Download Algorithms ・ 1.3 B

Algorithms
Stacks and queues
R OBERT S EDGEWICK | K EVIN W AYNE
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?
1.3 B AGS, QUEUES, AND STACKS
‣ stacks
stack
push
‣ resizing arrays
Algorithms
F O U R T H
E D I T I O N
R OBERT S EDGEWICK | K EVIN W AYNE
http://algs4.cs.princeton.edu
pop
‣ queues
queue
dequeue
enqueue
‣ generics
‣ iterators
‣ applications
Stack. Examine the item most recently added.
LIFO = "last in first out"
Queue. Examine the item least recently added.
FIFO = "first in first out"
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Client, implementation, interface
Separate interface and implementation.
Ex: stack, queue, bag, priority queue, symbol table, union-find, .…
Benefits.
・Client can't know details of implementation
1.3 B AGS, QUEUES, AND S TACKS
⇒
client has many implementation from which to choose.
・Implementation can't know details of client needs
‣ stacks
⇒
many clients can re-use the same implementation.
・
・Performance:
‣ resizing arrays
Design: creates modular, reusable libraries.
Algorithms
use optimized implementation where it matters.
R OBERT S EDGEWICK | K EVIN W AYNE
Client: program using operations defined in interface.
http://algs4.cs.princeton.edu
Implementation: actual code implementing operations.
Interface: description of data type, basic operations.
3
‣ queues
‣ generics
‣ iterators
‣ applications
Stack API
How to implement a stack with a linked list?
Warmup API. Stack of strings data type.
A.
push
Can't be done efficiently with a singly-linked list.
pop
public class StackOfStrings
StackOfStrings()
top of stack
create an empty stack
B.
void push(String item)
it
was
the
best
of
null
best
the
was
it
null
insert a new string onto stack
remove and return the string
most recently added
String pop()
is the stack empty?
boolean isEmpty()
number of strings on the stack
int size()
top of stack
C.
of
Warmup client. Reverse sequence of strings from standard input.
5
Stack: linked-list implementation
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Stack pop: linked-list implementation
・Maintain pointer first to first node in a singly-linked list.
・Push new item before first.
・Pop item from first.
save item to return
String item = first.item;
delete first node
inner class
private class Node
{
String item;
Node next;
}
first = first.next;
first
or
best
the
was
it
null
to
null
first
or
top of stack
of
be
be
to
null
return saved item
return item;
first
Removing the first node in a linked list
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Stack push: linked-list implementation
Stack: linked-list implementation in Java
public class LinkedStackOfStrings
{
private Node first = null;
save a link to the list
Node oldfirst = first;
oldfirst
first
inner class
or
be
private class Node
{
String item;
Node next;
}
to
null
create a new node for the beginning
public boolean isEmpty()
{ return first == null;
first = new Node();
private class Node
{
oldfirst
String item;
Node next;
first
or
be
}
to
null
first.item = "not";
first.next = oldfirst;
not
or
be
}
public void push(String item)
{
Node oldfirst = first;
first = new Node();
first.item = item;
first.next = oldfirst;
}
set the instance variables in the new node
first
private inner class
(access modifiers for instance
variables don't matter)
public String pop()
{
String item = first.item;
first = first.next;
return item;
}
to
null
}
Inserting a new node at the beginning of a linked list
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Stack: linked-list implementation performance
How to implement a fixed-capacity stack with an array?
Proposition. Every operation takes constant time in the worst case.
A.
Can't be done efficiently with an array.
Proposition. A stack with N items uses ~ 40 N bytes.
node object (inner class)
public class Node
{
String item;
inner class
Node next;
...
private class
Node
}
{
String item;
Node next;
}
top of stack
40 bytes
object
overhead
16 bytes (object overhead)
extra
overhead
B.
8 bytes (inner class extra overhead)
it
was
the
best
of
times
null
null
null
null
0
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times
of
best
the
was
it
null
null
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null
0
1
2
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9
8 bytes (reference to String)
item
references
next
8 bytes (reference to Node)
40 bytes per stack node
top of stack
Remark. This accounts for the memory for the stack
C.
(but not the memory for strings themselves, which the client owns).
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Fixed-capacity stack: array implementation
Fixed-capacity stack: array implementation
・Use array s[] to store N items on stack.
・push(): add new item at s[N].
・pop(): remove item from s[N-1].
public class FixedCapacityStackOfStrings
{
private String[] s;
private int N = 0;
a cheat
(stay tuned)
top of stack
public FixedCapacityStackOfStrings(int capacity)
{
s[]
it
was
the
best
of
times
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N
s = new String[capacity];
}
public boolean isEmpty()
{ return N == 0; }
capacity = 10
public void push(String item)
{
s[N++] = item;
}
public String pop()
{ return s[--N]; }
use to index into array;
then increment N
}
decrement N;
then use to index into array
Defect. Stack overflows when N exceeds capacity. [stay tuned]
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Stack considerations
Overflow and underflow.
・Underflow: throw exception if pop from an empty stack.
・Overflow: use resizing array for array implementation. [stay tuned]
Null items. We allow null items to be inserted.
1.3 B AGS, QUEUES, AND S TACKS
Loitering. Holding a reference to an object when it is no longer needed.
‣ stacks
‣ resizing arrays
public String pop()
{ return s[--N]; }
loitering
Algorithms
public String pop()
{
String item = s[--N];
s[N] = null;
return item;
}
R OBERT S EDGEWICK | K EVIN W AYNE
http://algs4.cs.princeton.edu
this version avoids "loitering":
garbage collector can reclaim memory for
an object only if no outstanding references
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‣ queues
‣ generics
‣ iterators
‣ applications
Stack: resizing-array implementation
Stack: resizing-array implementation
Problem. Requiring client to provide capacity does not implement API!
Q. How to grow array?
Q. How to grow and shrink array?
A. If array is full, create a new array of twice the size, and copy items.
public ResizingArrayStackOfStrings()
{ s = new String[1]; }
First try.
・push(): increase size of array s[] by 1.
・pop(): decrease size of array s[] by 1.
Too expensive.
public void push(String item)
{
if (N == s.length) resize(2 * s.length);
s[N++] = item;
}
infeasible for large N
・Need to copy all items to a new array, for each operation.
・Array accesses to insert first N items = N + (2 + 4 + … + 2(N – 1)) ~
1 array access
per push
"repeated doubling"
private void resize(int capacity)
{
String[] copy = new String[capacity];
for (int i = 0; i < N; i++)
copy[i] = s[i];
s = copy;
}
N 2.
2(k–1) array accesses to expand to size k
(ignoring cost to create new array)
Array accesses to insert first N = 2i items. N + (2 + 4 + 8 + … + N) ~ 3N.
Challenge. Ensure that array resizing happens infrequently.
1 array access
per push
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Stack: resizing-array implementation
Stack: resizing-array implementation
Q. How to shrink array?
Q. How to shrink array?
First try.
Efficient solution.
・push():
・pop():
・push():
・pop():
double size of array s[] when array is full.
halve size of array s[] when array is one-half full.
Too expensive in worst case.
・
・Each operation takes time proportional to N.
to
be
or
not
N = 4
to
be
or
not
N = 5
to
be
or
not
N = 4
to
be
or
not
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double size of array s[] when array is full.
halve size of array s[] when array is one-quarter full.
public String pop()
{
Consider push-pop-push-pop-… sequence when array is full.
N = 5
k array accesses to double to size k
(ignoring cost to create new array)
String item = s[--N];
s[N] = null;
if (N > 0 && N == s.length/4) resize(s.length/2);
return item;
to
null
null
null
to
null
null
null
}
Invariant. Array is between 25% and 100% full.
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Stack resizing-array implementation: performance
Stack resizing-array implementation: memory usage
Amortized analysis. Starting from an empty data structure, average
Proposition. Uses between ~ 8 N and ~ 32 N bytes to represent a stack
running time per operation over a worst-case sequence of operations.
with N items.
Proposition. Starting from an empty stack, any sequence of M push and
・~ 8 N when full.
・~ 32 N when one-quarter full.
pop operations takes time proportional to M.
best
worst
amortized
construct
1
1
1
push
1
N
1
pop
1
N
1
1
1
1
size
public class ResizingArrayStackOfStrings
{
8 bytes × array size
private String[] s;
private int N = 0;
…
}
doubling and
halving operations
order of growth of running time
for resizing stack with N items
Remark. This accounts for the memory for the stack
(but not the memory for strings themselves, which the client owns).
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Stack implementations: resizing array vs. linked list
Tradeoffs. Can implement a stack with either resizing array or linked list;
save a link to the list
client can use
Nodeinterchangeably.
oldfirst = first; Which one is better?
oldfirst
Linked-list implementation.
first
or
be
・
extra time and space to deal with the links.
・Usescreate
a new node for the beginning
1.3 B AGS, QUEUES, AND S TACKS
to
Every operation takes constant time
in the worst case.
null
‣ stacks
first = new Node();
Resizing-array implementation.
oldfirst
・
・Less wasted space.
‣ resizing arrays
Every operation
takes constant amortized time.
first
or
be
Algorithms
to
null
set the instance variables in the new node
N = 4
to= "not";
be
or
first.item
not
null
null
null
R OBERT S EDGEWICK | K EVIN W AYNE
null
http://algs4.cs.princeton.edu
first.next = oldfirst;
first
not
or
be
to
null
Inserting a new node at the beginning of a linked list
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‣ queues
‣ generics
‣ iterators
‣ applications
Queue API
How to implement a queue with a linked list?
enqueue
A.
Can't be done efficiently with a singly-linked list.
public class QueueOfStrings
QueueOfStrings()
void enqueue(String item)
create an empty queue
back of queue
B.
insert a new string onto queue
times
front of queue
of
best
the
was
it
null
remove and return the string
least recently added
String dequeue()
is the queue empty?
boolean isEmpty()
number of strings on the queue
int size()
front of queue
back of queue
dequeue
C.
it
was
the
best
of
times
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Queue: linked-list implementation
null
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Queue dequeue: linked-list implementation
・Maintain one pointer first to first node in a singly-linked list.
・Maintain another pointer last to last node.
・Dequeue from first.
・Enqueue after last.
save item to return
String item = first.item;
delete first node
first = first.next;
inner class
last
first
private class Node
{
String item;
Node next;
}
front of queue
it
first
to
be
or
null
first
last
to
be
or
null
back of queue
was
the
best
of
times
return saved item
null
return item;
last
Removing the first node in a linked list
Remark. Identical code to linked-list
stack pop().
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Queue enqueue: linked-list implementation
Queue: linked-list implementation in Java
save a link to the last node
public class LinkedQueueOfStrings
{
private Node first, last;
Node oldlast = last;
oldlast
private class Node
{ /* same as in LinkedStackOfStrings */
last
first
to
be
or
null
public boolean isEmpty()
{ return first == null;
create a new node for the end
inner class
oldlast
first
to
be
last
or
null
}
public void enqueue(String item)
{
Node oldlast = last;
last = new Node();
last.item = item;
last.next = null;
if (isEmpty()) first = last;
else
oldlast.next = last;
}
last = new Node();
last.item = "not";
private class Node
{
String item;
Node next;
}
}
not
null
special cases for
empty queue
link the new node to the end of the list
public String dequeue()
{
String item = first.item;
first
= first.next;
if (isEmpty()) last = null;
return item;
}
oldlast.next = last;
oldlast
to
first
be
last
or
not
null
}
Inserting a new node at the end of a linked list
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How to implement a fixed-capacity queue with an array?
Queue: resizing-array implementation
A.
・Use array q[] to store 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.
Can't be done efficiently with an array.
front of queue
B.
back of queue
it
was
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times
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front of queue
q[]
back of queue
null
null
the
best
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times
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front of queue
head
C.
back of queue
times
of
best
the
was
it
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tail
capacity = 10
Q. How to resize?
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Parameterized stack
We implemented: StackOfStrings.
We also want: StackOfURLs, StackOfInts, StackOfVans, ….
Attempt 1. Implement a separate stack class for each type.
・Rewriting code is tedious and error-prone.
・Maintaining cut-and-pasted code is tedious and error-prone.
1.3 B AGS, QUEUES, AND STACKS
‣ stacks
‣ resizing arrays
‣ queues
Algorithms
@#$*! most reasonable approach until Java 1.5.
‣ generics
‣ iterators
R OBERT S EDGEWICK | K EVIN W AYNE
‣ applications
http://algs4.cs.princeton.edu
34
Parameterized stack
Parameterized stack
We implemented: StackOfStrings.
We implemented: StackOfStrings.
We also want: StackOfURLs, StackOfInts, StackOfVans, ….
We also want: StackOfURLs, StackOfInts, StackOfVans, ….
Attempt 2. Implement a stack with items of type Object.
Attempt 3. Java generics.
・Casting is required in client.
・Casting is error-prone: run-time error if types mismatch.
・Avoid casting in client.
・Discover type mismatch errors at compile-time instead of run-time.
type parameter
StackOfObjects s = new StackOfObjects();
Stack<Apple> s = new Stack<Apple>();
Apple a = new Apple();
Orange b = new Orange();
Apple a = new Apple();
Orange b = new Orange();
s.push(a);
s.push(b);
s.push(a);
s.push(b);
a = (Apple) (s.pop());
a = s.pop();
compile-time error
run-time error
Guiding principles. Welcome compile-time errors; avoid run-time errors.
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Generic stack: linked-list implementation
public class LinkedStackOfStrings
{
private Node first = null;
the way it should be
public class Stack<Item>
{
private Node first = null;
private class Node
{
String item;
Node next;
}
public boolean isEmpty()
{ return first == null;
Generic stack: array implementation
private class Node
{
Item item;
Node next;
}
generic type name
public boolean isEmpty()
{ return first == null;
}
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 boolean isEmpty()
{ return N == 0; }
public boolean isEmpty()
{ return N == 0; }
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 void push(String item)
{
Node oldfirst = first;
first = new Node();
first.item = item;
first.next = oldfirst;
}
public void push(Item item)
{
Node oldfirst = first;
first = new Node();
first.item = item;
first.next = oldfirst;
}
public String pop()
{
String item = first.item;
first = first.next;
return item;
}
public Item pop()
{
Item item = first.item;
first = first.next;
return item;
}
}
public class FixedCapacityStackOfStrings
{
private String[] s;
private int N = 0;
}
}
}
}
@#$*! generic array creation not allowed in Java
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Generic stack: array implementation
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Unchecked cast
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 ..StackOfStrings(int capacity)
{ s = new String[capacity]; }
public FixedCapacityStack(int capacity)
{ s = (Item[]) new Object[capacity]; }
public boolean isEmpty()
{ return N == 0; }
public boolean isEmpty()
{ return N == 0; }
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];
}
% javac FixedCapacityStack.java
Note: FixedCapacityStack.java uses unchecked or unsafe operations.
Note: Recompile with -Xlint:unchecked for details.
% javac -Xlint:unchecked FixedCapacityStack.java
FixedCapacityStack.java:26: warning: [unchecked] unchecked cast
found
: java.lang.Object[]
required: Item[]
a = (Item[]) new Object[capacity];
^
1 warning
}
}
Q. Why does Java make me cast (or use reflection)?
Short answer. Backward compatibility.
Long answer. Need to learn about type erasure and covariant arrays.
the ugly cast
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Generic data types: autoboxing
Q. What to do about primitive types?
Wrapper type.
・Each primitive type has a wrapper object type.
・Ex: Integer is wrapper type for int.
1.3 B AGS, QUEUES, AND S TACKS
‣ stacks
Autoboxing. Automatic cast between a primitive type and its wrapper.
‣ resizing arrays
Algorithms
Stack<Integer> s = new Stack<Integer>();
s.push(17);
int a = s.pop();
// s.push(Integer.valueOf(17));
// int a = s.pop().intValue();
R OBERT S EDGEWICK | K EVIN W AYNE
http://algs4.cs.princeton.edu
‣ queues
‣ generics
‣ iterators
‣ applications
Bottom line. Client code can use generic stack for any type of data.
41
Iteration
Iterators
java.lang.Iterable interface
Design challenge. Support iteration over stack items by client,
Q. What is an Iterable ?
without revealing the internal representation of the stack.
A. Has a method that returns an Iterator.
i
s[]
it
was
0
1
first
the
best
2
3
N
of
4
times
null
null
null
null
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5
Q. What is an Iterator ?
java.util.Iterator interface
A. Has methods hasNext() and next().
current
Q. Why make data structures Iterable ?
times
of
best
public interface Iterable<Item>
{
Iterator<Item> iterator();
}
the
was
it
A. Java supports elegant client code.
null
“foreach” statement (shorthand)
for (String s : stack)
StdOut.println(s);
Java solution. Make stack implement the java.lang.Iterable interface.
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public interface Iterator<Item>
{
boolean hasNext();
Item next();
optional; use
void remove();
at your own risk
}
equivalent code (longhand)
Iterator<String> i = stack.iterator();
while (i.hasNext())
{
String s = i.next();
StdOut.println(s);
}
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Stack iterator: linked-list implementation
Stack iterator: array implementation
import java.util.Iterator;
import java.util.Iterator;
public class Stack<Item> implements Iterable<Item>
{
...
public class Stack<Item> implements Iterable<Item>
{
…
public Iterator<Item> iterator() { return new ListIterator(); }
public Iterator<Item> iterator()
{ return new ReverseArrayIterator(); }
private class ListIterator implements Iterator<Item>
{
private Node current = first;
private class ReverseArrayIterator implements Iterator<Item>
{
private int i = N;
public boolean hasNext() { return current != null; }
public void remove()
{ /* not supported */
}
public Item next()
{
throw UnsupportedOperationException
Item item = current.item;
throw NoSuchElementException
if no more items in iteration
current
= current.next;
return item;
}
public boolean hasNext() {
public void remove()
{
public Item next()
{
return i > 0;
/* not supported */
return s[--i];
}
}
}
}
}
}
}
i
first
s[]
times
N
current
of
best
the
was
it
null
it
was
the
best
of
times
null
null
null
null
0
1
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Iteration: concurrent modification
Q. What if client modifies the data structure while iterating?
A. A fail-fast iterator throws a java.util.ConcurrentModificationException.
concurrent modification
for (String s : stack)
stack.push(s);
1.3 B AGS, QUEUES, AND S TACKS
‣ stacks
‣ resizing arrays
Algorithms
Q. How to detect?
A.
・Count total number of push() and pop() operations in Stack.
・Save counts in *Iterator subclass upon creation.
・If, when calling next() and hasNext(), the current counts do not equal
R OBERT S EDGEWICK | K EVIN W AYNE
http://algs4.cs.princeton.edu
the saved counts, throw exception.
47
‣ queues
‣ generics
‣ iterators
‣ applications
Java collections library
Java collections library
List interface. java.util.List is API for an sequence of items.
java.util.Stack.
・Supports push(), pop(), and iteration.
・Extends java.util.Vector, which implements java.util.List
public interface List<Item> implements Iterable<Item>
interface from previous slide, including get() and remove().
create an empty list
List()
・Bloated and poorly-designed API (why?)
is the list empty?
boolean isEmpty()
number of items
int size()
void add(Item item)
append item to the end
Item get(int index)
return item at given index
Java 1.3 bug report (June 27, 2001)
The iterator method on java.util.Stack iterates through a Stack from
the bottom up. One would think that it should iterate as if it were
popping off the top of the Stack.
return and delete item at given index
Item remove(int index)
does the list contain the given item?
boolean contains(Item item)
iterator over all items in the list
Iterator<Item> iterator()
status (closed, will not fix)
...
It was an incorrect design decision to have Stack extend Vector ("is-a"
rather than "has-a"). We sympathize with the submitter but cannot fix
this because of compatibility.
Implementations. java.util.ArrayList uses resizing array;
java.util.LinkedList uses linked list.
caveat: only some
operations are efficient
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Java collections library
War story (from Assignment 1)
java.util.Stack.
Generate random open sites in an N-by-N percolation system.
・Supports push(), pop(), and iteration.
・Extends java.util.Vector, which implements java.util.List
・Jenny:
percolates
blocked
pick (i, j) at random; if already open, repeat.
does not percolate
site
Takes ~ c1 N 2 seconds.
・Kenny:
interface from previous slide, including get() and remove().
・Bloated and poorly-designed API (why?)
create a java.util.ArrayList of N 2 closed sites.
Pick an index at random and delete.
open
Takes ~ c2 N 4 seconds.
site
open site connected to top
no open site connected to top
Why is my program so slow?
Kenny
java.util.Queue. An interface, not an implementation of a queue.
Lesson. Don't use a library until you understand its API!
Best practices. Use our implementations of Stack, Queue, and Bag.
This course. Can't use a library until we've implemented it in class.
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Stack applications
Function calls
・Parsing in a compiler.
・Java virtual machine.
・Undo in a word processor.
・Back button in a Web browser.
・PostScript language for printers.
・Implementing function calls in a compiler.
・...
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 to remove recursion.
gcd (216, 192)
static int gcd(int p, int q) {
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;
gcd
else return gcd(q, p
% (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);
}
p = 216, q = 192
4.3 Stacks and Queues
573
53
it is easy to convince yourself that it computes the proper value: any time the algorithm encounters a subexpression consisting of two operands separated by an opArithmetic expressionerator,
evaluation
all surrounded by parentheses, it leaves the result of performing that operation on those operands on the operand stack. The result is the same as if that value
had appeared in the input instead of the subexpression, so we can think of replacing
valuethe
stack
Goal. Evaluate infix expressions.
(1+((2+3)*(4*5)))
subexpression by the value to get an
expression
operator
stack
that would yield the same result. We can apply
1
+((2+3)*(4*5)))
this argument again and again until we get a
( 1 + ( ( 2 + 3 ) * ( single
4 * 5value.
) )For) example, the algorithm com1
((2+3)*(4*5)))
putes the same value of all of these expres+
sions:
1 2
operand
operator
( 1
( 1
( 1
( 1
101
+
+
+
+
( (
( 5
( 5
100
2 + 3 ) * ( 4 * 5 ) ) )
* ( 4 * 5 ) ) )
* 20 ) )
)
Two-stack algorithm. [E. W. Dijkstra]
+
+3)*(4*5)))
1 2
+ +
3)*(4*5)))
1 2 3
+ +
)*(4*5)))
(PROGRAM 4.3.5) is an implemen1 5
Value: push onto the
stack.This code is a simple
tationvalue
of this method.
*(4*5)))
+
example of an interpreter : a program that in1 5
Operator: push onto
the
stack.by a given
terprets
theoperator
computation specified
(4*5)))
+ *
string and performs the computation to ar1 5 4
Left parenthesis: ignore.
rive at the result. A compiler is a program that
*5)))
+ *
the string into code on a lower-level
Right parenthesis: converts
pop
operator
and
two
values;
1 5 4
machine that can do the job. This conversion
5)))
+ * *
is a more complicated
process than the steppush the result of applying
that operator
1 5 4 5
by-step conversion used by an interpreter, but
)))
+ * *
it is based
the same underlying
to those values onto
the on
operand
stack.mechanism.
1 5 20
Initially, Java was based on using an interpret))
+ *
er. Now, however, the Java system includes a
1 100
compiler that converts arithmetic expressions
)
+
(and, more generally, Java programs) into code
101
for the Java virtual machine, an imaginary maContext. An interpreter!
chine that is easy to simulate on an actual computer.
Trace of expression evaluation (Program 4.3.5)
・
・
・
・
54
Dijkstra's two-stack algorithm demo
Evaluate
value stack
infix expression
operator stack
(fully parenthesized)
(
1
+
(
(
2
+
operand
55
3
)
*
(
4
*
5
)
)
)
operator
56
Arithmetic expression evaluation
Correctness
Q. Why correct?
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 + ( ( 2 + 3 ) * ( 4 * 5 ) ) )
101.0
A. When algorithm encounters an operator surrounded by two values
within parentheses, it leaves the result on the value stack.
( 1 + ( ( 2 + 3 ) * ( 4 * 5 ) ) )
as if the original input were:
( 1 + ( 5 * ( 4 * 5 ) ) )
Repeating the argument:
( 1 + ( 5 * 20 ) )
( 1 + 100 )
101
Extensions. More ops, precedence order, associativity.
57
Stack-based programming languages
Observation 1. Dijkstra's two-stack algorithm computes the same value if
the operator occurs after the two values.
( 1 ( ( 2 3 + ) ( 4 5 * ) * ) + )
Observation 2. All of the parentheses are redundant!
1 2 3 + 4 5 * * +
Jan Lukasiewicz
Bottom line. Postfix or "reverse Polish" notation.
Applications. Postscript, Forth, calculators, Java virtual machine, …
59
58