Algorithms. Algorithms 1.3 BAGS, QUEUES, AND STACKS. stacks resizing arrays. queue. generics. applications. Stacks and queues

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1 ROBERT SEDGEWICK KEVIN WAYNE Algithms ROBERT SEDGEWICK KEVIN WAYNE Stacks and queues Algithms F O U R T H E D I T I O N BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars see precept applications Abstract data types. Value: collection of objects. Operations: add, remove, iterate, test if empty. Intent is clear when we add. Which item do we remove? enqueue stack queue Stack. Examine the item. Queue. Examine the item. push pop dequeue LIFO = "last in out" FIFO = " in out" Last updated on 2/8/16 7:22 AM 2 Client, implementation, API Interfaces can ambiguous Separate client and implementation via API. An Abstract Data type is one type of API. Benefits. Design: creates modular, reusable libraries. Perfmance: substitute optimized implementation when it matters. Client API Implementation API: description of data type, basic operations. Client: program using operations defined in API. Implementation: actual code implementing operations. Ex. Stack, queue, bag, priity queue, symbol table, union-find,. 3 Stacks and queues. Value: collection of objects. Operations: add, remove, iterate, test if empty. Q. What are two ways in which the semantics of iteration can ambiguous? A. What der iterate in: same as removal der does client not care? What happens if collection is modified during iteration? 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 not fix) 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. 4

2 Ambiguity in the semantics of interfaces leads bugs Layers in a computer system Example: Mars climate biter. Lost due metric vs. imperial mishap in contract tween NASA & Lockheed Program API Libraries This lecture Programming language Operating system Hardware 5 6 Stack API Warmup API. Stack of strings data type. push pop public class StackOfStrings 1.3 BAGS, QUEUES, stacks resizing arrays Algithms R OBERT S EDGEWICK K EVIN W AYNE queues generics AND S TACKS StackOfStrings() void push(string item) String pop() boolean isempty() int size() create an empty stack add a new string stack remove and return the string is the stack empty? numr of strings on the stack iterars applications 8

3 Stacks quiz 1 How implement a stack with a singly-linked list? A. Stack: linked-list implementation Maintain pointer node in a singly-linked list. Push new item fe. Pop item from. it was the st of B. of st the was it C. None of the above. of st the was it D. I don't know Stack pop: linked-list implementation Stack push: linked-list implementation save item return String item =.item; delete node save a link the list Node old = ; old inner class String item; Node next; =.next; 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 saved item return item;.item = "not";.next = old; not 11 12

4 Stack: linked-list implementation in Java Stack: linked-list implementation perfmance public class LinkedStackOfStrings private Node = ; private String item; private Node next; public boolean isempty() return == ; public void push(string item) Node old = ; = new Node();.item = item;.next = old; public String pop() String item =.item; =.next; return item; private inner class (access modifiers f instance variables don't matter) Proposition. Every operation takes constant time in the wst case. Proposition. A stack with n items uses ~ 40 n bytes. inner class String item; Node next; object overhead extra overhead item next references Remark. This accounts f the memy f the stack 16 bytes (object overhead) 8 bytes (inner class extra overhead) 8 bytes (reference String) 8 bytes (reference Node) 40 bytes allocated per stack node (but not memy f the strings themselves, which the client owns) Stacks quiz 2 How implement a fixed-capacity stack with an array? A. B. it was the st of times times of st the was it B. None of the above. C. I don't know. 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]. s[] it was the st of times n capacity = 10 Defect. Stack overflows when n exceeds capacity. [stay tuned] 15 16

5 Fixed-capacity stack: array implementation Stack considerations public class FixedCapacityStackOfStrings private String[] s; private int n = 0; public FixedCapacityStackOfStrings(int capacity) s = new String[capacity]; public boolean isempty() 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 added. Duplicate items. We allow an item added me than once. 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; public String pop() return s[--n]; public String pop() return s[--n]; loitering public String pop() String item = s[--n]; s[n] = ; return item; decrement n; then use index in array this version avoids "loitering": garbage collecr can reclaim memy f an object only if no remaining references Stack: resizing-array implementation Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications Problem. Requiring client provide capacity does not implement API! Q. How grow and shrink array? 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, f each operation. Array accesses add n items = n + ( (n 1)) ~ n2. 1 array access per push Challenge. Ensure that array resizing happens infrequently. infeasible f large n 2(k 1) array accesses expand size k (igning cost create new array) 20

6 Why isn t array resizing easier? Stack: resizing-array implementation Q. Why no way increment array size except by creating new one? A. Because Java doesn t let us Because OS doesn t let us safely write past memy bounds Because memy is fragmented at the hardware level Q. How grow array? A. If array is full, create a new array of twice the size, and copy items. public ResizingArrayStackOfStrings() s = new String[1]; "repeated doubling" Program Library Programming language Operating system See Applications section Data structures (ArrayStack) Basic data types (Array) Memy allocation API / Abstract Data Type / Interface (Stack)* Memy fragmentation 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; Hardware Physical memy *Note: in this course Stack is a class and not an interface 21 Array accesses add n = 2 i items. n + ( n) ~ 3n. 1 array access per push k array accesses double size k (igning cost create new array) 22 Stack: resizing-array implementation Stack: resizing-array implementation Q. How shrink array? First try. Q. How shrink array? Efficient solution. 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. full not push(): double size of array s[] when array is full. pop(): halve size of array s[] when array is one-quarter full. public String pop() String item = s[--n]; s[n] = ; if (n > 0 && n == s.length/4) resize(s.length/2); return item; push("") not pop() not push("") not Invariant. Array is tween 25% and 100% full

7 Stack resizing-array implementation: perfmance Stack resizing-array implementation: perfmance 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. typical wst amtized construct push 1 n 1 pop 1 n 1 size der of growth of running time f resizing array stack with n items doubling and halving operations 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. Proof. Divide the operations in batches. Each batch starts after a resize has just completed ( at the ginning) and ends with the next resize ( at the end). Claim: the cost of processing each batch is proptional the numr of operations in that batch. Exercise: complete this argument. Exercise: would proof hold if we d had different constants instead of ½ and ¼ (say ⅔ & ⅓)? What are the facrs consider in picking the constants? Stack resizing-array implementation: memy usage Proposition. A ResizingArrayStackOfStrings uses ~ 8n ~ 32n bytes of memy f a stack with n items. ~ 8n when full. ~ 32n when one-quarter full. public class ResizingArrayStackOfStrings private String[] s; private int n = 0; 8 bytes array size 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 not 27 28

8 Queue API enqueue public class QueueOfStrings Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications QueueOfStrings() create an empty queue 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 dequeue 30 Queues quiz 1 Queue: linked-list implementation How implement a queue with a singly-linked linked list? A. times of st the was it Maintain one pointer node in a singly-linked list. Maintain another pointer last last node. Dequeue from. Enqueue after last. B. it was the st of times it was the st of times C. None of the above. last D. I don't know

9 Queue dequeue: linked-list implementation Queue enqueue: linked-list implementation save a link the last node save item return Node oldlast = last; String item =.item; oldlast last delete node inner class String item; Node next; =.next; last last inner class String item; Node next; create a new node f the end last = new Node(); last.item = "not"; oldlast last not link the new node the end of the list return saved item return item; oldlast.next = last; oldlast last not Remark. Identical code linked-list stack pop() Queue: linked-list implementation in Java Queues quiz 2 public class LinkedQueueOfStrings private Node, last; /* same as in LinkedStackOfStrings */ public boolean isempty() return == ; public void enqueue(string item) Node oldlast = last; last = new Node(); last.item = item; last.next = ; if (isempty()) = last; else oldlast.next = last; special cases f empty queue How implement a fixed-capacity queue with an array? A. B. it was the st of times times of st the was it public String dequeue() String item =.item; =.next; if (isempty()) last = ; return item; C. None of the above. D. I don't know

10 Queue: resizing-array implementation Queue with two stacks Use array q[] sre items in queue. Job interview problem. Implement a queue with two stacks so that: enqueue(): dequeue(): Each queue op uses a constant amtized numr of stack ops. At most constant extra memy (sides the two stacks). add new item at q[tail]. remove item from q[head]. Update head and tail modulo the capacity. Add resizing array. Solution. Call the two stacks incoming and outgoing. enqueue: push incoming dequeue: pop from outgoing if outgoing is empty, pour incoming in outgoing (O(N)). isempty: check if both stacks are empty q[] 0 1 the st of times head tail Analysis: crectness. Left as exercise. capacity = 10 Analysis: efficiency. Consider the lifecycle of each item: pushed in incoming, popped from incoming, pushed in outgoing, popped from outgoing. Q. How resize? At most 4 stack operations per item. [Exercise: complete this argument.] Parameterized stack We implemented: StackOfStrings. We also want: StackOfURLs, StackOfInts, StackOfApples, StackOfOranges,. Solution in Java: generics. 1.3 BAGS, QUEUES, stacks resizing arrays Algithms R OBERT S EDGEWICK K EVIN W AYNE queues generics iterars applications AND S TACKS 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); compile-time err... Guiding principle. Welcome compile-time errs; avoid run-time errs. 40

11 Generic stack: linked-list implementation Generic stack: array implementation public class LinkedStackOfStrings private Node = ; String item; Node next; public boolean isempty() return == ; public void push(string item) Node old = ; = new Node();.item = item;.next = old; public String pop() String item =.item; =.next; return item; public class LinkedStack<Item> private Node = ; Item item; Node next; public boolean isempty() 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 public class FixedCapacityStackOfStrings private String[] s; private int n = 0; public..stackofstrings(int capacity) s = new String[capacity]; public boolean isempty() return n == 0; public void push(string item) s[n++] = item; public String pop() return s[--n]; the way it should public class FixedCapacityStack<Item> private Item[] s; private int n = 0; public FixedCapacityStack(int capacity) s = new Item[capacity]; public boolean isempty() return n == 0; public void push(item item) s[n++] = item; public Item pop() return s[--n]; generic array creation not allowed in Java Generic stack: array implementation Generic data types: auboxing and unboxing the way it is Q. What do about primitive types? public class FixedCapacityStackOfStrings private String[] s; private int n = 0; public..stackofstrings(int capacity) s = new String[capacity]; public class FixedCapacityStack<Item> private Item[] s; private int n = 0; public FixedCapacityStack(int capacity) s = (Item[]) new Object[capacity]; Wrapper type. Each primitive type has a wrapper object type. Ex: Integer is wrapper type f int. public boolean isempty() return n == 0; public boolean isempty() return n == 0; Auboxing. Aumatic cast from primitive type wrapper type. public void push(string item) s[n++] = item; public void push(item item) s[n++] = item; Unboxing. Aumatic cast from wrapper type primitive type. public String pop() return s[--n]; public Item pop() return s[--n]; Stack<Integer> stack = new Stack<Integer>(); stack.push(17); // stack.push(integer.valueof(17)); int a = stack.pop(); // int a = stack.pop().intvalue(); ugly cast (will result in a compile-time warning, but it s not your fault) 43 Botm line. Client code can use generic stack f any type of data. 44

12 Iteration Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications see precept Design challenge. Suppt iteration over stack items by client, without revealing the internal representation of the stack. resizing-array representation s[] linked-list representation it was the st of current times of st the was it Java solution. Use a f-each loop. i time s n F-each loop Iterars Java provides elegant syntax f iteration over collections. F a user-defined collection, do this enable looping over it with f-each: Data type must have a method named iterar(). The iterar() method returns an object that has two ce methods. f-each loop (shthand) Stack<String> stack;... f (String s : stack)... equivalent code (longhand) Stack<String> stack;... Iterar<String> i = stack.iterar(); while (i.hasnext()) String s = i.next(); the hasnext() methods returns false when there are no me items the next() method returns the next item in the collection To suppt f-each 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(); void remove(); Type safety. optional; use at your own risk java.lang.iterable interface public interface Iterable<Item> Iterar<Item> iterar(); iterar(); Data type must use these interfaces suppt f-each loop. Client program won't compile if implementation doesn't. 48

13 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() throw UnsupptedOperationException Item item = current.item; throw NoSuchElementException current = current.next; if no me items in iteration return item; 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]; current times of st the was it 49 s[] i n it was the st of time s Iteration: concurrent modification Q. What if client modifies the data structure while iterating? A. A fail-fast iterar throws a java.util.concurrentmodificationexception. Q. How detect concurrent modification? A. Count tal numr of push() and pop() operations in Stack. Save counts in *Iterar subclass upon creation. concurrent modification f (String s : stack) stack.push(s); If, when calling either next() hasnext(), the current counts do not equal the saved counts, throw exception. Algithms ROBERT SEDGEWICK KEVIN WAYNE BAGS, QUEUES, AND STACKS stacks resizing arrays queues generics iterars applications 51

14 PSA F this course, use our Stack and Queue implementations instead of Java s. 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 Arithmetic expression evaluation Dijkstra's two-stack algithm demo Goal. Evaluate infix expressions using two stacks. value stack operar stack 1 ( 1 + ( ( ) * ( 4 * 5 ) ) ) + ( ( ) * ( 4 * 5 ) ) ) ( 1 + ( ( ) * ( 4 * 5 ) ) ) ( ( ) * ( 4 * 5 ) ) ) + 3 ) * ( 4 * 5 ) ) ) ) * ( 4 * 5 ) ) ) 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 apply operar the two values push the result value stack ) * ( 4 * 5 ) ) ) * ( 4 * 5 ) ) ) * ( 4 * 5 ) ) ) * * 5 ) ) ) * * 5 ) ) ) * * ) ) ) * ) ) ) 101 infix expression value stack operar stack (fully parenthesized) ( 1 + ( ( ) * ( 4 * 5 ) ) ) 55 operand operar 56

15 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("(")) /* 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 ) ) ) 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 Stack-based programming languages Observation 1. Dijkstra's two-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 Botm line. Postfix "reverse Polish" notation. Applications. Postscript, Fth, calculars, Java virtual machine, 59