enqueue(x, Q): adds x to the tail (=end) of the queue.

18 Queue (fifo) 177 A queue is an ADT with the following operations enqueue(x, Q): adds x to the tail (=end) of the queue. dequeue(q): removes x from the head of the queue and returns x (null otherwise)

19 Queue (fifo) 177 A queue is an ADT with the following operations enqueue(x, Q): adds x to the tail (=end) of the queue. dequeue(q): removes x from the head of the queue and returns x (null otherwise) head(q): returns the object from the head of the queue (null otherwise)

20 Queue (fifo) 177 A queue is an ADT with the following operations enqueue(x, Q): adds x to the tail (=end) of the queue. dequeue(q): removes x from the head of the queue and returns x (null otherwise) head(q): returns the object from the head of the queue (null otherwise) isempty(q): return true if the queue is empty, otherwise false

21 177 Queue (fifo) A queue is an ADT with the following operations enqueue(x, Q): adds x to the tail (=end) of the queue. dequeue(q): removes x from the head of the queue and returns x (null otherwise) head(q): returns the object from the head of the queue (null otherwise) isempty(q): return true if the queue is empty, otherwise false emptyqueue(): returns empty queue.

22 178 Implementation Queue x 1 x 2 x n 1 x n null head tail enqueue(x, S):

23 178 Implementation Queue x 1 x 2 x n 1 x n null x null head tail enqueue(x, S): 1 Create a new list element with x and pointer to null.

24 178 Implementation Queue x 1 x 2 x n 1 x n null x null head tail enqueue(x, S): 1 Create a new list element with x and pointer to null. 2 If tail null, then set tail.next to the node with x.

25 Implementation Queue 178 x 1 x 2 x n 1 x n x null head tail enqueue(x, S): 1 Create a new list element with x and pointer to null. 2 If tail null, then set tail.next to the node with x. 3 Set tail to the node with x.

26 Implementation Queue x 1 x 2 x n 1 x n x null head tail enqueue(x, S): 1 Create a new list element with x and pointer to null. 2 If tail null, then set tail.next to the node with x. 3 Set tail to the node with x. 4 If head = null, then set head to tail. 178

27 Invariants 179 x 1 x 2 x n 1 x n null head tail With this implementation it holds that

28 Invariants 179 x 1 x 2 x n 1 x n null head tail With this implementation it holds that either head = tail = null,

29 Invariants 179 x 1 x 2 x n 1 x n null head tail With this implementation it holds that either head = tail = null, or head = tail null and head.next = null

30 179 Invariants x 1 x 2 x n 1 x n null head tail With this implementation it holds that either head = tail = null, or head = tail null and head.next = null or head null and tail null and head tail and head.next null.

31 Implementation Queue 180 x 1 x 2 x n 1 x n null head tail dequeue(s):

32 Implementation Queue 180 x 1 x 2 x n 1 x n null r head tail dequeue(s): 1 Store pointer to head in r. If r = null, then return r.

33 180 Implementation Queue x 1 x 2 x n 1 x n null r head tail dequeue(s): 1 Store pointer to head in r. If r = null, then return r. 2 Set the pointer of head to head.next.

34 180 Implementation Queue x 1 x 2 x n 1 x n null r head tail dequeue(s): 1 Store pointer to head in r. If r = null, then return r. 2 Set the pointer of head to head.next. 3 Is now head = null then set tail to null.

35 180 Implementation Queue x 1 x 2 x n 1 x n null r head tail dequeue(s): 1 Store pointer to head in r. If r = null, then return r. 2 Set the pointer of head to head.next. 3 Is now head = null then set tail to null. 4 Return the value of r.

36 Analysis 181 Each of the operations enqueue, dequeue, head and isempty on the queue can be executed in O(1) steps.

37 Implementation Variants of Linked Lists 182 List with dummy elements (sentinels). x 1 x 2 x n 1 x n head tail Advantage: less special cases

38 Implementation Variants of Linked Lists 182 List with dummy elements (sentinels). x 1 x 2 x n 1 x n head tail Advantage: less special cases Variant: like this with pointer of an element stored singly indirect. (Example: pointer to x 3 points to x 2.)

39 Implementation Variants of Linked Lists 183 Doubly linked list null x 1 x 2 x n 1 x n null head tail

40 Overview 184 enqueue delete search concat (A) Θ(1) Θ(n) Θ(n) Θ(n) (B) Θ(1) Θ(n) Θ(n) Θ(1) (C) Θ(1) Θ(1) Θ(n) Θ(1) (D) Θ(1) Θ(1) Θ(n) Θ(1) (A) = singly linked (B) = Singly linked with dummy element at the beginning and the end (C) = Singly linked with indirect element addressing (D) = doubly linked

Linked Structures. See Section 3.2 of the text.

Linked Structures. See Section 3.2 of the text. Linked Structures See Section 3.2 of the text. First, notice that Java allows classes to be recursive, in the sense that a class can have an element which is itself an object of that class: class Person