Stacks and Queues. stacks dynamic resizing queues generics iterators applications. Stacks and queues

Transcription

1 Stacks and queues Stacks and Queues stacks dynamic resizing queues generics iterars applications Fundamental data types. Values: sets of objects Operations: insert, remove, test if empty. Intent is clear when we insert. Which item do we remove? Stack. Remove the item most recently added. Analogy. Cafeteria trays, Web surfing. Queue. Remove the item least recently added. Analogy. Registrar's line. LIFO = "last in out" FIFO = " in out" push pop Reference: Algithms in Java, Chapter 3, 4 enqueue dequeue Algithms in Java, 4 th Edition Rt Sedgewick and Kevin Wayne Copyright 2008 February 11, :54:15 PM 2 Client, implementation, interface Client, Implementation, Interface Separate interface and implementation so as : Build layers of abstraction. Reuse software. Ex: stack, queue, symbol table, union-find,... Benefits. Client can't know details of implementation client has many implementation from which choose. Implementation can't know details of client needs many clients can re-use the same implementation. Design: creates modular, reusable libraries. Perfmance: use optimized implementation where it matters. Client: program using operations defined in interface. Implementation: actual code implementing operations. Interface: description of data type, basic operations. Client: program using operations defined in interface. Implementation: actual code implementing operations. Interface: description of data type, basic operations. 3 4

2 Stacks Stack operations. push() Insert a new item on stack. pop() Remove and return the item most recently added. isempty() Is the stack empty? push pop stacks dynamic resizing queues generics iterars applications public static void main(string[] args) StackOfStrings stack = new StackOfStrings(); while (!StdIn.isEmpty()) String item = StdIn.readString(); if (item.equals("-")) StdOut.print(stack.pop()); else stack.push(item); % me.txt not that is % java StackOfStrings <.txt not that 5 6 Stack pop: linked-list implementation Stack push: linked-list implementation st the was it of st the was it String item =.item; old st the was it Node old = ; st the was it =.next; old st the was it Node = new Node(); st the was it return item; old of st the was it.item = "of";.next = old; 7 8

3 Stack: linked-list implementation Stack: linked-list trace public class StackOfStrings private Node = ; StdIn StdOut private class Node String item; Node next; "inner class" not not return == ; - not not public void push(string item) Node old = ; = new Node();.item = item;.next = old; - - that not not that not public String pop() if (isempty()) throw new RuntimeException(); String item =.item; =.next; return item; is that is 9 10 Stack: array implementation Stack: array implementation Array implementation of a stack. Use array s[] sre N items on stack. push(): add new item at s[n]. pop(): remove item from s[n-1]. s[] it was the st of times N capacity = 10 public class StackOfStrings private String[] s; private int N = 0; public StackOfStrings(int capacity) s = new String[capacity]; return N == 0; public void push(string item) s[n++] = item; public String pop() String item = s[--n]; s[n] = ; return item; this version avoids "loitering" garbage collecr only reclaims memy if no outstanding references public String pop() return s[--n]; 11 12

4 Stack: dynamic array implementation Q. How grow array? Q. How shrink array? First try. push(): increase size of s[] by 1. pop(): decrease size of s[] by 1. stacks dynamic resizing queues generics iterars applications Too expensive. Need copy all item a new array. Inserting N items takes time proptional N ~ N 2 /2. infeasible f large N Goal. Ensure that array resizing happens infrequently Stack: dynamic array implementation Q. How grow array? A. If array is full, create a new array of twice the size, and copy items. public StackOfStrings() s = new String[1]; public void push(string item) if (N == s.length) resize(2 * s.length); s[n++] = item; private void resize(int capacity) String[] dup = new String[capacity]; f (int i = 0; i < N; i++) dup[i] = s[i]; s = dup; "repeated doubling" Stack: dynamic array implementation Q. How shrink array? First try. push(): double size of s[] when array is full. pop(): halve size of s[] when array is half full. Too expensive Consider push-pop-push-pop- sequence when array is full. Time proptional N per operation. N = 5 N = 4 "thrashing" it was the st of it was the st N/2 + N ~ 2N N = 5 it was the st of Consequence. Inserting N items takes time proptional N (not N 2 ). N = 4 it was the st 15 16

5 Stack: dynamic array implementation Stack: dynamic array implementation trace Q. How shrink array? Efficient solution. push(): double size of s[] when array is full. pop(): halve size of s[] when array is one-quarter full. public String pop() String item = s[n-1]; s[n-1] = ; N--; if (N == s.length/4) resize(s.length / 2); s[n++] = item; return item; Invariant. Array is always tween 25% and 100% full. Consequence. Starting from empty data structure, any sequence of M ops takes time proptional M. "amtized" bound a StdIn StdOut N a.length not 4 4 not 5 8 not not 5 8 not not - not 3 8 that 4 8 that - that is 2 2 is Stack implementations: dynamic array vs. linked List Tradeoffs. Can implement with either array linked list; client can use interchangeably. Which is tter? Linked list. Every operation takes constant time in wst-case. Uses extra time and space deal with the links. Array. Every operation takes constant amtized time. Less wasted space. stacks dynamic resizing queues generics iterars applications 19 20

6 Queues Queue dequeue: linked list implementation Queue operations. enqueue() dequeue() isempty() Insert a new item on queue. Delete and return the item least recently added. Is the queue empty? it last was the st of String item =.item; public static void main(string[] args) QueueOfStrings q = new QueueOfStrings(); while (!StdIn.isEmpty()) String item = StdIn.readString(); if (item.equals("-")) StdOut.print(q.dequeue()); else q.enqueue(item); else % me.txt not that is % java QueueOfStrings <.txt not last was the st of =.next; last was the st of return item; Queue enqueue: linked list implementation Queue: linked list implementation last public class QueueOfStrings private Node, last; it was the st private class Node String item; Node next; oldlast last return == ; it was the st of oldlast last it was the st of oldlast last it was the st of Node oldlast = last; Node last = new Node(); last.item = "of"; last.next = ; oldlast.next = last; public void enqueue(string item) Node oldlast = last; last = new Node(); last.item = item; last.next = ; if (isempty()) = last; else oldlast.next = last; public String dequeue() String item =.item; =.next; if (isempty()) last = ; return item; 23 24

7 Queue: dynamic array implementation Array implementation of a queue. Use array q[] sre items in queue. enqueue(): add new item at q[tail]. dequeue(): remove item from q[head]. Update head and tail modulo the capacity. Add repeated doubling and shrinking. q[] the st of times head tail capacity = 10 stacks dynamic resizing queues generics iterars applications Parameterized stack Parameterized stack We implemented: StackOfStrings. We also want: StackOfURLs, StackOfCusmers, StackOfInts, etc? We implemented: StackOfStrings. We also want: StackOfURLs, StackOfCusmers, StackOfInts, etc? Attempt 1. Implement a separate stack class f each type. Rewriting code is tedious and err-prone. Maintaining cut-and-pasted code is tedious and err-prone. Attempt 2. Implement a stack with items of type Object. Casting is required in client. Casting is err-prone: run-time err if types most reasonable approach until Java 1.5. [hence, used in AlgsJava] StackOfObjects s = new StackOfObjects(); Apple a = new Apple(); Orange b = new Orange(); s.push(a); s.push(b); a = (Apple) (s.pop()); run-time err 27 28

8 Parameterized stack Generic stack: linked list implementation We implemented: StackOfStrings. We also want: StackOfURLs, StackOfCusmers, StackOfInts, etc? public class StackOfStrings private Node = ; public class Stack<Item> private Node = ; Attempt 3. Java generics. Avoid casting in both client and implementation. Discover type mismatch errs at compile-time instead of run-time. Stack<Apple> s = new Stack<Apple>(); Apple a = new Apple(); Orange b = new Orange(); s.push(a); s.push(b); a = s.pop(); type parameter compile-time err Guiding principles. Welcome compile-time errs; avoid run-time errs. private class Node String item; Node next; return == ; public void push(string item) Node old = ; = new Node();.item = item;.next = old; public String pop() String item =.item; =.next; return item; private class Node Item item; Node next; return == ; public void push(item item) Node old = ; = new Node();.item = item;.next = old; public Item pop() Item item =.item; =.next; return item; generic type name Generic stack: array implementation Generic stack: array implementation public class StackOfStrings private String[] s; private int N = 0; public class Stack<Item> private Item[] s; private int N = 0; public class StackOfStrings private String[] s; private int N = 0; public class Stack<Item> private Item[] s; private int N = 0; public StackOfStrings(int capacity) s = new String[capacity]; public Stack(int capacity) s = new Item[capacity]; public StackOfStrings(int capacity) s = new String[capacity]; public Stack(int capacity) s = (Item[]) new Object[capacity]; return N == 0; return N == 0; return N == 0; return N == 0; public void push(string item) s[n++] = item; public void push(item item) s[n++] = item; public void push(string item) s[n++] = item; public void push(item item) s[n++] = item; public String pop() return s[--n]; public Item pop() return s[--n]; public String pop() return s[--n]; public Item pop() return s[--n]; the way it should the way it generic array creation not allowed in Java the ugly cast 31 32

9 Generic data types: auboxing Q. What do about primitive types? Wrapper type. Each primitive type has a wrapper object type. Ex: Integer is wrapper type f int. Auboxing. Aumatic cast tween a primitive type and its wrapper. Syntactic sugar. Behind-the-scenes casting. Stack<Integer> s = new Stack<Integer>(); s.push(17); // s.push(new Integer(17)); int a = s.pop(); // int a = s.pop().intvalue(); stacks dynamic resizing queues generics iterars applications Botm line. Client code can use generic stack f any type of data Iteration Iterars Design challenge. Suppt iteration over stack items by client, without revealing the internal representation of the stack. Q. What is an Iterable? A. Has a method that returns an Iterar. public interface Iterable<Item> Iterar<Item> iterar(); s[] i N it was the st of times current of st the was it Q. What is an Iterar? A. Has methods hasnext() and next(). Q. Why make data structures Iterable? A. Java suppts elegant client code. public interface Iterar<Item> boolean hasnext(); Item next(); void remove(); optional; use at your own risk feach statement equivalent code Java solution. Make stack Iterable. f (String s : stack) StdOut.println(s); Iterar<String> i = stack.iterar(); while (i.hasnext()) String s = i.next(); StdOut.println(s); 35 36

10 Stack iterar: linked list implementation Stack iterar: array implementation impt java.util.iterar; public class Stack<Item> implements Iterable<Item>... public Iterar<Item> iterar() return new ListIterar(); private class ListIterar implements Iterar<Item> private Node current = ; public boolean hasnext() return current!= ; public void remove() /* not suppted */ public Item next() Item item = current.item; current = current.next; return item; impt java.util.iterar; public class Stack<Item> implements Iterable<Item> public Iterar<Item> iterar() return new ArrayIterar(); private class ArrayIterar implements Iterar<Item> private int i = N; public boolean hasnext() return i > 0; public void remove() /* not suppted */ public Item next() return s[--i]; i N current s[] it was the st of times of st the was it Stack applications stacks dynamic resizing queues generics iterars applications Real wld applications. Parsing in a compiler. Java virtual machine. Undo in a wd process. Back butn in a Web browser. PostScript language f printers. Implementing function calls in a compiler

11 Function calls Arithmetic expression evaluation How a compiler implements a function. Function call: push local environment and return address. Return: pop return address and local environment. Recursive function. Function that calls itself. Note. Can always use an explicit stack remove recursion. p = 216, q = 192 p = 192, q = 24 gcd (216, 192) static int gcd(int p, int q) if (q == 0) return p; else return gcd(q, p % q); gcd (192, 24) p = 24, q = 0 static int gcd(int p, int q) if (q == 0) return p; else return gcd(q, p % q); gcd (24, 0) static int gcd(int p, int q) if (q == 0) return p; else return gcd(q, p % q); Goal. Evaluate infix expressions. ( 1 + ( ( ) * ( 4 * 5 ) ) ) operand Two-stack algithm. [E. W. Dijkstra] Value: push on the value stack. Operar: push on the operar stack. Left parens: igne. Right parens: pop operar and two values; push the result of applying that operar those values on the operand stack. Context. An interpreter! operar value stack operar stack * * * * * * * ( 1 + ( ( ) * ( 4 * 5 ) ) ) + ( ( ) * ( 4 * 5 ) ) ) ( ( ) * ( 4 * 5 ) ) ) + 3 ) * ( 4 * 5 ) ) ) 3 ) * ( 4 * 5 ) ) ) ) * ( 4 * 5 ) ) ) * ( 4 * 5 ) ) ) ( 4 * 5 ) ) ) * 5 ) ) ) 5 ) ) ) ) ) ) ) ) ) Arithmetic expression evaluation Crectness public class Evaluate public static void main(string[] args) Stack<String> ops = new Stack<String>(); Stack<Double> vals = new Stack<Double>(); while (!StdIn.isEmpty()) String s = StdIn.readString(); if (s.equals("(")) ; else if (s.equals("+")) ops.push(s); else if (s.equals("*")) ops.push(s); else if (s.equals(")")) String op = ops.pop(); if (op.equals("+")) vals.push(vals.pop() + vals.pop()); else if (op.equals("*")) vals.push(vals.pop() * vals.pop()); else vals.push(double.parsedouble(s)); StdOut.println(vals.pop()); % java Evaluate ( 1 + ( ( ) * ( 4 * 5 ) ) ) Q. Why crect? A. When algithm encounters an operar surrounded by two values within parentheses, it leaves the result on the value stack. as if the iginal input were: Repeating the argument: ( 1 + ( ( ) * ( 4 * 5 ) ) ) ( 1 + ( 5 * ( 4 * 5 ) ) ) ( 1 + ( 5 * 20 ) ) ( ) 101 Extensions. Me ops, precedence der, associativity

12 Stack-based programming languages PostScript Observation 1. The 2-stack algithm computes the same value if the operar occurs after the two values. ( 1 ( ( ) ( 4 5 * ) * ) + ) Observation 2. All of the parentheses are redundant! * * + Jan Lukasiewicz Page description language. Explicit stack. Full computational model Graphics engine. Basics. %!: I am a PostScript program. Literal: push me on the stack. Function calls take arguments from stack. Turtle graphics built in. a PostScript program %! move 0 72 rline 72 0 rline 0-72 rline rline 2 setlinewidth stroke its output Botm line. Postfix "reverse Polish" notation. Applications. Postscript, Fth, calculars, Java virtual machine, PostScript PostScript Data types. basic: integer, floating point, boolean,... Graphics: font, path, curve,... Full set of built-in operars. Text and strings. Full font suppt. System.out.print() show (display a string, using current font). cvs (convert anything a string). String() Variables (and functions). Identifiers start with /. def operar associates id with value. Braces. args on stack. function definition %! /box /sz exch def 0 sz rline sz 0 rline 0 sz neg rline sz neg 0 rline def %! /Helvetica-Bold findfont 16 scalefont setfont move (Square root of 2:) show move 2 sqrt 10 string cvs show Square root of 2: function calls move 72 box move 144 box 2 setlinewidth stroke 47 48

13 PostScript PostScript F loop. from, increment, on stack. Loop body in braces. f operar. If-else conditional. Boolean on stack. Alternatives in braces. if operar.... (hundreds of operars) mul dup 2 add move 72 box f Application 1. All figures in Algithms in Java Application 2. Deluxe version of StdDraw also saves PostScript f vecr graphics. %! translate /kochr 2 copy ge dup 0 rline 3 div 2 copy kochr 60 rotate 2 copy kochr -120 rotate 2 copy kochr 60 rotate 2 copy kochr ifelse pop pop def 0 0 move kochr 0 81 move kochr move kochr move kochr stroke See page Queue applications M/M/1 queuing model Familiar applications. itunes playlist. Data buffers (ipod, TiVo). Asynchronous data transfer (file IO, pipes, sockets). Dispensing requests on a shared resource (printer, process). M/M/1 queue. Cusmers arrive accding Poisson process at rate of λ per minute. Cusmers are serviced with rate of µ per minute. interarrival time has exponential distribution Pr[X x] = 1 - e - λ x service time has exponential distribution Pr[X x] = 1 - e - µ x Simulations of the real wld. Traffic analysis. Waiting times of cusmers at call center. Determining numr of cashiers have at a supermarket. Arrival rate Infinite queue Server Departure rate Q. What is average wait time W of a cusmer in system? Q. What is average numr of cusmers L in system? 51 52

14 M/M/1 queuing model: example simulation M/M/1 queuing model: event-based simulation time (seconds) 0 public class MM1Queue public static void main(string[] args) double lambda = Double.parseDouble(args[0]); // arrival rate double mu = Double.parseDouble(args[1]); // service rate double nextarrival = StdRandom.exp(lambda); double nextservice = nextarrival + StdRandom.exp(mu); arrival departure wait Queue<Double> queue = new Queue<Double>(); Hisgram hist = new Hisgram("M/D/1 Queue", 60); while (true) // next event is an arrival while (nextarrival < nextservice) queue.enqueue(nextarrival); nextarrival += StdRandom.exp(lambda); // next event is a service completion double arrival = queue.dequeue(); double wait = nextservice - arrival; hist.adddatapoint(math.min(60, (int) (Math.round(wait)))); if (queue.isempty()) nextservice = nextarrival + StdRandom.exp(mu); else nextservice = nextservice + StdRandom.exp(mu); 54 M/M/1 queuing model: experiments M/M/1 queuing model: experiments Observation. If service rate µ is much larger than arrival rate λ, cusmers gets good service. Observation. As service rate µ approaches arrival rate λ, services goes h***. % java MM1Queue % java MM1Queue

15 M/M/1 queuing model: experiments M/M/1 queuing model: analysis Observation. As service rate µ approaches arrival rate λ, services goes h***. M/M/1 queue. Exact fmulas known. % java MM1Queue.2.21 wait time W and queue length L approach infinity as service rate approaches arrival rate W = λ 2µ (µ λ) + 1 µ, L = λ W Little s Law Me complicated queueing models. Event-based simulation essential! Queueing they. See ORFE