Stacks, Queues, Deques
Hsuan-Tien Lin
Dept. of CSIE, NTU
April 7, 2015
A Simple Application: Parentheses Balancing
in C, the following characters show up in pairs: (), [], {}, ""
good: {xxx(xxxxxx)xxxxx"xxxx"x}
bad: {xxx(xxxxxx}xxxxx"xxxx"x}
the LISP programming language
(append (pow (* (+ 3 5) 2) 4) 3) how can we check parentheses balancing?
Stack Solution to Parentheses Balancing
inner-most parentheses pair =⇒ top-most plate
’(’: 堆盤子上去 ; ’)’: 拿盤子下來 Parentheses Balancing Algorithm
for each c in the input do if c is a left character
push c to the stack else if c is a right character
pop d from the stack and check if match end if
end for
many more sophisticated use in compiler design
System Stack
recall: function call ⇔ 拿新的草稿紙來算
old (original) scrap paper: temporarily not used, 可以壓在下面 System Stack: 一疊草稿紙 , each paper (stack frame) contains
return address: where to return to the previous scrap paper local variables (including parameters): to be used for calculating within this function
previous frame pointer: to be used when escaping from this function
some related issues: stack overflow? security attack?
Stacks Implemented on Array (5.1.4)
Reading Assignment
be sure to go ask the TAs or me if you are still confused
Stacks Implemented on Linked List (5.1.5)
Reading Assignment
be sure to go ask the TAs or me if you are still confused
Stack for Expression Evaluation (Supplementary)
a/b − c + d ∗ e − a ∗ c precedence: {∗, /} first; {+, −} later steps
f = a/b g = f − c h = d ∗ e i = g + h j = a ∗ c
` =i − j
Postfix Notation
same operand order, but put “operator”afterneeded operands
—can “operate” immediately when seeing operator
—no need to look beyond for precedence
Postfix from Infix (Usual) Notation
infix:
3 / 4 − 5 + 6 ∗ 7 − 8 ∗ 9
parenthesize:
3 / 4 − 5 + 6 ∗ 7 − 8 ∗ 9
for every triple in parentheses, switch orders
remove parentheses
difficult to parenthesize efficiently
Evaluate Postfix Expressions
34/5 − 67 ∗ +89 ∗ −
how to evaluate? left-to-right, “operate” when see operator 3, 4, / ⇒ 0.75
0.75, 5, - ⇒ -4.25
-4.25, 6, 7, * ⇒ -4.25, 42 (note: -4.25 stored for latter use) -4.25, 42, + ⇒ 37.75
37.75, 8, 9, * ⇒ 37.75, 72 (note: 37.75 stored for latter use) 37.75, 72, - ⇒ ...
stored where?
stackso closest operands will be considered first!
Stack Solution to Postfix Evaluation
Postfix Evaluation
for each token in the input do if token is a number
push token to the stack else if token is an operator
sequentially pop operands at−1, · · · ,a0from the stack push token(a0,a1,at−1)to the stack
end if end for
return the top of stack
matches closely with the definition of postfix notation
One-Pass Algorithm for Infix to Postfix
infix ⇒ postfix efficiently?
at/, not sure of what to do (need later operands) sostore a/b − c + d ∗ e − a ∗ c
at-, know that a / b can be a b / because-is of lower precedence a/b−c + d ∗ e − a ∗ c
at+, know that ? - c can be ? c - because+is of same precedence but {-,+} is left-associative
a/b − c+d ∗ e − a ∗ c
at*, not sure of what to do (need later operands) sostore a/b − c + d∗e − a ∗ c
stored where?stackso closest operators will be considered first!
Stack Solution to Infix-Postfix Translation
for each token in the input do if token is a number
output token
else if token is an operator
while top of stack is of higher (or same) precedence do pop and output top of stack
end while
push token to the stack end if
end for
here: infix to postfix with operator stack
—closest operators will be considered first recall: postfix evaluation with operand stack
—closest operands will be considered first
mixing the two algorithms (say, use two stacks): simple calculator
Some More Hints on Infix-Postfix Translation
for each token in the input do if token is a number
output token
else if token is an operator
while top of stack is of higher (or same) precedence do pop and output top of stack
end while
push token to the stack end if
end for
for left associativity and binary operators
right associativity? same precedence needs to wait unary/trinary operator? same
parentheses? higest priority
at ’(’, cannot pop anything from stack
—like seeing ’*’ while having ’+’ on the stack at ’)’, can pop until ’(’ —like parentheses matching
