02-mathematics.pdf

8/13/2019 02-mathematics.pdf

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CS 97SI: INTRODUCTION TO

PROGRAMMING CONTESTS

Jaehyun Park


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Todays Lecture

Algebra

Number Theory

Combinatorics

(non-computational) Geometry

Emphasis on how to compute


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Sum of Powers

=

( 1)(2 1)

1

Pretty useful in many random situations

Memorize above!


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Fast Exponentiation

1, if 0 , if 1

/ , if is even

()/

, if is odd

Can be computed recursively


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Implementation (recursive)

double pow(double a, int n) {

if(n == 0) return 1;

if(n == 1) return a;

double t = pow(a, n/2);return t * t * pow(a, n%2);

}

Running time: log


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Implementation (non-recursive)

double pow(double a, int n) {

double ret = 1;

while(n) {

if(n%2 == 1) ret *= a;a *= a; n /= 2;

}

return ret;

}

You should understand how it works


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Linear Algebra

Gaussian Elimination

Solve a system of linear equations

Invert a matrix

Find the rank of a matrix

Compute the determinant of a matrix

Cramers Rule


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Greatest Common Divisor (GCD)

gcd(,): greatest integer divides both and

Used very frequently in number theoretical

problems

Some facts:

gcd , gcd(, )

gcd , 0

gcd(,)is the smallest positive number in {

| , }


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Euclidean Algorithm

Repeated use of gcd , gcd(, )

gcd 1989, 867 gcd 1989 2 867, 867 gcd 255, 867 gcd 255, 867 3 255 gcd 255, 102 gcd 255 2 102, 102 gcd 51, 102

gcd 51, 102 2 51 gcd 51, 0 51


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Implementation

int gcd(int a, int b) {

while(b){int r = a % b; a = b; b = r;}

return a;

}

Running time: log

Be careful: a % b follows the sign of a 5 % 3 == 2

-5 % 3 == -2


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Congruence & Modulo Operation

(mod )means and have the same

remainder when divided by

Multiplicative inverse

is the inverse of modulo if 1 (mod )

5 3 (mod 7)because 5 3 15 1 (mod 7)

May not exist (e.g. Inverse of 2 mod 4) Exists iff gcd , 1


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Multiplicative Inverse

All intermediate numbers computed by Euclidean

algorithm are integer combinations of and

Therefore, gcd , for some integers ,

If gcd , 1, then 1for some ,

Taking modulo gives 1 (mod )

We will be done if we can find such and


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Extended Euclidean Algorithm

Main idea: keep the original algorithm, but write all

intermediate numbers as integer combinations of

and

Exercise: implementation!


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Chinese Remainder Theorem

Given ,,,such that and are coprime

Find such that mod , mod

Solution:

Let be the inverse of modulo

Let be the inverse of modulo

Set (check this yourself)

Extension: solving for more simultaneous equations


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Binomial Coefficients

is the number of ways to choose objects out

of distinguishable objects

or, the coefficient of in the expansion of

Appears everywhere in combinatorics


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Computing

Solution 1: Compute using the following formula

( 1) ( 1)

!

Solution 2: Use Pascals triangle

Case 1: Both and are small

Use either solution

Case 2: is big, but or is small

Use Solution 1 (carefully)


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Fibonacci sequence

Definition:

0, 1

, where 2

Appears in many different contexts


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Closed Form

, where

+

Bad because and 5are irrational

Cannot compute the exact value of for large

There is a more stable way to compute and any other recurrence of a similar form


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Better Closed Form

+

1 1

1 0

1 1

1 0

Use fast exponentiation to compute the matrix

power

Can be extended to support any linear recurrencewith constant coefficients


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Geometry

In theory: not that hard

In programming contests: avoid if you can

Will cover basic stuff today

Computational geometry in week 9


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When Solving Geometry Problems

Precision, precision, precision!

If possible, dont use floating-point numbers

If you have to, always use double and never use float

Avoid division whenever possible

Introduce small constant in (in)equality tests

e.g. Instead of if(x == 0), write if(abs(x) < EPS)

No hacks! In most cases, randomization, probabilistic methods,

small perturbations wont help


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2D Vector Operations

Have a vector (,)

Norm (distance from the origin):

Counterclockwise rotation by :

Left-multiply bycos sin

sin cos

Make sure to use correct units (degrees, radians)

Normal vectors: (, ), ,

Memorize all of them!


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Line-Line Intersection

Have two lines: 0, 0

Write in matrix form:

Invert the matrix on the left using the following:

Memorize this!

Edge case:


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Circumcircle of a Triangle

Have three points, ,

Want to compute that is equidistance from, ,

Dont try to solve the system of quadratic equations!

Instead, do the following:

Find the bisectors ofand

Compute their intersection


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Area of a Triangle

Have three points, ,

Use cross product: 2

Cross product:

, ,

Very important in computational geometry. Memorize!


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Area of a Simple Polygon

Given , , , around perimeter of polygon

If is convex, we can decompose into triangles:

2 + =

It turns out that the formula above works for non-

convex polygons too

Area is the absolute value of the sum of signed area

Alternative formula:

2 (+ +)=


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Conclusion

No need to look for one-line closed form solutions

Multi-step algorithms are good enough

This is a CS class, after all

Have fun with the exercise problems

and come to the practice contest if you can!

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