ADT Implementation:
Recursion, Algorithm
Analysis, and Standard
Algorithms
Chapter 10
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Chapter Contents
10.1 Recursion
10.2 Examples of Recursion: Towers of Hanoi;
Parsing
10.3 Implementing Recursion
10.4 Algorithm Efficiency
10.5 Standard Algorithms in C++
10.6 Proving Algorithms Correct (Optional)
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Chapter Objectives
• Review recursion by looking at examples
• Show how recursion is implemented using a
run-time stack
• Look at the important topic of algorithm
efficiency and how it is measured
• Describe some of the powerful and useful
standard C++ function templates in STL
• (Optional) Introduce briefly the topic of
algorithm verification
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Recursion
A function is defined recursively if it has the
following two parts
• An anchor or base case
– The function is defined for one or more specific
values of the parameter(s)
• An inductive or recursive case
– The function's value for current parameter(s) is
defined in terms of previously defined function
values and/or parameter(s)
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Recursive Example
• Consider a recursive power function
double power (double x, unsigned n)
{ if ( n == 0 )
return 1.0;
else
return x * power (x, n-1); }
• Which is the anchor?
• Which is the inductive or recursive part?
• How does the anchor keep it from going forever?
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Recursive Example
• Note the
results of
a call
– Recursive
calls
– Resolution
of the
calls
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A Bad Use of Recursion
• Fibonacci numbers
1, 1, 2, 3, 5, 8, 13, 21, 34
f1 = 1, f2 = 1 … fn = fn -2 + fn -1
– A recursive function
double Fib (unsigned n)
{ if (n <= 2)
return 1;
else
return Fib (n – 1) + Fib (n – 2);
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A Bad Use of Recursion
• Why is this inefficient?
– Note the recursion tree
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Uses of Recursion
• Binary Search
– See source code
– Note results of
recursive call
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Uses of Recursion
• Palindrome checker
– A palindrome has same value with characters reversed
1234321
racecar
• Recursive algorithm for an integer
– If numDigiths <= 1 return true
– Else check first and last digits
num/10numDigits-1 and num % 10
• if they do not match return false
– If they match, check more digits
Apply algorithm recursively to:
num % 10numDigits-1 and numDigits - 2
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Recursion Example:
Towers of Hanoi
• Recursive algorithm especially appropriate for
solution by recursion
• Task
–
–
–
–
Move disks from left peg to right peg
When disk moved, must be placed on a peg
Only one disk (top disk on a peg) moved at a time
Larger disk may never be placed on a smaller disk
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Recursion Example:
Towers of Hanoi
• Identify base case:
If there is one disk move from A to C
• Inductive solution for n > 1 disks
– Move topmost n – 1 disks from A to B, using C
for temporary storage
– Move final disk remaining on A to C
– Move the n – 1 disk from B to C using A for
temporary storage
• View code for solution, Fig 10.4
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Recursion Example:
Towers of Hanoi
• Note the graphical steps to the solution
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Recursion Example: Parsing
• Examples so far are direct recursion
– Function calls itself directly
• Indirect recursion occurs when
– A function calls other functions
– Some chain of function calls eventually results in
a call to original function again
• An example of this is the problem of
processing arithmetic expressions
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Recursion Example: Parsing
• Parser is part of the compiler
• Input to a compiler is
characters
– Broken up into meaningful groups
– Identifiers, reserved words, constants, operators
• These units are called tokens
– Recognized by lexical analyzer
– Syntax rules applied
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Recursion Example: Parsing
• Parser generates a
parse tree using the
tokens according to
rules below:
• An expression:
term + term | term – term | term
• A term:
factor * factor | factor / factor | factor
• A factor:
( expression ) | letter | digit
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Note the indirect
recursion
16
Implementing Recursion
• Recall section 7.4, activation record created
for function call
• Activation records placed on run-time stack
• Recursive calls generate stack of similar
activation records
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Implementing Recursion
• When base case reached and successive
calls resolved
– Activation records are popped off the stack
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Algorithm Efficiency
• How do we measure efficiency
– Space utilization – amount of memory required
– Time required to accomplish the task
• Time efficiency depends on :
– size of input
– speed of machine
– quality of source code
– quality of compiler
These vary from one
platform to another
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Algorithm Efficiency
• We can count the number of times
instructions are executed
– This gives us a measure of efficiency of an
algorithm
• So we measure computing time as:
T(n)= computing time of an algorithm for input of size n
= number of times the instructions are executed
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Example: Calculating the Mean
Task
1.
2.
3.
4.
5.
6.
# times executed
Initialize the sum to 0
Initialize index i to 0
While i < n do following
a) Add x[i] to sum
b) Increment i by 1
Return mean = sum/n
Total
1
1
n+1
n
n
1
3n + 4
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Computing Time Order of
Magnitude
• As number of inputs increases
 T(n) = 3n + 4 grows at a rate proportional to n
• Thus T(n) has the "order of magnitude" n
• The computing time of an algorithm on input
of size n,
 T(n) said to have order of magnitude f(n),
 written T(n) is O(f(n))
if … there is some constant C such that
 T(n) < Cf(n) for all sufficiently large values of n
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Big Oh Notation
Another way of saying this:
• The complexity of the algorithm is O(f(n)).
•
Example: For the Mean-Calculation
Algorithm:
T(n) is O(n)
•
Note that constants and multiplicative
factors are ignored.
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Big Oh Notation
• f(n) is usually simple:
n, n2, n3, ...
2n
1, log2n
n log2n
log2log2n
• Note graph
of common
computing times
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Big Oh Notation
• Graphs of common computing times
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Common Computing Time
Functions
log2n
n
n log2n
n2
n3
2n
0
1
0
1
1
2
0.00
1
2
2
4
8
4
1.00
2
4
8
16
64
16
1.58
3
8
24
64
512
256
2.00
4
16
64
256
4096
65536
2.32
5
32
160
1024
32768
4294967296
2.58
6
64
384
4096
262144
1.84467E+19
3.00
8
256
2048
65536
16777216
1.15792E+77
3.32
10
1024
10240
1048576
1.07E+09 1.8E+308
4.32
20
1048576
20971520
1.1E+12
1.15E+18 6.7E+315652
log2log2n
---
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Computing in Real Time
• Suppose each instruction can be done in 1
microsecond
• For n = 256 inputs how long for various f(n)
Function
Time
log2log2n
3 microseconds
Log2n
8 microseconds
n
.25 milliseconds
n log2n
2 milliseconds
n2
65 milliseconds
n3
17 seconds
2n
3.7+E64 centuries!!
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STL's Algorithms
• STL has a collection of more than 80 generic
algorithms.
– not member functions of STL's container classes
– do not access containers directly.
• They are stand-alone functions
– operate on data by means of iterators .
• Makes it possible to work with regular C-style
arrays as well as containers.
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Example of sort Algorithm
• Comes in several different forms
• One uses the < operator to compare elements
– Requires that operator<() be defined on the data
type being sorted
– Example program, Fig 10.7
• The sort algorithm can be used with arrays
– See example, Fig. 10.8
• Sort algorithm can have a third parameter
– The name of a boolean function to be used for a less
than
– See Fig. 10.9
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A Sample of STL Algorithms
• Designed to operate on a sequence of
elements
• Designate a sequence by using two iterators
– One positioned at the first element of the
sequence
– One positioned after the last element in the
sequence
• Note Table 10-4, Page 571 of text
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Algorithms from <numeric> Library
• Contains function templates that operate on
sequential containers
• Intended for use with numeric sequences
– Such as valarrays
• See Table 10-5, page 572
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Using Algorithms
• Consider the judging of figure skaters
– When judges give scores the high and low score
are discarded
– Then the remaining scores are averaged
• Given the vector of scores shown below
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Using Algorithms
• Use min_element algorithms to find and
discard minimum score
scores.erase(min_element(scores.begin(),
scores.end())));
• Discard maximum element similarly
• Now use accumulate() algorithm
double avg = accumulate(scores.begin(),
scores.end(),0.0)/scores.size();
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Proving Algorithms Correct
• Deductive proof of correctness may be
required
– In safety-critical systems where lives at risk
• Must specify
– The "given" or preconditions
– The "to show" or post conditions
Pre and Algorithm => Post
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Deduction (演繹法; indicator, “⇨” )
• Deduction, a kind of inference form
– by reasoning from the general to the specific.
• Deduction (in logic) form of inference such that the
conclusion must be true if the premises are true.
– The famous Aristotelian syllogism is one species of
deductive reasoning.
– For example, if we know that all men have two legs
and that John is a man, it is then logical to deduce
that John has two legs.
All men have two legs.
John is a man.
⇨ John has two legs.
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Induction ( 歸納法 )
• Induction (logic):
– The process of deriving general principles (i.e.,
rules) from particular facts or instances.
– A conclusion reached by this process.
• induction, in logic, a form of argument in which
the premises give grounds for the conclusion but
do not necessitate it.
– An important form of induction is the process of
reasoning from the particular to the general.
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Example: Recursive Power
Function
• Function:
double power (double x, unsigned n)
{ if ( n == 0 )
return 1.0;
else
return x * power (x, n-1); }
• Precondition:
– Input consists of a real number x and a nonnegative
integer n
• Postcondition:
– Execution of function terminates
– When it terminates value returned is xn
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Example: Recursive Power
Function
• Use mathematical induction on n
– Show postcondition follows if n = 0
• Assume for n = k, execution terminates and
returns correct value
– When called with n = k + 1, inductive case
return x * power (x, n – 1) is
executed
– Value of n – 1 is k
– It follows that with n = k + 1, returns x * xk
which equals xk+1
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Proving Algorithms Correct
• Notation could be used to state assertions
Pre
S
{ Post = Pre (v, e ) }
"If precondition Pre holds before an assignment
statement S of the form v = e is executed, then the
postcondition Post is obtained from Pre by replacing
each occurrence of variable v by expression e"
• Formal deductive system can be used to reason
from one assertion to next
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Proving Algorithms Correct (cont.)
• A Very Simple case
Pre  A  Post
• More complex case
Ai
S
Ai+1
Pre  A1  A2  …  An  Post
• General cases might even contain many
selection paths (e.g., even more complex)
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Logic – validity vs. truth
• Logic, concerned with reasoning and validity of arguments
– in general, in logic, we are not concerned with the truth of
statements, but rather are concerned with their validity.
– That is, although the following arguments (syllogism; 三段
論) is clearly logically, it is not something that we would
considered to be true.
All lemons are blue.
Mary is lemon.
Therefore, Mary is blue.
– This set of statements is considered valid because the
conclusions (Mary is blue) follows logically from the other
two statements, which we often call the premises.
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Logic – validity vs. truth (cont.)
 inference is not directly related to truth
– i.e. we can infer a sentence provided we have rules of
inference that produce the sentence from the original sentences.
For example,
• Black list: blocking all SMTP relays without reverse DNS
( heuristic; false positive).
• While list: accepting all messages with known envelope From
address (heuristic; false negative)
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Deduction vs. Induction
• Deduction has to do with necessity; induction has to do
with probability.
– Logicians contrast deduction with induction, in which the
conclusion might be false even when the premises are true.
• Facts: Lots of spam mails are originated from Internet
sites without reverse DNS mappings.
– Induction rule (valid but not necessarily true):
• E-mail messages relayed through incoming MTAs/MUAs, without
reverse DNS mappings, will be considered as SPAM messages.
• Specific-to-general; maybe false positive
– Deduction rule (general principle, or policy):
• E-mail messages relayed through incoming MTAs (or from NCTU
MUAs), without reverse DNS mappings, should be blocked (i.e.,
considered as SPAM messages. )
• General-to-specific
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