Lecture 16. Lecture

Transcription

1 Recursive lists D0010E Variants of lists Doubly linked lists Binary trees Circular lists - Håkan Jonsson 1 - Håkan Jonsson 2 - Håkan Jonsson 3 1

2 1. Circular lists A singly linked list has a beginning and an end. In a circular list there is a link also from the end to the beginning. Makes it possible to move between elements in circle. Useful in situawons where a program should pay axenwon to several things (objects) one a\er another over and over again. Process control in operawng systems. Round robin. Lecture 17 - Håkan Jonsson 4 OperaWons - Håkan Jonsson 5 - Håkan Jonsson 6 2

3 2. Doubly linked lists A double linked list is a linked list where internal nodes have two links. One leading forward ("rightwards") to the next node and one leading backwards ("le\wards") to the previous node. Such a list can be seen as two single linked lists going opposite direcwons merged into one. It has two headers, one at each end of the list. It provides much flexibility to the programmer since it is easy to move among the elements and in any direcwon. Add mulwple iterators using the Iterator design pa7ern - to enable clients to iterate over the elements of the list. - Håkan Jonsson 7 ImplementaWon view Headers - Håkan Jonsson 8 UML - Håkan Jonsson 9 3

4 ImplementaWon - Håkan Jonsson 10 Method: insertbefore - Håkan Jonsson 11 - Håkan Jonsson 12 4

5 3. Lists as recursive structures We have so far seen two ways to implement lists: Using links and internal nodes. A list has an inner structure that consists of node objects linked to each other. Singly and doubly linked lists. Using arrays. List elements are placed in an array. Indices are used as links. These implementawons are based on the view that a list is simply a sequence. An alternawve view is to describe a list recursively. What follows is a parallel between how data organiza>on can be view as either iterawve (sequenwal) or recursive to how computa>ons can be view as either iterawve or recursive. - Håkan Jonsson 13 - Håkan Jonsson Rooms reconsidered In Lab assignment 2 we programmed dynamic objects of type Room. A room had four corridors leading out of the room to other rooms, or just back again. (Technically, they were null to start with but assume for now they were all valid references.) W N S E Here all four corridors have been assigned the value this. - Håkan Jonsson 15 5

6 N W E S public class Room { Color c; Room north; Room east; Room south; Room west;... - Håkan Jonsson 16 W E public class Room { Color c; Room east; Room west;... - Håkan Jonsson 17 E public class Room { Color c; Room east;... - Håkan Jonsson 18 6

7 E public class Room { Color c; Room east;... - Håkan Jonsson 19 E public class Room { Color c; Room east;... - Håkan Jonsson 20 - Håkan Jonsson 21 7

8 5. A recursive stack This kind of stack is either empty, lacking content, or, non- empty, containing one element and a stack. 2: A non- empty stack containing "wine- red" and this stack. 1 d 2 d 1: A non- empty stack containing "blue" and this stack. 3 d 3: An empty stack. - Håkan Jonsson 22 - Håkan Jonsson Example: IPStack public class IPStack<E> implements Iterable<E> { private E value; private IPStack<E> stack; public IPStack() { this.stack = this; public IPStack(E value, IPStack<E> stack) { this.value = value; this.stack = stack; IPStackIterator<E> <<interface>> Iterator<E> + hasnext() : boolean + next() : void + remove() : void <<interface>> Iterable<E> + iterator() : Iterator<E> IPStack<E> + <<constructor>> IPStack() + <<constructor>> IPStack(value : E, stack : IPStack<E>) + isempty() : boolean + push(item : E) : IPStack<E> + pop() : IPStack<E> + peek() : E + iterator() : Iterator<E> - s : IP + isem + push + pop( + peek + itera - Håkan Jonsson 24 8

9 ImplementaWon public boolean isempty() { return stack == this; public IPStack<E> push(e value) { return new IPStack<E>(value, this); public IPStack<E> pop() { if (!isempty()) { return stack; else { throw new NoSuchElementExcepWon(); public E peek() { if (!isempty()) { return value; else { throw new NoSuchElementExcepWon(); - Håkan Jonsson 25 ImplementaWon private class IPStackIterator implements Iterator<E> { IPStack<E> stack = IPStack.this; public boolean hasnext() { return!stack.isempty(); public E next() { E element = stack.peek(); stack = stack.pop(); return element; public void remove() { throw new UnsupportedOperaWonExcepWon(); public Iterator<E> iterator() { return new IPStackIterator(); - Håkan Jonsson 26 - Håkan Jonsson 27 9

10 7. About IPStack An IPStack object never changes so it is immutable. All operawons create new stacks or returns parts of exiswng stacks. Moreover, it is persistent: A persistent data structure is a data structure that always preserves the previous version of itself when it is modified. Actually, persistence follows since it is immutable. Each iterator gets a reference to the current stack, that can not change. - Håkan Jonsson 28 UML <<interface>> Iterator<E> + hasnext() : boolean + next() : void + remove() : void IPStackIterator<E> <<interface>> Iterable<E> + iterator() : Iterator<E> I = Immutable P = Persistent IPStack<E> + <<constructor>> IPStack() + <<constructor>> IPStack(value : E, stack : IPStack<E>) + isempty() : boolean + push(item : E) : IPStack<E> + pop() : IPStack<E> + peek() : E + iterator() : Iterator<E> MPStack<E> - s : IPStack<E> + isempty() : boolean + push(item : E) : void + pop() : void + peek() : E + iterator() : Iterator<E> M = Mutable P = Persistent - Håkan Jonsson 29 - Håkan Jonsson 30 10

11 8. Natural numbers Peano's axioms gives a recursive definiwon of natural numbers: 1) Zero is a natural number. 2) The successor of a natural number is a natural number. 3) Natural numbers with the same successor are idenwcal. 4) Zero is not the successor of any natural number. 5) If P is a property such that P holds for zero, and P also holds for the successor of a natural number n whenever P holds for n, P holds for all natural numbers. Natural numbers and the way we usually refer to them in the decimal system: "Zero" 0 (by Axiom 1) "The successor of zero" 1 (by Axiom 2) "The successor of the successor of zero" 2 ( - "- ) and so on for 3, 4, 5,... - Håkan Jonsson 31 RepresenWng natural numbers If we use "Zero" for 0 and "Succ" for "successor" we could write: Zero 1 Succ Zero 2 Succ (Succ Zero) 3 Succ (Succ (Succ Zero)) 4 Succ (Succ (Succ (Succ Zero))) 5... and so on. To introduce natural numbers into Java we define a class Nat with subclasses Zero and Succ. We can then define operawons like addi>on and mul>plica>on: x + Zero = x x * Zero = Zero x + (Succ y) = Succ (x + y) x * (Succ y) = x + x * y - Håkan Jonsson 32 AddiWon Example: x = Succ (Succ Zero) = 2 y = Succ (Succ (Succ Zero)) = 3 x + y = Succ (Succ Zero) + Succ (Succ (Succ Zero)) = Succ (Succ (Succ Zero) + Succ (Succ Zero)) = Succ (Succ (Succ (Succ Zero) + Succ Zero)) = Succ (Succ (Succ (Succ (Succ Zero) + Zero))) = Succ (Succ (Succ (Succ (Succ Zero)))) = 5. - Håkan Jonsson 33 11

12 MulWplicaWon Example: x = Succ (Succ Zero) = 2 y = Succ (Succ (Succ Zero)) = 3 x * y = Succ (Succ Zero) * Succ (Succ (Succ Zero)) = Succ (Succ Zero) + (Succ (Succ Zero) * Succ (Succ Zero)) = Succ (Succ Zero) + (Succ (Succ Zero) + (Succ (Succ Zero) * Succ Zero)) = Succ (Succ Zero) + (Succ (Succ Zero) + (Succ (Succ Zero) + (Succ (Succ Zero) * Zero))) = Succ (Succ Zero) + (Succ (Succ Zero) + (Succ (Succ Zero) + (Zero))) = 2 + (2 + (2 + 0)) = 2 + (2 + 2) = = 6 by the addiwon rule. - Håkan Jonsson 34 ImplementaWon l16.peano.core l16.peano Peano + main(string[] args) : void + f(nat n) : Nat <<abstract>> Nat <<abstract>> + add(nat n) : Nat <<abstract>> + mul(nat n) : Nat <<abstract>> + pred() : Nat <<abstract>> + nat2int() : int <<abstract>> + iszero() : boolean + int2nat(int i) : Nat Zero + add(nat n) : Nat + add(nat n) : Nat + pred() : Nat + nat2int() : int + iszero() : boolean + tostring() : String Succ - n : Nat + add(nat n) : Nat + add(nat n) : Nat + pred() : Nat + nat2int() : int + iszero() : boolean + tostring() : String - Håkan Jonsson 35 - Håkan Jonsson 36 12

13 9. Binary search trees A binary tree is linked structure similar to a list but in which each node has not one but two reference variables that refer to the rest of the structure. A list can be seen as the special case of a binary tree in which one of the two variables are null in all nodes. A binary tree is a rooted singly- connected acyclic graph. A binary tree has two children called the lef subtree and the right subtree. private class TreeNode<E> {! E content;! TreeNode left, right;!! private class ListNode<E> {! E content;! ListNode next;!! - Håkan Jonsson 37 Binary search trees Let B be a binary tree with element E in the root, le\ subtree L, and right subtree R. If either B is empty or if the elements in L are all smaller than E, and the elements in R are at least as large as E, B is a binary search tree if, and only if, also L and R are binary search trees. Binary search trees are used to order things. They can be made very efficient ("balanced"). B L E R - Håkan Jonsson 38 - Håkan Jonsson 39 13