stacks infix postfix
TRANSCRIPT
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Stacks
Briana B. Morrison
Adapted from Alan Eugenio
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Topics Define Stack
APIs
Applications Create Hex Numer
!ecursion Con"erting from Infix to Postfix
E"aluating Postfix
Con"erting from Infix to Prefix
Implementation Arra# ased
$inked list ased
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Stacks A stack is a se%uence of items t&at are
accessile at onl# one end of t&e se%uence.
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Pushing/Popping a Stack
Because a pop remo"es t&e item last added tot&e stack' (e sa# t&at a stack &as $I)* +last,
in-first,out ordering.
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TOP – THE MOST RECENTLY INSERTED
ITEM
PUSH - TO INSERT ONTO THE TOP OF A
STACK
POP - TO REMOVE THE TOP ITEM IN A
STACK
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THE stack CLASS IS TEMPLATED:
template /class 0, class Container 1 de%ue/02 2
0 IS THE TEMPLATE PARAMETER
FOR THE ITEM TYPE. Container IS THE
TEMPLATE PARAMETER FOR THE
CLASS THAT WILL HOLD THE ITEMS.
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METHOD INTERFACES
FOR THE
stack CLASS
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CLASS stack Constructor stack!
stack+3Create an empt# stack
CLASS stack O"#rat$ons stack!
ool empty+3 const
C&eck (&et&er t&e stack is empt#. !eturn true if it isempt# and false ot&er(ise.
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CLASS stack O"#rat$ons stack!
"oid pop+3!emo"e t&e item from t&e top of t&e stack.Precondition4 0&e stack is not empt#.
Postcondition4 Eit&er t&e stack is empt# ort&e stack &as a ne( topmost
item from a pre"ious pus&."oid push+const 05 item3
Insert t&e argument item at t&e top of t&e stack.Postcondition4 0&e stack &as a ne( item at
t&e top.
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CLASS stack O"#rat$ons stack!
int size+ const3!eturn t&e numer of items on t&e stack.
05 top+ const3
!eturn a reference to t&e "alue of t&e item at t&e
top of t&e stack.Precondition4 0&e stack is not empt#.
const 05 top+ const3
Constant "ersion of top+.
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THERE ARE NO ITERATORS%
THE ONLY ACCESSI&LE ITEM IS THE
FRONT ITEM.
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DETERMINE THE O'TP'T FROM
THE FOLLOWIN(:
stack/int2 m#6stack3
for +int i 1 73 i / 873 i99 m#6stack.pus& +i : i3
while +;m#6stack.empt#+
< cout // m#6stack.top+ // endl3
m#6stack.pop+3
= -- (&ile
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SO THE "ector ) de%ue) AND list CLASSES
CAN SERVE AS THE 'NDERLYIN(
CONTAINER CLASS.
NOTE: THE TOP ITEM ON THE
STACK IS AT THE BACK OF THE
'NDERLYIN( CONTAINER%
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HERE IS PART OF THE DEFINITION OF
THE stack CLASS:
template /class 0' class Container 1 de%ue/02 2class stack< protected4
Container c3
public4
@
void pus& +const "alue6t#pe5 x < c.pus&6ack +x3 ==
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stack %oe
template void stack::push(const Item_Type& item) {
container.push_back(item)
!
template Item_Type& stack::top() {
return container.back()
!
template
const Item_Type& stack::top() const {
return container.back()
!
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STACK APPLICATIONS
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EACH ACTIVATION RECORD CONTAINS:
*. A VARIA&LE THAT CONTAINS THE RET'RN ADDRESS
IN THE CALLIN( METHOD+
,. FOR EACH VAL'E FORMAL PARAMETER) A
VARIA&LE THAT CONTAINS A COPY OF THE
AR('MENT+
-. FOR EACH REFERENCE FORMAL PARAMETER) A
VARIA&LE THAT CONTAINS THE ADDRESS OF THE
AR('MENT+
. FOR EACH VARIA&LE DEFINED IN THE METHOD/S
&LOCK) A VARIA&LE THAT CONTAINS A COPY OF
THAT DEFINED VARIA&LE.
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THERE IS A R'N0TIME STACK TO
HANDLE THESE ACTIVATIONRECORDS.
P'SH: WHEN F'NCTION IS CALLED
POP: WHEN E1EC'TION OFF'NCTION IS COMPLETED
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AN ACTIVATION RECORD IS SIMILAR
TO AN E1EC'TION FRAME) E1CEPT
THAT AN ACTIVATION RECORD HAS
VARIA&LES ONLY) NO CODE.
YO' CAN REPLACE REC'RSIONWITH ITERATION &Y CREATIN(
YO'R OWN STACK.
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%.. untime Stack
0&e C99 run,time s#stem keepstrack of t&e c&ain of acti"e functions(it& a stack
?&en a function is called' t&e run,
time s#stem pus&es on t&e stack aframe containing $ocal "ariales and return "alue Program counter' keeping track of
t&e statement eing executed
?&en a function returns' its frame ispopped from t&e stack and controlis passed to t&e met&od on top oft&e stack
main() {
int i = 5;
foo(i);
}
foo(int j) {
int k;
k = j+1;
bar(k);
}
bar(int m) {
…
}
bar PC = 1 m = 6
foo PC = 3 j = 5 k = 6
main PC = 2 i = 5
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A r g u m e n t s
i n t n
R e t u r n A d d r e s s
R e t L o c o r R e t L o c 2
A c t $ 2 a t $ o n R # c o r 3
" n m a i n # $ %
c a & & a c t # 4 $ A r g u m e n t 4 R e t u r n ( e t ) o c 1
S 4 s t # 5 S t a c k
" n a c t # 4 $
c a & & a c t # 3 $
A r g u m e n t 3
A r g u m e n t 4
R e t u r n ( e t ) o c 2
R e t u r n ( e t ) o c 1
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STACK APPLICATION
CONVERTIN( FROM
INFI1 TO POSTFI1
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IN INFIX NOTATION) AN OPERATOR
IS PLACED &ETWEEN ITSOPERANDS.
a 6 7
c 8 3 6 9# ; 8 < => ? $
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OLD COMPILERS:
INFI1 MACHINE LAN('A(E
THIS (ETS MESSY &ECA'SE OF
PARENTHESES.
NEWER COMPILERS:
INFI1 POSTFI1 MACHINE LAN(.
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In POSTFIX Notat$on) An OPERATOR $s
"@ac#3 IMMEDIATELY AFTER $ts
OPERANDS.
INFI1
POSTFI1
a 6 7
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LET/S CONVERT AN INFI1 STRIN( TO
A POSTFI1 STRIN(.
8 4 B
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INFI1 POSTFI1
8 4 B
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INFI1 POSTFI1
8 4 B
THE OPERANDS FOR 0/ ARE NOT YET
IN POSTFI1) SO 0/ M'ST &E
TEMPORARILY SAVED SOMEWHERE.9STACK>
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INFI1 POSTFI1
8 4 B 4
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INFI1 POSTFI1
8 4 B 4
THE OPERANDS FOR / ARE NOT YET
IN POSTFI1) SO / M'ST &E
TEMPORARILY SAVED SOMEWHERE)AND RESTORED BEFORE ‘-’ .
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INFI1 POSTFI1
8 4 B 4B
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Su""os#) $nst#a3) # start#3 $t=
40B
A;t#r 5o2$n< / to "ost;$) / $s
t#5"orar$@4 sa2#3) an3 t=#n 4/ $s
a""#n3#3 to "ost;$. W=at =a""#ns =#n
0/ $s acc#ss#3
INFI1 POSTFI1
4 8 B 4
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%one!ting In$ to Post$ Anal#sis4 Operands are in same order in infix and postfix
Operators occur later in postfix
Strateg#4 Send operands straig&t to output
Send &ig&er precedence operators first
If same precedence' send in left to rig&t order Hold pending operators on a stack
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FOR EACH OPERATOR IN INFI1:
LOOP 'NTIL OPERATOR P'SHED:
IF OPERATOR STACK IS EMPTY)P'SH
ELSE IF INFIX OPERATOR HAS
GREATER PRECEDENCE THAN TOP OPERATOR
ON STACK)
PUSHELSE
POP AND APPEND TO POSTFI1
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INFIX GREATER, PUSH
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CONVERT FROM INFI1 TO POSTFI1:
INFI1 POSTFI1
a 6 7 c ? 3 0 #
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INFI1 POSTFI1
a 6 7 c ? 3 8 # a7c3?6# 8
0
?
6
OPERATOR STACK
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WHAT A&O'T PARENTHESES
LEFT PARENTHESIS: P'SH) &'T
WITH LOWEST PRECEDENCE.
RI(HT PARENTHESIS:KEEP POPPIN(
AND APPENDIN( TO POSTFI1 'NTIL
9 POPPED+ PITCH 9 AND PROCEED.
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CONVERT TO POSTFI1:
94 6 B>
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INFI1 POSTFI1
94 6 B> 4B6
6 9
OPERATOR STACK
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# i
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#$e!cise
Con"ert t&e follo(ing expression to postfix4
a 9 , + c 9 d : e - + f g : + & 9 i
# l i P
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#aluating a Post$#$p!ession ?e descrie &o( to parse and e"aluate a
postfix expression.
?e read t&e tokens in one at a time.
If it is an integer' pus& it on t&e stack
If it is a inar# operator' pop t&e top t(o
elements from t&e stack' appl# t&e operator'
and pus& t&e result ack on t&e stack.
Stacks 58
5 9 3 . 4 2 7
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5 9 3 . 4 2 7. Stack *perations *utput
pus&+3 pus&+3
pus&+3
pus&+pop+ 9 pop+ 8F
pus&+G3 8F Gpus&+F3 8F G F
pus&+pop+ : pop+ 8F
pus&+pop+ : pop+ pus&+J Jpus&+pop+ 9 pop+ 87
pus&+pop+ : pop+ 8
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THE FOLLOWIN( TRANSITION
MATRI1 SPECIFIES THE TRANSITION
FROM INFI1 NOTATION TO POSTFI1
NOTATION:
TOP CHARACTER ON STACK
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9 6)0 )? #5"t4I
N $3#nt$;$#rF
I >1 9
6)0C
H )?
AR #5"t4
APPEND APPEND APPEND APPEND
TO TO TO TOPOSTFI1 POSTFI1 POSTFI1 POSTFI1
POP+ POP TO POP TO ERROR
PITCH 9 POSTFI1 POSTFI1
P'SH P'SH P'SH P'SH
P'SH POP TO POP TO P'SH
POSTFI1 POSTFI1
P'SH P'SH POP TO P'SH
POSTFI1
ERROR POP TO POP TO DONE
POSTFI1 POSTFI1
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STACK APPLICATION
EVAL'ATIN( A
% , ith 1 t k
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%an ,e one ith 1 stack
?&ene"er an operand is encountered' pus&
onto stack
?&ene"er operator is encountered' pop
re%uired numer of arguments from operandstack and e"aluate
Pus& result ack onto stack
(l ith " l
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(lgo!ithm "o! eal
8. Empt# t&e operand stack
#. $hile t&ere are more tokens
. Ket t&e next token
G. i% t&e first c&aracter of t&e token is a digit
.
Pus& t&e integer onto t&e stack. else i% t&e token is an operator
J. Pop t&e rig&t operand off t&e stack
. Pop t&e left operand off t&e stack
. E"aluate t&e operation87. Pus& t&e result onto t&e stack
88. Pop t&e stack and return t&e result
R P N 9 R # 2 # r s # P o @ $ s = N o t a t $ o n > # " r # s s $ o n , - 6
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R P N 9 R # 2 # r s # P o @ $ s = N o t a t $ o n > # " r # s s $ o n , - 6
3
2
o p e r a n * t a c + e m p t ,
3 - " * e n t i , . a s a n o p e r a t o r / e g i n t h e p r o c e s s o e 0 a & u a t i n g . -
4 - g e t p e r a n * s # $ p o p s s t a c + t i c e a n * a s s i g n s 3 t o r i g h t a n * 2 t o & e t -
5 - c o m p u t e # $ e 0 a & u a t e s & e t . r i g h t a n * r e t u r n s t h e 0 a & u e 5 - ( e t u r n 0 a & u e i s p u s h e * o n t h e s t a c + -
5
1 - " * e n t i , 2 a s a n o p e r a n * -
P u s h i n t e g e r 2 o n t h e s t a c + -
C u r r e n t o p e r a n * t a c +
2
3
2
2 - " * e n t i , 3 a s a n o p e r a n * - P u s h i n t e g e r 3 o n t h e s t a c + -
c a n o p r e s s i o n a n * A c t i o n
( th # l
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(nothe! #$ampleExpression4 , 9 F - 8 F 9 , :
F
Execute , Execute F 9
F
Execute - F
G
8
G
F
8
G
Execute
8 9 F
G
Execute
G ,
8
Execute
: 8
F
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IN PREFIX NOTATION) AN OPERATOR
IMMEDIATELY PRECEDES ITSOPERANDS.
INFI1 PREFI1
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INFI1 PREFI1
a 6 7 6a7
a 97 6 c> a67c
a 7 6 c 6a7c
IN PREFI1 NOTATION) AS IN POSTFI1)THERE ARE NO PARENTHESES.
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TWO STACKS ARE 'SED:
OPERATOR STACK: SAME R'LES AS
FOR POSTFI1 STACK
OPERAND STACK: TO HOLD THE
OPERANDS
WHENEVER o"t IS POPPED FROM
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WHENEVER o"t IS POPPED FROM
OPERATOR STACK) o"3* AND THEN
o"3, ARE POPPED FROM OPERAND
STACK. THE STRIN( o"t 6 o"3, 6 o"3*
IS P'SHED ONTO OPERAND STACK.
NOTE: o"3, WAS P'SHED &EFORE o"3*.
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CONVERT FROM INFI1 TO PREFI1:
INFI1
a 6 97 c 8 3> ? #
INFI1 PREFI1
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a 6 97 c 8 3> ? # 6a?8 7c3#
6a?8 7c3#
?8 7c3#
#
87c3 3 ?
7c 8
c
7 9 a 6
OPERAND OPERATOR
STACK STACK
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#$e!cise
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#$e!cise
Con"ert t&e follo(ing expression to prefix4
a 9 , + c 9 d : e - + f g : + & 9 i
Sel" Test
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Sel"Test
?rite t&e code to re"erse an input string
using a stack.
Assuming t&at #ou &a"e a stack t&at contains
colors4 stack/color2(rite t&e code to remo"e all t&e "alues of a
specific color (&ile lea"ing t&e remaining
"alues in t&e same order.
Stack Implementations
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Stack Implementations
Arra# ased ?&ere is topL
Ho( are elements added' remo"edL
$inked $ist ased ?&ere is topL
Ho( are elements added' remo"edL
Efficienc# of operations
(!!a ase Stack
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(!!a ase StackImplementation?&ere s&ould top eL
8F
8
F
G
stack
7 arra#
?&at are arra# "aluesL
s.pus& +F7
s.pop+
!oa,le (!!a ,ase Stack
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!oa,le (!!a,ase Stack
In a pus& operation' (&ent&e arra# is full' instead oft&ro(ing an exception' (ecan replace t&e arra# (it& a
larger one Ho( large s&ould t&e ne(
arra# eL incremental strateg#4 increase
t&e si>e # a constant c douling strateg#4 doule t&e
si>e
A@
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Stacks 81
%ompa!ison o" the St!ategies
?e compare t&e incremental strateg# and t&edouling strateg# # anal#>ing t&e total time T #%$
needed to perform a series of % pus& operations
?e assume t&at (e start (it& an empt# stack
represented # an arra# of si>e 1
?e call amorti>ed time of a pus& operation t&e
a"erage time taken # a pus& o"er t&e series of
operations' i.e.' T #%$/%
Inc!emental St!ateg
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Inc!emental St!ateg (nalsis
?e replace t&e arra# ( = %-c times 0&e total time T #%$ of a series of % pus&
operations is proportional to
% + c + 2c . 3c . 4c . ) . (c =
% + c#1 . 2 . 3 . 6 . ( $ =
% + c( #( . 1$/2 Since c is a constant' T #%$ is O#% + ( 2$' i.e.' O#%2$ 0&e amorti>ed time of a pus& operation is O#%$
ou,ling St!ateg (nalsis
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ou,ling St!ateg (nalsis
?e replace t&e arra# ( = &og2 %times
0&e total time T #%$ of a series of % pus& operations is proportional to
% + 1 . 2 . 4 . 9 . 6. 2( =
% + 2( . 1 −1 = 2% −1 T #%$ is O#%$
0&e amorti>ed time of a pus&operation is O#1$
geometric series
1
2
14
9
:inke :ist Implementation
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Stacks 84
:inke :ist Implementation
D
C
t o p
A
t a c +
D AC
) i n + e * ) i s t
r o n t
s.pus& +O)
s.pop+
Stack ith a Singl :inke
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Stacks 85
Stack ith a Singl :inke:ist ?e can implement a stack (it& a singl# linked list
0&e top element is stored at t&e first node of t&e list
0&e space used is O#%$ and eac& operation of t&e Stack AD0
takes O#1$ time
!
nodes
elements
%ompa!ison o" stack
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Stacks 86
%ompa!ison o" stack Implementations
0&e code for t&e "ector "ersion is "er# similar to t&e
implementation in t&e C99 standard lirar#. B# using delegation' (e a"oid &a"ing to implement t&e
operations oursel"es and are assured of *+8performance.
sing t&e standard containers &as a space penalt#. "ector allocates additional space to pro"ide ammorti>ed constant
insertion time. list uses a doule linked list (&ic& &as unneeded pointers.
sing a single linked list allocates space for t&e links' utt&ese are necessar#.
sing a single linked list also pro"ides *+8 performance.
Summa! Slie 1
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Stacks 87 87
Summa! Slie 1Q, Stack
, Storage Structure (it& insert +pus& and erase +popoperations occur at one end' called t&e top of t&estack.
, 0&e last element in is t&e first element out of t&estack' so a stack is a $I)* structure.
Summa! Slie 2
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Stacks 88 88
Summa! Slie 2Q, !ecursion
, 0&e s#stem maintains a stack of acti"ation recordst&at specif#4
8 t&e function arguments
F t&e local "ariales-oRects t&e return address
, 0&e s#stem pus&es an acti"ation record (&encalling a function and pops it (&en returning.
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Summa! Slie 4
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Summa! Slie 4Q, Infix notation
, A inar# operator appears et(een its operands.
, More complex t&an postfix' ecause it re%uires t&euse of operator precedence and parent&eses.
, In addition' some operators are left,associati"e' anda fe( are rig&t,associati"e.