Algorithms. Algorithms. Algorithms 1.3 BAGS, QUEUES, AND STACKS. stacks resizing arrays queues generics iterators applications

Transcription

1 Algithms ROBERT SEDGEWICK KEVIN WAYNE Stacks and queues 1.3 BAGS, QUEUES, AND STACKS Fundamental data types. Value: collection of objects. Operations: add, remove, iterate, test if empty. Intent is clear when we add. Which item do we remove? Algithms F O U R T H E D I T I O N ROBERT SEDGEWICK KEVIN WAYNE stacks resizing arrays queues generics iterars applications stack queue enqueue Stack. Examine the item. push pop dequeue LIFO = "last in out" Queue. Examine the item. FIFO = " in out" Last updated on Feb 9, 2015, 5:18 AM 2 Client, implementation, interface Separate interface and implementation. Ex: stack, queue, bag, priity queue, symbol table, union-find,. Benefits. Client cannot know details of implementation client has many implementation from which choose. Implementation cannot know details of client needs many clients can re-use the same implementation. Design: creates modular, reusable libraries. Perfmance: substitute optimized implementation when it matters. Client: program using operations defined in interface. Implementation: actual code implementing operations. Interface: description of data type, basic operations. Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications 3

2 Stack API Stacks quiz 1 Warmup API. Stack of strings data type. How implement a stack with a singly-linked list? public class StackOfStrings push pop A. StackOfStrings() create an empty stack it was the st of void push(string item) add a new string stack String pop() remove and return the string B. boolean isempty() is the stack empty? int size() numr of strings on the stack of st the was it C. None of the above. Warmup client. Reverse sequence of strings from standard input. 5 D. I don't know. 6 Stack: linked-list implementation Maintain pointer node in a singly-linked list. Push new item fe. Pop item from. Stack pop: linked-list implementation save item return String item =.item; delete node inner class =.next; String item; Node next; of st the was it return saved item 7 8

3 Stack push: linked-list implementation Stack: linked-list implementation in Java save a link the list Node old = ; public class LinkedStackOfStrings private Node = ; old private String item; private Node next; private inner class (access modifiers f instance variables don't matter) inner class String item; Node next; create a new node f the ginning = new Node(); old set the instance variables in the new node return == ; public void push(string item) Node old = ; = new Node();.item = item;.next = old;.item = "not";.next = old; not String item =.item; =.next; 9 10 Stack: linked-list implementation perfmance Stacks quiz 2 Proposition. Every operation takes constant time in the wst case. How implement a fixed-capacity stack with an array? Proposition. A stack with N items uses ~ 40 N bytes. A. it was the st of times inner class object overhead 16 bytes (object overhead) String item; Node next; extra overhead item next references 8 bytes (inner class extra overhead) 8 bytes (reference String) 8 bytes (reference Node) B. times of st the was it 40 bytes allocated per stack node Remark. This accounts f the memy f the stack C. None of the above. (but not memy f the strings themselves, which the client owns). D. I don't know

4 Fixed-capacity stack: array implementation Use array s[] sre N items on stack. push(): add new item at s[n]. pop(): remove item from s[n-1]. Fixed-capacity stack: array implementation public class FixedCapacityStackOfStrings private String[] s; private int N = 0; a cheat (stay tuned) public FixedCapacityStackOfStrings(int capacity) s = new String[capacity]; s[] it was the st of times N capacity = 10 return N == 0; public void push(string item) s[n++] = item; use index in array; then increment N return s[--n]; Defect. Stack overflows when N exceeds capacity. [stay tuned] decrement N; then use index in array Stack considerations Overflow and underflow. Underflow: throw exception if pop from an empty stack. Overflow: use "resizing array" f array implementation. [stay tuned] Null items. We allow items added. Duplicate items. We allow an item added me than once. Loitering. Holding a reference an object when it is no longer needed. return s[--n]; loitering String item = s[--n]; s[n] = ; this version avoids "loitering": garbage collecr can reclaim memy f an object only if no remaining references Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications 15

5 Stack: resizing-array implementation Stack: resizing-array implementation Problem. Requiring client provide capacity does not implement API! Q. How grow array? "repeated doubling" Q. How grow and shrink array? A. If array is full, create a new array of twice the size, and copy items. First try. push(): increase size of array s[] by 1. pop(): decrease size of array s[] by 1. Too expensive. infeasible f large N Need copy all items a new array, f each operation. Array accesses add N items = N + ( (N 1)) ~ N 2. 1 array access per push 2(k 1) array accesses expand size k (igning cost create new array) public ResizingArrayStackOfStrings() 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[] copy = new String[capacity]; f (int i = 0; i < N; i++) copy[i] = s[i]; s = copy; Array accesses add N = 2 i items. N + ( N) ~ 3N. Challenge. Ensure that array resizing happens infrequently array access per push k array accesses double size k (igning cost create new array) 18 Stack: resizing-array implementation Stack: resizing-array implementation Q. How shrink array? Q. How shrink array? First try. push(): double size of array s[] when array is full. pop(): halve size of array s[] when array is one-half full. Efficient solution. push(): double size of array s[] when array is full. pop(): halve size of array s[] when array is one-quarter full. Too expensive in wst case. Consider push-pop-push-pop- sequence when array is full. Each operation takes time proptional N. full not String item = s[--n]; s[n] = ; if (N > 0 && N == s.length/4) resize(s.length/2); push("") not pop() not push("") not Invariant. Array is tween 25% and 100% full

6 Stack resizing-array implementation: perfmance Stack resizing-array implementation: memy usage Amtized analysis. Starting from an empty data structure, average running time per operation over a wst-case sequence of operations. Proposition. Starting from an empty stack, any sequence of M push and pop operations takes time proptional M. Proposition. A ResizingArrayStackOfStrings uses ~ 8 N ~ 32 N bytes of memy f a stack with N items. ~ 8 N when full. ~ 32 N when one-quarter full. construct push typical wst amtized N 1 public class ResizingArrayStackOfStrings private String[] s; private int N = 0; 8 bytes array size pop size 1 N doubling and halving operations der of growth of running time f resizing array stack with N items Remark. This accounts f the memy f the stack (but not the memy f strings themselves, which the client owns) Stack implementations: resizing array vs. linked list Tradeoffs. Can implement a stack with either resizing array linked list; client can use interchangeably. Which one is tter? Linked-list implementation. Every operation takes constant time in the wst case. Uses extra time and space deal with the links. Resizing-array implementation. Every operation takes constant amtized time. Less wasted space. N = 4 not Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications not 23

7 Queue API Queues quiz 1 enqueue How implement a queue with a singly-linked linked list? public class QueueOfStrings A. QueueOfStrings() create an empty queue times of st the was it void enqueue(string item) add a new string queue String dequeue() remove and return the string boolean isempty() is the queue empty? int size() numr of strings on the queue B. it was the st of times dequeue C. None of the above. 25 D. I don't know. 26 Queue: linked-list implementation Maintain one pointer node in a singly-linked list. Maintain another pointer last last node. Dequeue from. Enqueue after last. Queue dequeue: linked-list implementation save item return String item =.item; delete node =.next; inner class String item; Node next; last last it was the st of times last return saved item Remark. Identical code linked-list stack pop()

8 Queue enqueue: linked-list implementation Queue: linked-list implementation in Java save a link the last node Node oldlast = last; oldlast last public class LinkedQueueOfStrings private Node, last; /* same as in LinkedStackOfStrings */ return == ; inner class String item; Node next; create a new node f the end last = new Node(); last.item = "not"; link the new node the end of the list oldlast.next = last; oldlast oldlast last last not not 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 = ; special cases f empty queue Queues quiz 2 How implement a fixed-capacity queue with an array? A. it was the st of times Queue: resizing-array implementation 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 resizing array. B. times of st the was it q[] the st of times head tail capacity = 10 C. None of the above. D. I don't know. Q. How resize? 31 32

9 QUEUE WITH TWO STACKS Problem. Implement a queue with two stacks so that: Each queue op uses a constant amtized numr of stack ops. At most constant extra memy (sides two stacks). 1.3 B AGS, QUEUES, AND S TACKS Applications. Job interview. Implement an immutable persistent queue. Implement a queue in a (purely) functional programming language. stacks resizing arrays queues Algithms generics iterars R OBERT S EDGEWICK K EVIN W AYNE applications 33 Parameterized stack Generic stack: linked-list implementation We implemented: StackOfStrings. public class LinkedStackOfStrings private Node = ; We also want: StackOfURLs, StackOfInts, StackOfApples, StackOfOranges,. String item; Node next; Solution in Java: generics. return == ; type parameter (use both specify type and call construcr) Stack<Apple> stack = new Stack<Apple>(); Apple apple = new Apple(); Orange ange = new Orange(); stack.push(apple); stack.push(ange); public class Stack<Item> private Node = ; compile-time err Item item; Node next; return == ; generic type name public void push(string item) Node old = ; = new Node();.item = item;.next = old; public void push(item item) Node old = ; = new Node();.item = item;.next = old; String item =.item; =.next; public Item pop() Item item =.item; =.next; 36

10 Generic stack: array implementation Generic stack: array implementation the way it should the way it is public class FixedCapacityStackOfStrings private String[] s; private int N = 0; public class FixedCapacityStack<Item> private Item[] s; private int N = 0; public class FixedCapacityStackOfStrings private String[] s; private int N = 0; public class FixedCapacityStack<Item> private Item[] s; private int N = 0; public..stackofstrings(int capacity) s = new String[capacity]; public FixedCapacityStack(int capacity) s = new Item[capacity]; public..stackofstrings(int capacity) s = new String[capacity]; public FixedCapacityStack(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; return s[--n]; public Item pop() return s[--n]; return s[--n]; public Item pop() return generic array creation not allowed in Java the ugly cast Unchecked cast Generic data types: auboxing Q. What do about primitive types? % javac FixedCapacityStack.java Note: FixedCapacityStack.java uses unchecked unsafe operations. Note: Recompile with -Xlint:unchecked f details. % javac -Xlint:unchecked FixedCapacityStack.java FixedCapacityStack.java:26: warning: [unchecked] unchecked cast found : java.lang.object[] required: Item[] 1 warning a = (Item[]) new Object[capacity]; ^ Q. Why does Java make me cast ( use reflection)? Sht answer. Backward compatibility. Long answer. Need learn about type erasure and covariant arrays. 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. Stack<Integer> stack = new Stack<Integer>(); stack.push(17); int a = stack.pop(); // stack.push(integer.valueof(17)); // int a = stack.pop().intvalue(); Botm line. Client code can use generic stack f any type of data

11 Stacks quiz 5 Which of the following is the crect way declare and initialize an empty stack of characters? A. Stack<Character> stack = new Stack(); 1.3 BAGS, QUEUES, AND STACKS B. Stack stack = new Stack<Character>(); C. Stack<Character> stack = new Stack<Character>(); D. Stack<char> stack = new Stack<char>(); Algithms ROBERT SEDGEWICK KEVIN WAYNE stacks resizing arrays queues generics iterars applications see precept E. None of the above. 41 Iteration Feach loop Design challenge. Suppt iteration over stack items by client, without revealing the internal representation of the stack. Java provides elegant syntax f iteration over collections. feach loop (shthand) equivalent code (longhand) resizing-array representation i N Stack<String> stack;... Stack<String> stack;... s[] it was the st of times linked-list representation current f (String s : stack)... Iterar<String> i = stack.iterar(); while (i.hasnext()) String s = i.next();... times of st the was it Java solution. Use a feach loop. To make user-defined collection suppt feach loop: Data type must have a method named iterar(). The iterar() method returns an object that has two ce method. the hasnext() methods returns false when there are no me items the next() method returns the next item in the collection 43 44

12 Iterars Stack iterar: linked-list implementation To suppt feach loops, Java provides two interfaces. Iterar interface: next() and hasnext() methods. Iterable interface: iterar() method that returns an Iterar. Both should used with generics. java.util.iterar interface public interface Iterar<Item> boolean hasnext(); Item next(); optional; use void remove(); at your own risk Type safety. java.lang.iterable interface public interface Iterable<Item> Iterar<Item> iterar(); iterar(); Data type must use these interfaces suppt feach loop. Client program won't compile if implementation doesn't. 45 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; current throw UnsupptedOperationException throw NoSuchElementException if no me items in iteration times of st the was it 46 Stack iterar: array implementation Bag API impt java.util.iterar; public class Stack<Item> implements Iterable<Item>... public Iterar<Item> iterar() return new ReverseArrayIterar(); private class ReverseArrayIterar implements Iterar<Item> private int i = N; public boolean hasnext() return i > 0; public void remove() /* not suppted */ public Item next() return s[--i]; Main application. Adding items a collection and iterating (when der doesn't matter). public class Bag<Item> implements Iterable<Item> Bag() create an empty bag void add(item x) add a new item bag int size() numr of items in bag Iterar<Item> iterar() iterar f all items in bag a bag of marbles add( ) add( ) f (Marble m : bag) i N process each marble m (in any der) s[] it was the st of times 47 Implementation. Stack (without pop) queue (without dequeue). 48

13 Java collections library List interface. java.util.list is API f a sequence of items. Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications public interface List<Item> extends Iterable<Item> List() create an empty list boolean isempty() is the list empty? int size() numr of items void add(item item) add item the end Item get(int index) return item at given index Item remove(int index) return and delete item at given index boolean contains(item item) does the list contain the given item? Iterar<Item> iterar() iterar over all items in the list Implementations. java.util.arraylist uses a resizing array; java.util.linkedlist uses doubly-linked list. Caveat: only some operations are efficient. 50 Java collections library Java collections library java.util.stack. Suppts push(), pop(), and iteration. Inherits from java.util.vecr, which implements java.util.list interface. java.util.stack. Suppts push(), pop(), and iteration. Inherits from java.util.vecr, which implements java.util.list interface. Java 1.3 bug rept (June 27, 2001) The iterar method on java.util.stack iterates through a Stack from the botm up. One would think that it should iterate as if it were popping off the p of the Stack. java.util.queue. An interface, not an implementation of a queue. status (closed, will not fix) Best practices. Use our implementations of Stack and Queue. It was an increct design decision have Stack extend Vecr ("is-a" rather than "has-a"). We sympathize with the submitter but cannot fix this cause of compatibility

14 War sry (from Assignment 1) Generate random open sites in an N-by-N percolation system. Jenny: pick (i, j) at random; if already open, repeat. Takes ~ c1 N 2 seconds. Kenny: create a java.util.arraylist of N 2 closed sites. Pick an index at random and delete. Takes ~ c2 N 4 seconds. Why is my program so slow? Stack 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.... Kenny Lesson. Don't use a library until you understand its API! This course. Cannot use a library until we've implemented it in class 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. Goal. Evaluate infix expressions. ( 1 + ( ( ) * ( 4 * 5 ) ) ) value stack operar stack ( 1 + ( ( ) * ( 4 * 5 ) ) ) + ( ( ) * ( 4 * 5 ) ) ) ( ( ) * ( 4 * 5 ) ) ) Recursive function. Function that calls itself. Note. Can always use an explicit stack remove recursion. operand operar ) * ( 4 * 5 ) ) ) 3 ) * ( 4 * 5 ) ) ) gcd (216, 192) static int gcd(int p, int q) p = 216, q = 192 if (q == 0) return p; else return gcd(q, gcd p %(192, q); 24) static int gcd(int p, int q) p = 192, q = 24 if (q == 0) return p; else return gcd(q, p gcd % (24, q); 0) static int gcd(int p, int q) if (q == 0) return p; p = 24, q = 0 else return gcd(q, p % q); Two-stack algithm. [E. W. Dijkstra] Value: push on the value stack. Operar: push on the operar stack. Left parenthesis: igne. Right parenthesis: pop operar and two values; push the result of applying that operar those values on the operand stack. Context. An interpreter! * * * * * * * ) * ( 4 * 5 ) ) ) * ( 4 * 5 ) ) ) ( 4 * 5 ) ) ) * 5 ) ) ) 5 ) ) ) ) ) ) ) ) ) 55 56

15 Dijkstra's two-stack algithm demo Arithmetic expression evaluation infix expression value stack operar stack (fully parenthesized) ( 1 + ( ( ) * ( 4 * 5 ) ) ) operand operar 57 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("(")) /* noop */ ; 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 ) ) ) Crectness Stack-based programming languages Q. Why crect? Observation 1. Dijkstra's two-stack algithm computes the same value if A. When algithm encounters an operar surrounded by two values the operar occurs after the two values. within parentheses, it leaves the result on the value stack. ( 1 + ( ( ) * ( 4 * 5 ) ) ) ( 1 ( ( ) ( 4 5 * ) * ) + ) as if the iginal input were: ( 1 + ( 5 * ( 4 * 5 ) ) ) Observation 2. All of the parentheses are redundant! Repeating the argument: * * + Jan Lukasiewicz ( 1 + ( 5 * 20 ) ) ( ) 101 Extensions. Me ops, precedence der, associativity. Botm line. Postfix "reverse Polish" notation. Applications. Postscript, Fth, calculars, Java virtual machine, 59 60