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Introduction. Thinking about Algorithms Abstractly. Credits: Steve Rudich, Jeff Edmond, Ping Xuan. So you want to be a bioinformatician /mathematician /computer scientist? What is an Algorithm ? Grade school revisited: How to multiply two numbers. So you want to be a computer scientist?.

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Introduction

Introduction

Thinking about Algorithms Abstractly

Credits: Steve Rudich, Jeff Edmond, Ping Xuan




Is your goal to be a mundane programmer
Is your goal to be /computer scientist?a mundane programmer?


Or a great leader and thinker? /computer scientist?


Original thinking

Original Thinking /computer scientist?


Boss assigns task
Boss assigns task: /computer scientist?

  • Given today’s prices of pork, grain, sawdust, …

  • Given constraints on what constitutes a hotdog.

  • Make the cheapest hotdog.

Everyday industry asks these questions.


Your answer
Your answer: /computer scientist?

  • Um? Tell me what to code.

With more suffocated software engineering systems,the demand for mundane programmers will diminish.


Your answer1
Your answer: /computer scientist?

  • I learned this great algorithm that will work.

Soon all known algorithms will be available in libraries.


Your answer2
Your answer: /computer scientist?

  • I can develop a new algorithm for you.

Great thinkers will always be needed.


The future belongs to the computer scientist who has
The future belongs to the computer scientist who has /computer scientist?

  • Content: An up to date grasp of fundamental problems and solutions

  • Method: Principles and techniques to solve the vast array of unfamiliar problems that arise in a rapidly changing field


Course content
Course Content /computer scientist?

  • A list of algoirthms.

    • Learn their code.

    • Trace them until you are convenced that they work.

    • Impliment them.

class InsertionSortAlgorithm extends SortAlgorithm {

void sort(int a[]) throws Exception {

for (int i = 1; i < a.length; i++) {

int j = i;

int B = a[i];

while ((j > 0) && (a[j-1] > B)) {

a[j] = a[j-1];

j--;}

a[j] = B;

}}


Course content1
Course Content /computer scientist?

  • A survey of algorithmic design techniques.

  • Abstract thinking.

  • How to develop new algorithms for any problem that may arise.


Study
Study: /computer scientist?

  • Many experienced programmers were asked to code up binary search.


Study1
Study: /computer scientist?

  • Many experienced programmers were asked to code up binary search.

80% got it wrong

Good thing is was not for a nuclear power plant.


What did they lack
What did they lack? /computer scientist?


What did they lack1
What did they lack? /computer scientist?

  • Formal proof methods?


What did they lack2
What did they lack? /computer scientist?

  • Formal proof methods?

Yes, likely

Industry is starting to realize that formal methods are important.

But even without formal methods …. ?


What did they lack3
What did they lack? /computer scientist?

  • Fundamental understanding of the algorithmic design techniques.

  • Abstract thinking.


Course content2
Course Content /computer scientist?

Notations, analogies, and abstractions

for developing,

thinking about,

and describing algorithms


You will see some math

Time Complexity /computer scientist?

t(n) = Q(n2)

Recurrence Relations

T(n) = a T(n/b) + f(n)

You will see some Math …


What is an algorithm
What is an Algorithm? /computer scientist?

  • A step-by-step description of procedures performing certain task.

    • Example: Sorting.

      Given a list of numbers, put them in increasing

      (non-decreasing) order.

  • Properties: Generality, Termination, Correctness, Feasibility.

  • Feasibility  Analysis of Algorithm

    Complexity theory


Analysis of algorithm
Analysis of Algorithm /computer scientist?

  • Running time analysis

    • Input size N

    • Running time is a function of N: T(N)

      • Worst case, average case, … , etc

      • Time measured by number of computer operations

        • Each basic operation (add, load, write, etc) count as 1

      • Actual clock time differs by a constant factor

      • Usually very complex

    • The use of asymptotic bounds

      • To study the rate of growth of T(N) compared to simpler functions f(N), such as N, N2, log(N), etc

      • A constant factor can be ignored


Some definitions big o notation
Some Definitions: Big O Notation /computer scientist?

  • T(N) = O( f(N) )

    • Exists constant c and n0 such that when N>n0, T(N) <= c * f(N)

    • Asymptotic Upper bound

c f(N)

T(N)

n0

N


Some definitions big omega
Some Definitions: Big Omega /computer scientist?

  • T(N) = (g(N))

    • Exists constant c and n0 such that when N>n0, T(N) >= c * g(N)

    • Asymptotic Lower bound

T(N)

c g(N)

n0

N


Some definitions big theta
Some Definitions: Big Theta /computer scientist?

  • T(N) = ( h(N) )

    • if and only if T(N)=O(h(N)) and T(N)= (h(N))

    • tight bound

c1 h(N)

T(N)

c2 h(N)

N


Some definitions little o

c p(N) /computer scientist?

T(N)

N

Some Definitions: Little “o”

  • T(N) = o(p(N))

    if lim N  ∞ = o. E.g. log(N) = o(N).


Example insertion sort algorithm
Example: Insertion Sort Algorithm /computer scientist?

  • class InsertionSortAlgorithm extends

    SortAlgorithm {

  • void sort(int a[ ]) throws Exception {

  • for (int i = 1; i < a.length; i++) {

  • int j = i;

  • int B = a[i];

  • while ((j > 0) && (a[j-1] > B)) {

  • a[j] = a[j-1];

  • j--;}

  • a[j] = B;

  • }}


Iterative algorithms

0 /computer scientist?

i-1

i

T+1

i

i

9 km

5 km

Iterative Algorithms

<preCond>

codeA

loop

exit when <exit Cond>

codeB

codeC

<postCond>

One step at a time

Relay Race

Code


52 /computer scientist?

88

14

14,23,25,30,31,52,62,79,88,98

31

98

25

30

23

62

79

Problem Specification

  • Precondition: The input is a list of n values with the same value possibly repeated.

  • Post condition: The output is a list consisting of the same n values in non-decreasing order.


Define step

30 /computer scientist?

14

25

23,31,52,88

98

79

Define Step

  • Select arbitrary element from side.

  • Insert it where it belongs.

30

14

62

25

98

79

23,31,52,62,88


Making progress while maintaining the loop invariant

Exit /computer scientist?

79 km

75 km

6 elements

to school

5 elements

to school

30

14

25

23,31,52,88

98

79

Making progress while Maintaining the loop invariant

30

14

62

25

98

79

23,31,52,62,88

Sorted sub-list


52 /computer scientist?

52

88

88

14

14

31

31

62

62

25

25

30

30

Exit

Exit

0 km

23

23

98

98

79

79

n elements

to school

0 elements

to school

14,23,25,30,31,52,62,79,88,98

14,23,25,30,31,52,62,79,88,98

Beginning &

Ending


Running time

30 /computer scientist?

14

25

23,31,52,88

98

79

Running Time

Inserting an element into a list of size i

takes O(i) time in the worst case.

Total = 1+2+3+…+n = n(n+1)/2 = (n2)

30

14

62

25

98

79

23,31,52,62,88


Explaining insertion sort
Explaining Insertion Sort /computer scientist?

We maintain a subset of elements sorted within alist. The remaining elements are off to the sidesomewhere. Initially,think of the first element in the array as a sorted list of lengthone. One at a time, we take one of the elements that is off to theside and we insert it into the sorted list where itbelongs. This gives a sorted list that is one element longer than itwas before. When the last element has been inserted, the array iscompletely sorted.


Insertion sort

23,25,31,52,88 /computer scientist?

52,23,88,31,25,30,98,62,14,79

23,31,52,88

Insertion Sort

The input consists of an array of integers

We read in the i+1st object.

We will pretend that this larger prefix is the entire input.

We extend the solution we have by one for this larger prefix.


Insertion sort algorithm pseudo code
Insertion Sort Algorithm: /computer scientist?Pseudo-Code

  • Input: an array (or list) a of numbers

    // data structure: a[0] …a[n-1]

  • Operations:

    • Leave the first element (location 0) untouched.

    • for each element from location 1 on

      insert it to the appropriate place in the

      (sorted) sub-array to its left.

a

0

1

2

n-1


Operations in an algorithm
Operations in an Algorithm /computer scientist?

  • Decision Making

    • branching operation

    • if … then … else …

  • Repetition

    • loops: for each element do { … }

    • conditional loops: while ( …. ) do { … }


Insertion sort algorithm code
Insertion Sort Algorithm: Code /computer scientist?

  • class InsertionSortAlgorithm extends

    SortAlgorithm {

  • void sort(int a[ ]) throws Exception {

  • for (int i = 1; i < a.length; i++) {

  • int j = i;

  • int b = a[i];

  • while ( (j > 0) && (a[j-1] > b) ) {

  • a[j] = a[j-1];

  • j--;}

  • a[j] = B;

  • }}}


Grade school revisited how to multiply two numbers

2 X 2 = 5 /computer scientist?

Another Algorithm

Grade School Revisited:How To Multiply Two Numbers


Complex numbers
Complex Numbers /computer scientist?

  • Remember how to multiply 2 complex numbers?

  • (a+bi)(c+di) = [ac –bd] + [ad + bc] i

  • Input: a,b,c,d Output: ac-bd, ad+bc

  • If a real multiplication costs $1 and an addition cost a penny. What is the cheapest way to obtain the output from the input?

  • Can you do better than $4.02?


Gauss 3 05 method input a b c d output ac bd ad bc
Gauss’ $3.05 Method: /computer scientist?Input: a,b,c,d Output: ac-bd, ad+bc

  • m1 = ac

  • m2 = bd

  • A1 = m1 – m2 = ac-bd

  • m3 = (a+b)(c+d) = ac + ad + bc + bd

  • A2 = m3 – m1 – m2 = ad+bc


Question
Question: /computer scientist?

  • The Gauss “hack” saves one multiplication out of four. It requires 25% less work.

  • Could there be a context where performing 3 multiplications for every 4 provides a more dramatic savings?


Odette /computer scientist?

Bonzo


How to add 2 n bit numbers
How to add 2 n-bit numbers. /computer scientist?

**

**

**

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+


How to add 2 n bit numbers1
How to add 2 n-bit numbers. /computer scientist?

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+


How to add 2 n bit numbers2
How to add 2 n-bit numbers. /computer scientist?

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*

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+


How to add 2 n bit numbers3
How to add 2 n-bit numbers. /computer scientist?

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* **

* **

*

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*

**

*

+


How to add 2 n bit numbers4
How to add 2 n-bit numbers. /computer scientist?

**

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**

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* **

* **

*

* **

*

* **

*

**

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+


How to add 2 n bit numbers5
How to add 2 n-bit numbers. /computer scientist?

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*

* **

*

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*

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***

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Time complexity of grade school addition

* /computer scientist?

*

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Time complexity of grade school addition

On any reasonable computer adding 3 bits can be done in constant time.

T(n) = The amount of time grade school addition uses to add two n-bit numbers

= θ(n) = linear time.


f = /computer scientist?θ(n) means that f can be sandwiched between two lines

time

# of bits in numbers


Please feel free /computer scientist?

to ask questions!


Is there a faster way to add
Is there a faster way to add? /computer scientist?

  • QUESTION: Is there an algorithm to add two n-bit numbers whose time grows sub-linearly in n?


Any algorithm for addition must read all of the input bits
Any algorithm for addition must read all of the input bits /computer scientist?

  • Suppose there is a mystery algorithm that does not examine each bit

  • Give the algorithm a pair of numbers. There must be some unexamined bit position i in one of the numbers

  • If the algorithm is not correct on the numbers, we found a bug

  • If the algorithm is correct, flip the bit at position i and give the algorithm the new pair of numbers. It give the same answer as before so it must be wrong since the sum has changed


So /computer scientist?any algorithm for addition must use time at least linear in the size of the numbers.

Grade school addition is essentially as good as it can be.


How to multiply 2 n bit numbers

n /computer scientist?2

How to multiply 2 n-bit numbers.

* * * * * * * *

X

* * * * * * * *

* * * * * * * *

* * * * * * * *

* * * * * * * *

* * * * * * * *

* * * * * * * *

* * * * * * * *

* * * * * * * *

* * * * * * * *

* * * * * * * * * * * * * * * *



Grade school addition linear time grade school multiplication quadratic time
Grade School Addition: Linear time /computer scientist?Grade School Multiplication: Quadratic time

  • No matter how dramatic the difference in the constants the quadratic curve will eventually dominate the linear curve

time

# of bits in numbers


Neat! We have demonstrated that multiplication is a harder problem than addition.Mathematical confirmation of our common sense.


Don’t jump to conclusions! problem than addition.We have argued that grade school multiplication uses more time than grade school addition. This is a comparison of the complexity of two algorithms.

To argue that multiplication is an inherently harder problem than addition we would have to show that no possible multiplication algorithm runs in linear time.


Grade school addition n time grade school multiplication n 2 time
Grade School Addition: problem than addition.θ(n) timeGrade School Multiplication: θ(n2) time

Is there a clever algorithm to multiply two numbers in linear time?


Despite years of research, no one knows! If you resolve this question,

Tunghai will give you a PhD!



Divide and conquer an approach to faster algorithms
Divide and Conquer you learned in grade school?(an approach to faster algorithms)

  • DIVIDE my instance to the problem into smaller instances to the same problem.

  • Have a friend (recursively) solve them.Do not worry about it yourself.

  • GLUEthe answers together so as to obtain the answer to your larger instance.


Multiplication of 2 n bit numbers
Multiplication of 2 n-bit numbers you learned in grade school?

a

b

  • X =

  • Y =

  • X = a 2n/2 + b Y = c 2n/2 + d

  • XY = ac 2n + (ad+bc) 2n/2 + bd

c

d


Multiplication of 2 n bit numbers1
Multiplication of 2 n-bit numbers you learned in grade school?

a

b

c

d

  • X =

  • Y =

  • XY = ac 2n + (ad+bc) 2n/2 + bd

MULT( X, Y ) :

If |X| = |Y| = 1 then RETURN XY

Break X into a;b and Y into c;d

RETURN

MULT(a,c) 2n + (MULT(a,d) + MULT(b,c)) 2n/2 + MULT(b,d)


Time required by mult
Time required by MULT you learned in grade school?

  • T( n ) = time taken by MULT on two n-bit numbers

  • What is T(n)? What is its growth rate? Is it θ(n2)?


Recurrence relation
Recurrence Relation you learned in grade school?

  • T(1) = k for some constant k

  • T(n) = 4 T(n/2) + k n + l for some constants k and l

MULT(X,Y):

If |X| = |Y| = 1 then RETURN XY

Break X into a;b and Y into c;d

RETURN

MULT(a,c) 2n + (MULT(a,d) + MULT(b,c)) 2n/2 + MULT(b,d)


Let s be concrete
Let’s be concrete you learned in grade school?

  • T(1) = 1

  • T(n) = 4 T(n/2) + n

  • How do we unravel T(n) so that we can determine its growth rate?


Technique 1 guess and verify
Technique 1 you learned in grade school?Guess and Verify

  • Recurrence Relation:

    T(1) = 1 & T(n) = 4T(n/2) + n

  • Guess: G(n) = 2n2 – n

  • Verify:


Technique 2 decorate the tree

Technique 2: Decorate The Tree

T(1)

=

1

  • T(n) = n + 4 T(n/2)

  • T(n) = n + 4 T(n/2)

T(n)

T(n)

n

n

=

=

T(n/2)

T(n/2)

T(n/2)

T(n/2)

T(n/2)

T(n/2)

T(n/2)

T(n/2)


T(n) you learned in grade school?

n

=

T(n/2)

T(n/2)

T(n/2)

T(n/2)


n/2 you learned in grade school?

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n)

n

=

T(n/2)

T(n/2)

T(n/2)


n/2 you learned in grade school?

n/2

n/2

n/2

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n)

n

=


T(n) you learned in grade school?

n

=

n/2

n/2

n/2

n/2

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

n/4

11111111111111111111111111111111 . . . . . . 111111111111111111111111111111111


n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4

0

1

2

i


=1 n/4 + n/4 + n/4 + n/4 + n/4 + n/4×n

= 4×n/2

= 16×n/4

= 4i ×n/2i

= 4logn×n/2logn =nlog4×1

Total: θ(nlog4) = θ(n2)


Divide and conquer mult n 2 time grade school multiplication n 2 time
Divide and Conquer MULT: n/4 + n/4 + n/4 + n/4 + n/4 + n/4θ(n2) time Grade School Multiplication: θ(n2) time

All that work for nothing!


Mult revisited
MULT revisited n/4 + n/4 + n/4 + n/4 + n/4 + n/4

MULT(X,Y):

If |X| = |Y| = 1 then RETURN XY

Break X into a;b and Y into c;d

RETURN

MULT(a,c) 2n + (MULT(a,d) + MULT(b,c)) 2n/2 + MULT(b,d)

  • MULT calls itself 4 times. Can you see a way to reduce the number of calls?


Gauss hack input a b c d output ac ad bc bd
Gauss’ Hack: n/4 + n/4 + n/4 + n/4 + n/4 + n/4Input: a,b,c,d Output: ac, ad+bc, bd

  • A1 = ac

  • A3 = bd

  • m3 = (a+b)(c+d) = ac + ad + bc + bd

  • A2 = m3– A1- A3 = ad + bc


Gaussified mult karatsuba 1962
Gaussified MULT n/4 + n/4 + n/4 + n/4 + n/4 + n/4(Karatsuba 1962)

MULT(X,Y):

If |X| = |Y| = 1 then RETURN XY

Break X into a;b and Y into c;d

e = MULT(a,c) and f =MULT(b,d)

RETURN e2n + (MULT(a+b, c+d) – e - f) 2n/2 + f

  • T(n) = 3 T(n/2) + n

  • Actually: T(n) = 2 T(n/2) + T(n/2 + 1) + kn


n/2 n/4 + n/4 + n/4 + n/4 + n/4 + n/4

n/2

n/2

n/2

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n)

n

=


T(n) n/4 + n/4 + n/4 + n/4 + n/4 + n/4

n

=

T(n/2)

T(n/2)

T(n/2)


n/2 n/4 + n/4 + n/4 + n/4 + n/4 + n/4

T(n/4)

T(n/4)

T(n/4)

T(n)

n

=

T(n/2)

T(n/2)


n/2 n/4 + n/4 + n/4 + n/4 + n/4 + n/4

n/2

n/2

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n/4)

T(n)

n

=


0 n/4 + n/4 + n/4 + n/4 + n/4 + n/4

1

2

n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4 + n/4

i


=1 n/4 + n/4 + n/4 + n/4 + n/4 + n/4×n

= 3×n/2

= 9×n/4

= 3i ×n/2i

= 3logn×n/2logn =nlog3×1

Total: θ(nlog3) = θ(n1.58..)


Dramatic improvement for large n
Dramatic improvement for large n n/4 + n/4 + n/4 + n/4 + n/4 + n/4

Not just a 25% savings!

θ(n2) vs θ(n1.58..)


Multiplication algorithms

Homework n/4 + n/4 + n/4 + n/4 + n/4 + n/4

Multiplication Algorithms

3*4=3+3+3+3


You’re cool! Are you free sometime this weekend? n/4 + n/4 + n/4 + n/4 + n/4 + n/4

Not interested, Bonzo. I took the initiative and asked out a guy in my 4277 class.


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