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

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1 Algithms Stacks and queues 1.3 BAGS, QUEUES, AND STACKS Fundamental data types. Value: collection of objects. Operations: insert, remove, iterate, test if empty. Intent is clear when we insert. Which item do we remove? Algithms F O U R T H E D I T I O N stacks resizing arrays queues generics iterars applications stack queue enqueue Stack. Examine the item most recently added. push pop dequeue LIFO = "last in out" Queue. Examine the item least recently added. FIFO = " in out" 2 Client, implementation, interface Separate interface and implementation. Ex: stack, queue, bag, priity 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. Algithms BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications 3

2 Stack API Stack test client Warmup API. Stack of strings data type. public class StackOfStrings push pop Read strings from standard input. If string equals "-", pop string from stack and print. Otherwise, push string on stack. push pop StackOfStrings() create an empty stack void push(string item) insert a new string on stack String pop() remove and return the string most recently added boolean isempty() is the stack empty? int size() numr of strings on the stack public static void main(string[] args) StackOfStrings stack = new StackOfStrings(); while (!StdIn.isEmpty()) String s = StdIn.readString(); if (s.equals("-")) StdOut.print(stack.pop()); else stack.push(s); % me.txt that is Warmup client. Reverse sequence of strings from standard input. % java StackOfStrings <.txt that 5 6 Stack: linked-list representation Stack pop: linked-list implementation Maintain pointer node in a linked list; insert/remove from front. StdIn StdOut save item return String item =.item; delete node insert at front of linked list remove from front of linked list inner class String item; =.next; 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 String item; private inner class (access modifiers don't matter) inner class String item; 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 = "";.next = old; String item =.item; =.next; 9 10 Stack: linked-list implementation perfmance Stack: array implementation Proposition. Every operation takes constant time in the wst case. Proposition. A stack with N items uses ~ 40 N bytes. 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]. inner class String item; object overhead extra overhead item next references 16 bytes (object overhead) 8 bytes (inner class extra overhead) 8 bytes (reference String) 8 bytes (reference Node) 40 bytes per stack node s[] N capacity = 10 Remark. This accounts f the memy f the stack (but the memy f strings themselves, which the client owns). Defect. Stack overflows when N exceeds capacity. [stay tuned] 11 12

4 Stack: array implementation Stack considerations public class FixedCapacityStackOfStrings private String[] s; private int N = 0; public FixedCapacityStackOfStrings(int capacity) s = new String[capacity]; return N == 0; a cheat (stay tuned) 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 inserted. Loitering. Holding a reference an object when it is no longer needed. use index in array; then increment N public void push(string item) s[n++] = item; return s[--n]; return s[--n]; loitering String item = s[--n]; s[n] = ; decrement N; then use index in array this version avoids "loitering": garbage collecr can reclaim memy only if no outstanding references Stack: resizing-array implementation Problem. Requiring client provide capacity does implement API! Q. How grow and shrink array? Algithms BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications First try. push(): increase size of array s[] by 1. pop(): decrease size of array s[] by 1. Too expensive. Need copy all items a new array. Inserting N items takes time proptional N ~ N 2 / 2. infeasible f large N Challenge. Ensure that array resizing happens infrequently. 16

5 Stack: resizing-array implementation Q. How grow array? "repeated doubling" A. If array is full, create a new array of twice the size, and copy items. public ResizingArrayStackOfStrings() s = new String[1]; Stack: amtized cost of adding a stack Cost of inserting N items. N + ( N) ~ 3N. 1 array access per push k array accesses double size k (igning cost create new array) 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; see next slide 128 cost (array accesses) 0 one gray dot f each operation red dots give cumulative average 3 0 numr of push() operations 128 Consequence. Inserting N items takes time proptional N ( N 2 ) 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. Too expensive in wst case. Consider push-pop-push-pop- sequence when array is full. Each operation takes time proptional N. N = 5 Efficient solution. push(): double size of array s[] when array is full. pop(): halve size of array s[] when array is one-quarter full. String item = s[--n]; s[n] = ; if (N > 0 && N == s.length/4) resize(s.length/2); N = 4 N = 5 N = 4 Invariant. Array is tween 25% and 100% full

6 Stack: resizing-array implementation trace Stack resizing-array implementation: perfmance push() pop() N a.length a[] that 4 8 that - that is 2 2 is Trace of array resizing during a sequence of push() and pop() operations Amtized analysis. 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. construct push pop size st wst amtized N 1 1 N der of growth of running time f resizing stack with N items doubling and halving operations Stack resizing-array implementation: memy usage Stack implementations: resizing array vs. linked list Proposition. Uses tween ~ 8 N and ~ 32 N bytes represent a stack with N items. ~ 8 N when full. ~ 32 N when one-quarter full. public class ResizingArrayStackOfStrings private String[] s; private int N = 0; 8 bytes (reference array) 24 bytes (array overhead) 8 bytes array size 4 bytes (int) 4 bytes (padding) 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 Remark. This accounts f the memy f the stack (but the memy f strings themselves, which the client owns)

7 Queue API enqueue public class QueueOfStrings 1.3 BAGS, QUEUES, AND STACKS QueueOfStrings() create an empty queue void enqueue(string item) insert a new string on queue Algithms stacks resizing arrays queues generics String dequeue() remove and return the string least recently added boolean isempty() is the queue empty? int size() numr of strings on the queue iterars dequeue applications 26 Queue: linked-list representation Queue dequeue: linked-list implementation Maintain pointer and last nodes in a linked list; insert/remove from opposite ends. save item return String item =.item; StdIn StdOut delete node =.next; last insert at end of linked list inner class String item; last last remove from front of linked list - 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 StackOfStrings */ return == ; inner class String item; create a new node f the end last = new Node(); last.item = ""; link the new node the end of the list oldlast.next = last; oldlast oldlast last 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 = ; special cases f empty queue Queue: resizing array implementation Array implementation of a queue. Use array q[] sre items in queue. enqueue(): add new item at q[tail]. q[] dequeue(): remove item from q[head]. Update head and tail modulo the capacity. Add resizing array. the st of times head tail capacity = 10 Algithms BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications Q. How resize? 31

9 Parameterized stack Parameterized stack We implemented: StackOfStrings. We implemented: StackOfStrings. We also want: StackOfURLs, StackOfInts, StackOfVans,. We also want: StackOfURLs, StackOfInts, StackOfVans,. Attempt 1. Implement a separate stack class f each type. Attempt 2. Implement a stack with items of type Object. Rewriting code is tedious and err-prone. Maintaining cut-and-pasted code is tedious and err-prone. Casting is required in client. Casting is err-prone: run-time err if types mismatch. StackOfObjects s = new most reasonable approach until Java 1.5. Apple a = new Apple(); Orange b = new Orange(); s.push(a); s.push(b); a = (Apple) (s.pop()); run-time err 33 Parameterized stack 34 Generic stack: linked-list implementation We implemented: StackOfStrings. public class LinkedStackOfStrings private Node = ; We also want: StackOfURLs, StackOfInts, StackOfVans,. String item; Attempt 3. Java generics. Avoid casting in client. Discover type mismatch errs at compile-time instead of run-time. return == ; type parameter Stack<Apple> s = new Stack<Apple>(); Apple a = new Apple(); Orange b = new Orange(); s.push(a); s.push(b); public class Stack<Item> private Node = ; compile-time err a = s.pop(); Item item; 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; Guiding principles. Welcome compile-time errs; avoid run-time errs

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 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]; ^ 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> s = new Stack<Integer>(); s.push(17); // s.push(integer.valueof(17)); int a = s.pop(); // int a = s.pop().intvalue(); Botm line. Client code can use generic stack f any type of data

11 Iteration Design challenge. Suppt iteration over stack items by client, without revealing the internal representation of the stack. 1.3 BAGS, QUEUES, AND STACKS s[] i it was the st of times N Algithms stacks resizing arrays queues generics iterars applications current times of st the was it Java solution. Make stack implement the java.lang.iterable interface. 42 Iterars Q. What is an Iterable? A. Has a method that returns an Iterar. Iterable interface public interface Iterable<Item> Iterar<Item> iterar(); Stack iterar: linked-list implementation impt java.util.iterar; public class Stack<Item> implements Iterable<Item>... Q. What is an Iterar? A. Has methods hasnext() and next(). Q. Why make data structures Iterable? A. Java suppts elegant client code. feach statement (shthand) f (String s : stack) StdOut.println(s); Iterar interface public interface Iterar<Item> boolean hasnext(); Item next(); void remove(); equivalent code (longhand) optional; use at your own risk Iterar<String> i = stack.iterar(); while (i.hasnext()) String s = i.next(); StdOut.println(s); 43 public Iterar<Item> iterar() return new ListIterar(); private class ListIterar implements Iterar<Item> private Node current = ; public boolean hasnext() return current!= ; public void remove() /* 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 44

12 Stack iterar: array implementation Iteration: concurrent modification impt java.util.iterar; public class Stack<Item> implements Iterable<Item> public Iterar<Item> iterar() return new ReverseArrayIterar(); Q. What if client modifies the data structure while iterating? A. A fail-fast iterar throws a java.util.concurrentmodificationexception. concurrent modification f (String s : stack) stack.push(s); private class ReverseArrayIterar implements Iterar<Item> private int i = N; public boolean hasnext() return i > 0; public void remove() /* suppted */ public Item next() return s[--i]; i N To detect: Count tal numr of push() and pop() operations in Stack. Save counts in *Iterar subclass upon creation. If, when calling next() and hasnext(), the current counts do equal the saved counts, throw exception. s[] it was the st of times Java collections library List interface. java.util.list is API f an sequence of items. Algithms BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications public interface List<Item> implements Iterable<Item> List() create an empty list boolean isempty() is the list empty? int size() numr of items void add(item item) append 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? Itearr<Item> iterar() iterar over all items in the list... Implementations. java.util.arraylist uses resizing array; java.util.linkedlist uses linked list. caveat: only some operations are efficient 48

13 Java collections library Java collections library java.util.stack. java.util.stack. Suppts push(), pop(), and and iteration. Extends java.util.vecr, which implements java.util.list Suppts push(), pop(), and and iteration. Extends java.util.vecr, which implements java.util.list interface from previous slide, including get() and remove(). interface from previous slide, including get() and remove(). Bloated and poly-designed API (why?) Bloated and poly-designed API (why?) 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. status (closed, will fix) java.util.queue. An interface, an implementation of a queue. It was an increct design decision have Stack extend Vecr ("is-a" rather than "has-a"). We sympathize with the submitter but can fix this cause of compatibility. Best practices. Use our implementations of Stack, Queue, and Bag. 49 War sry (from Assignment 1) Stack applications Generate random open sites in an N-by-N percolation system. Jenny: percolates blocked pick (i, j) at random; if already open, repeat. does percolate site Takes ~ c1 N 2 seconds. Kenny: create a java.util.arraylist of N 2 closed sites. Pick an index at random and delete. open Takes ~ c2 N 4 seconds. 50 site open site connected p no open site connected p 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.... Why is my program so slow? Kenny Lesson. Don't use a library until you understand its API! This course. Can't use a library until we've implemented it in class

14 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. 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); Goal. Evaluate infix expressions. ( 1 + ( ( ) * ( 4 * 5 ) ) ) Two-stack algithm. [E. W. Dijkstra] Value: push on the value stack. Operar: push on the operar stack. operand 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! 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 ) ) ) ) ) ) ) ) ) Dijkstra's two-stack algithm demo Arithmetic expression evaluation infix expression value stack operar stack (fully parenthesized) ( 1 + ( ( ) * ( 4 * 5 ) ) ) operand operar 55 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 ) ) )

15 Crectness Stack-based programming languages Q. Why crect? A. When algithm encounters an operar surrounded by two values within parentheses, it leaves the result on the value stack. Observation 1. Dijkstra's two-stack algithm computes the same value if the operar occurs after the two values. ( 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" ation. Applications. Postscript, Fth, calculars, Java virtual machine, 57 58