Syllabus1. Go to2. www.cs.siena.edu/~ebreimer3. Click Analysis of Algorithms link4. bookmark course website Important stuff

– Schedule all due dates– Quizzes details, solutions– Homework details, solutions

Syllabus

Highlights Quiz every Friday HW due every Monday No exams except a final

– Combination of quizzes and HWs No huge projects Some HW will be collaborative

Syllabus Late HWs get a zero

– Due at the beginning of class– Can drop 2 HWs

No quiz makeups– Miss a quiz, get a zero– Can drop 2 quizzes

Attendance 10%– 2 unexcused absences– -2% for each additional unexcused absence.

SyllabusGrading

40% Quizzes

30% HWs

20% Final

10% Attendance

A 93 or higher

A- 90 or higher

B+ 87

B 83

B- 80

C+ 77

C 73

C- 70

Chapter 1 – Intro What is an algorithm?

– 7 properties Two Examples

– Max of 3– Closest Pair

Pseudo-code Present Future

Algorithms? Definition: step-by-step method for solving a

problem We will concentrate on algorithms that can

be executed on modern computers– like the one you probably have

This field existed before computer existed– 900 AD in Persia

7 Properties

1. Input

2. Output

3. Precision

4. Determinism

5. Finiteness

6. Correctness

7. Generality

Examples

Problem: Given three integer values A, B, and C return the maximum value.

Input: 3 integers

Output: 1 integer

Examples

Problem: Given a list of N points, find the closest pair of points (2D)

Input: N points, where a point is a pair of real values (x,y).

Output: 2 points

3. Precisionmax(a,b,c) {

compare a, b, and c and return the largest value;

}

Not precise!

3. Precisionmax(a,b,c) {

if (a > b && a > c)return a;

else if (b > c)return b;

elsereturn c;

}

Very precise

3. Precision A precise algorithm is one that can be

described by pseudo code or a high-level programming language like C++ or Java.

Pseudo code is like a high-level programming language where the syntax doesn’t have to be perfect.

4. Determinism Not random

– Given the same conditions, you expect the same outcome.

Computer are inherently deterministic. When computers behave randomly it is

usually a result of– human errors– environmental conditions– pseudo-random number generation

5. Finiteness Given finite input, an algorithm should

not run infinitely (i.e., forever). An algorithm that runs forever is not

really an algorithm because it will never solve the problem.

Recall that an algorithm is a step-by-step method for solving a problem.

What if the input is infinite?

6. Correctnessmax(a,b,c) {

if (a > b)

return a;

else if (b > c)

return b;

else

return c;

}

What if a = 5b = 4c = 6

6. Correctness

Algorithms that return incorrect answers aren’t really algorithms

Recall that an algorithm is a step-by-step method for solving a problem.

7. Generalitymax(a, b, c) {

if (a > 10 && b < 10 && c < 10)return a;

if (b > 10 && a < 10 && c < 10)return b;

if (c > 10 && b < 10 && a < 10)return c;

}Will never return an incorrect answer, but does not

apply to a general set of input (like all integers or all real numbers).

Closest Pair Design an algorithm to find the closest

pair of points

Closest Pair Designing algorithms is more of an art

than a science.

Closest Pair Visually this problem is very simple to

solve, but what if I gave you 1 million points?

Closest Pair What if I asked you to solve this

problem for 1 million different cases?

Closest Pair What is the input? What is the output? What are the precise step-by-step

instructions? Will the algorithm terminate? Will it produce correct answers all the time? How much time will it take to solve? How much memory is required to solve the

problem?

Closest PairThe input: 2D point, a pair values (x, y)

– Point p;– p.x;– p.y;

An array of n Points– Point P[n]– P[0], P[1], P[2], P[3], …, P[n-1]

Closest PairThe output: Two points P[a] and P[b] where the

distance between P[a] and P[b] is minimum among all possible pairs of points.

Sometimes describing the output with precision gives you clues about how to solve the problem.

Closest Pair

1. How do you compute the distance between to points?

2. How do you find a distance is minimum among all possible pairs of points?

Closest Pair

dist(a,b){

d = sqrt[

pow((a.x – b.x),2) +

pow((a.y – b.y),2) ];

return d;

}

22 )()( baba yyxxd

Closest Paird = dist(P[0], P[1]);

Compare all possible pairs of points, i.e.,compare P[0] with P[1], P[2], P[3], …, P[n-1]compare P[1] with P[2], P[3], P[4], …, P[n-1]compare P[2] with P[3], P[4], P[5], …, P[n-1]compare P[3] with P[4], P[5], P[6], …, P[n-1]…compare P[n-3] with P[n-2], P[n-1]compare P[n-2] with P[n-1]

Closest PairClosestPair(P[ ], n) {

min_dist = dist(P[0],P[1]);for (x = 1 to n-1) {

d = dist(P[0], P[x]);if (d < min_dist) {

min_dist = d;}

}return min_dist;

}This is not correct!

Closest PairClosestPair(P[ ], n) {

min_dist = dist(P[0],P[1]);for (y = 0 to n-1) {

for (x = 1 to n-1) {d = dist(P[y], P[x]);if (d < min_dist) {

min_dist = d;}

}}return min_dist;

}

Closest Pair Compare P[a] and P[b]

a

b

0 1 2 3 ... n-1

0

1

2

3

...

n-1

Closest Pair Compare P[a] and P[b]

a

b

0 1 2 3 ... n-1

0 X1 X2 X3 X

... Xn-1 X

Closest Pair Compare P[a] and P[b]

a

b

0 1 2 3 ... n-1

0 X1 X2 X3 X

... Xn-1 X

Closest PairClosestPair(P[ ], n) {

min_dist = dist(P[0],P[1]);for (y = 0 to n-1) {

for (x = y+1 to n-1) {d = dist(P[y], P[x]);if (d < min_dist) {

min_dist = d;}

}}return min_dist;

}

Closest Pair This matrix is n x n n2 entries. How many of these n2

entries must we compute?

a

b

0 1 2 3 ... n-1

0

1

2

3

...

n-1

Summations n-1 +

n-2 +n-3 +...+3 +2 +1

The Present Remember the 7 properties? Many algorithms used in practice

aren’t – General– Deterministic, or– Finite

The Present There are problems that are just too

difficult to solve. Compromises must be made.

1. Some algorithms only work on certain “types” of input.

2. Some algorithms require pseudo-random processing

3. Some algorithms will run forever on certain “types” or problems.

The Present Today, an algorithm doesn’t have to satisfy

all the properties if the problem is very difficult and there is no other know way to get good answers.

Algorithms can be approximate, fuzzy, and even un-predicable.

Some Algorithm purist disagree with this. But, I think a half-baked algorithm is better

than no algorithm at all.

The Future Quantum computing.

– Uses a quantum bit which can store an manipulate information in a massivle parallel way.

– Theoretical only– Might be impossible to implement

DNA computing– DNA bases (A,C,G,T) will be used to represent

data– Biological processes can manipulate large

amounts of data in one step.