Discrete Structures

Li Tak Sing ( 李德成 ) mt263f-9.ppt

Pigeonhole principle

 If m pigeons fly into n pigeonholes where m>n, then one pigeonhole will have two or more pigeons.

Examples in pigeonhole principle

 How many people are needed in a group to say that three were born on the same day of the week?

 How many people are needed in a group to say that four were born in the same month?

 Why does any set of 10 nonempty strings over {a,b,c} contain two different strings whose starting agree and whose ending letters agree?

 Find the size needed for a set o nonempty strings over {a,b,c,d} to contain two strings whose starting letters agree and whose ending letters agree.

The Mod function and inverses

 Let n>1 and let f:N n  N n be defined as follows, where a and b are integers.

f(x)=(ax+b)mod n.

Then f is a bijection iff gcd(a,n)=1. When this is the case, the inverse function f -1 is defined by f -1 (x)=(kx+c) mod n where c is an integer such that f(c)=0, and k is an integer such that 1=ak+nm for some integer m.

1=ax+by

 Given that gcd(a,b)=1, then there are x,y such that 1=ax+by  For example, gcd(10,7)=1, first we have 10=1*7+3 then we have 7=3*2+1 therefore we have 2*10=2*7+(7-1) 2*10=3*7-1 1=3*7-2*10

Example

 Find a,b so that 7a+5b=1  Find a,b so that 7a+11b=1

Simple ciphers

 N 26 ={0,1,..25} represents letters {a,...,z}  We can use a function to transform a letter to another. For example f: N 26  N 26 , f(x)=(x+5) mod 26 The inverse of f, denoted as f -1 , is: f -1 (x)=(x-5) mod 26  f is known as a cipher which would transform a letter into another letter to hide secret.

Multiplicative cipher

 The previous cipher is called an additive cipher because addition is use.

 A multiplicative cipher is one that multiplication is used.

 An example is: g(x)=3x mod 26 the inverse is g -1 (x)=9x mod 26

Example

 Determine where the following functions define a cipher. If a function is a cipher, then find the inverse function  f(x)=2x mod 6  f(x)=2x mod 5  f(x)=5x mod 6  f(x)=(3x+2) mod 6  f(x)=(2x+3) mod 7

Hash function

 A hash function is a function that maps a set S of keys to a finite set of table indexes, which we'll assume are 0,1,...,n 1. A table whose information is found by a hash function is called a hash table.

Hash example

 For example, let S be the set of three letter abbreviations for the months of the year. We might define a hash function f:S  {0,1,..,11} in the following way.

f(XYZ)=(ord(X)+ord(Y)+ord(Z))mod 12.

where ord(X) represents the ASCII code for X.

One use of hash functions

 Assume that you have a table of m entry and you want to store some items in the table. Each item has a key which can be used to uniquely identify it. Now, we can use a hash function so that it will generate an index for the item to be stored in the corresponding entry.  This method has an advantage that when we know of the key, we can find the position where the item is stored.

Collisions

 If the numbers of objects to be stored is greater than the size of the table, then collision must occur.  One technique to solve the problem of collision is called linear probing. With this technique the program searches the remaining locations in a linear manner.

Hash examples

 Let S={Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} and suppose that f:S  N 7 is the hash function defined by f(x)=lenght(x) mod 7 where length(x) is the number of letters in x. Use h to place each day of S into a hash table indexed by {0,1,..,6} starting with Monday, then Tuesday, and so on until Sunday. Resolve collisions by linear probing with a gap of 2.