Stacks

1. Stacks

2. Outline and Reading • The Stack ADT (§4.1) • Applications of Stacks • Array-based implementation (§4.1.2) • Growable array-based stack • Think of a spring-loaded plate dispenser week 2a

3. An abstract data type (ADT) is an abstraction of a data structure An ADT specifies: Data stored Operations on the data Error conditions associated with operations Example: ADT modeling a simple stock trading system The data stored are buy/sell orders The operations supported are order buy(stock, shares, price) order sell(stock, shares, price) void cancel(order) Error conditions: Buy/sell a nonexistent stock Cancel a nonexistent order Abstract Data Types (ADTs) week 2a

4. The Stack ADT stores arbitrary objects Insertions and deletions follow the last-in first-out scheme Main stack operations: push(object): inserts an element object pop(): removes and returns the last inserted element Auxiliary stack operations: object top(): returns the last inserted element without removing it integer size(): returns the number of elements stored boolean isEmpty(): indicates whether no elements are stored The Stack ADT week 2a

5. Attempting the execution of an operation of ADT may sometimes cause an error condition, called an exception Exceptions are said to be “thrown” by an operation that cannot be executed In the Stack ADT, operations pop and top cannot be performed if the stack is empty Attempting the execution of pop or top on an empty stack throws an EmptyStackException Exceptions week 2a

6. Applications of Stacks • Direct applications • Page-visited history in a Web browser • Undo sequence in a text editor • Chain of method calls in the Java Virtual Machine • Indirect applications • Auxiliary data structure for algorithms • Component of other data structures week 2a

7. Method Stack in the JVM main() { int i = 5; foo(i); } foo(int j) { int k; k = j+1; bar(k); } bar(int m) { … } • The Java Virtual Machine (JVM) keeps track of the chain of active methods with a stack • When a method is called, the JVM pushes on the stack a frame containing • Local variables and return value • Program counter, keeping track of the statement being executed • When a method ends, its frame is popped from the stack and control is passed to the method on top of the stack bar PC = 1 m = 6 foo PC = 3 j = 5 k = 6 main PC = 2 i = 5 week 2a

8. A simple way of implementing the Stack ADT uses an array We add elements from left to right A variable keeps track of the index of the top element Array-based Stack Algorithmsize() returnt +1 Algorithmpop() ifisEmpty()then throw EmptyStackException else tt 1 returnS[t +1] … S 0 1 2 t week 2a

9. … S 0 1 2 t Array-based Stack (cont.) • The array storing the stack elements may become full • A push operation will then throw a FullStackException • Limitation of the array-based implementation • Not intrinsic to the Stack ADT Algorithmpush(o) ift=S.length 1then throw FullStackException else tt +1 S[t] o week 2a

10. Performance and Limitations • Performance • Let n be the number of elements in the stack • The space used is O(n) • Each operation runs in time O(1) • Limitations • The maximum size of the stack must be defined a priori and cannot be changed • Trying to push a new element into a full stack causes an implementation-specific exception week 2a

11. Computing Spans • We show how to use a stack as an auxiliary data structure in an algorithm • Given an an array X, the span S[i] of X[i] is the maximum number of consecutive elements X[j] immediately preceding X[i] and such that X[j]  X[i] • Spans have applications to financial analysis • E.g., stock at 52-week high X S week 2a

12. Quadratic Algorithm Algorithmspans1(X, n) Inputarray X of n integers Outputarray S of spans of X # S new array of n integers n fori0ton 1do n s 1n while s i X[i - s]X[i]1 + 2 + …+ (n 1) ss+ 11 + 2 + …+ (n 1) S[i]sn returnS 1 • Algorithm spans1 runs in O(n2) time week 2a

13. Computing Spans with a Stack • We keep in a stack the indices of the elements visible when “looking back” • We scan the array from left to right • Let i be the current index • We pop indices from the stack until we find index j such that X[i] X[j] • We set S[i]i - j • We push i onto the stack week 2a

14. Linear Algorithm • Each index of the array • Is pushed into the stack exactly one • Is popped from the stack at most once • The statements in the while-loop are executed at most n times • Algorithm spans2 runs in O(n) time Algorithmspans2(X, n)# S new array of n integers n A new empty stack 1 fori0ton 1do n while(A.isEmpty()  X[top()]X[i] )do n A.pop()n if A.isEmpty()thenn S[i] i +1 n else S[i]i - A.top() n A.push(i) n returnS 1 week 2a

15. Growable Array-based Stack Algorithmpush(o) ift=S.length 1then A new array of size … fori0tot do A[i]  S[i] S A tt +1 S[t] o • In a push operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one • How large should the new array be? • incremental strategy: increase the size by a constant c • doubling strategy: double the size week 2a

16. Comparison of the Strategies • We compare the incremental strategy and the doubling strategy by analyzing the total time T(n) needed to perform a series of n push operations • We assume that we start with an empty stack represented by an array of size 1 • We call amortized time of a push operation the average time taken by a push over the series of operations, i.e., T(n)/n week 2a

17. Incremental Strategy Analysis • We replace the array k = n/c times • The total time T(n) of a series of n push operations is proportional to n + c + 2c + 3c + 4c + … + kc = n + c(1 + 2 + 3 + … + k) = n + ck(k + 1)/2 • Since c is a constant, T(n) is O(n +k2),i.e., O(n2) • The amortized time of a push operation is O(n) week 2a

18. geometric series 2 4 1 1 8 Doubling Strategy Analysis • We replace the array k = log2n times • The total time T(n) of a series of n push operations is proportional to n + 1 + 2 + 4 + 8 + …+ 2k= n+ 2k + 1-1 = 2n -1 • T(n) is O(n) • The amortized time of a push operation is O(1) week 2a

19. Stack Interface in Java public interfaceStack{ public int size(); public boolean isEmpty(); public Object top()throwsEmptyStackException; public voidpush(Object o); public Object pop()throwsEmptyStackException;} • Java interface corresponding to our Stack ADT • Requires the definition of class EmptyStackException • Different from the built-in Java class java.util.Stack week 2a

20. Array-based Stack in Java public classArrayStackimplements Stack{ // holds the stack elementsprivate Object S[ ]; // index to top elementprivate int top = -1; // constructorpublicArrayStack(int capacity){S = new Object[capacity]);} public Object pop()throwsEmptyStackException{if isEmpty()throw newEmptyStackException(“Empty stack: cannot pop”);Object temp = S[top];// facilitates garbage collection S[top] =null; top = top – 1;returntemp;} week 2a

21. Queues

22. Outline and Reading • The Queue ADT (§4.2) • Implementation with a circular array (§4.2.2) • Growable array-based queue • Queue interface in Java week 2a

23. The Queue ADT stores arbitrary objects Insertions and deletions follow the first-in first-out scheme Insertions are at the rear of the queue and removals are at the front of the queue Main queue operations: enqueue(object): inserts an element at the end of the queue object dequeue(): removes and returns the element at the front of the queue Auxiliary queue operations: object front(): returns the element at the front without removing it integer size(): returns the number of elements stored boolean isEmpty(): indicates whether no elements are stored Exceptions Attempting the execution of dequeue or front on an empty queue throws an EmptyQueueException The Queue ADT week 2a

24. Applications of Queues • Direct applications • Waiting lists, bureaucracy • Access to shared resources (e.g., printer) • Multiprogramming • Indirect applications • Auxiliary data structure for algorithms • Component of other data structures week 2a

25. Use an array of size N in a circular fashion Two variables keep track of the front and rear f index of the front element r index immediately past the rear element Array location r is kept empty Q 0 1 2 f r Q 0 1 2 r f Array-based Queue normal configuration wrapped-around configuration week 2a

26. Q 0 1 2 f r Q 0 1 2 r f Queue Operations Algorithmsize() return(N-f +r) mod N AlgorithmisEmpty() return(f=r) • We use the modulo operator (remainder of division) week 2a

27. Q Q 0 0 1 1 2 2 f f r Queue Operations (cont.) Algorithmenqueue(o) ifsize()=N 1then throw FullQueueException else Q[r] o r(r + 1) mod N • Operation enqueue throws an exception if the array is full • This exception is implementation-dependent r week 2a

28. Q Q 0 0 1 1 2 2 f r r Queue Operations (cont.) Algorithmdequeue() ifisEmpty()then throw EmptyQueueException else oQ[f] f(f + 1) mod N returno • Operation dequeue throws an exception if the queue is empty • This exception is specified in the queue ADT f week 2a

29. Growable Array-based Queue • In an enqueue operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one • Similar to what we did for an array-based stack • The enqueue operation has amortized running time • O(n) with the incremental strategy • O(1) with the doubling strategy week 2a

30. Queue Interface in Java public interfaceQueue{ public int size(); public boolean isEmpty(); public Object front()throwsEmptyQueueException; public voidenqueue(Object o); public Object dequeue()throwsEmptyQueueException;} • Java interface corresponding to our Queue ADT • Requires the definition of class EmptyQueueException • No corresponding built-in Java class week 2a

31. Creativity: • Describe in pseudo-code a linear-time algorithm for reversing a queue Q. To access the queue, you are only allowed to use the methods of queue ADT. week 2a

32. Answer: Algorithm reverseQueue(Queue Q) Create new stack S while (not Q.isEmpty()) do S.push(Q.dequeue()) while (not S.isEmpty()) do Q.enqueue(S.pop()) week 2a