2x4a64 pigeonhole principle

7/24/2019 2x4a64 Pigeonhole Principle

1/14

11

The PigeonholeThe Pigeonhole

PrinciplePrincipleCS/APMA 202CS/APMA 202

Rosen section 4.2Rosen section 4.2Aaron BloomfieldAaron Bloomfield


2/14

2

The pigeonhole principleThe pigeonhole principle

Suppose a floc of pigeons fl! into a set ofSuppose a floc of pigeons fl! into a set ofpigeonholes to roostpigeonholes to roost

"f there are more pigeons than pigeonholes# then"f there are more pigeons than pigeonholes# thenthere must $e at least % pigeonhole that hasthere must $e at least % pigeonhole that hasmore than one pigeon in itmore than one pigeon in it

"f"f kk&% or more o$'ects are placed into&% or more o$'ects are placed into kk$o(es#$o(es#then there is at least one $o( containing t)o orthen there is at least one $o( containing t)o ormore of the o$'ectsmore of the o$'ects This is Theorem %This is Theorem %


3/14

*

Pigeonhole principle e(amplesPigeonhole principle e(amples

"n a group of *+, people# there must $e"n a group of *+, people# there must $e

t)o people )ith the same $irthda!t)o people )ith the same $irthda! As there are *++ possi$le $irthda!sAs there are *++ possi$le $irthda!s

"n a group of 2, -nglish )ords# at least"n a group of 2, -nglish )ords# at least

t)o )ords must start )ith the same lettert)o )ords must start )ith the same letter As there are onl! 2+ lettersAs there are onl! 2+ letters


4/14

4

eneralied pigeonhole principleeneralied pigeonhole principle

"f"f NNo$'ects are placed intoo$'ects are placed into kk$o(es# then$o(es# then

there is at least one $o( containingthere is at least one $o( containing NN//kko$'ectso$'ects This is Theorem 2This is Theorem 2


5/14

eneralied pigeonhole principleeneralied pigeonhole principle

e(amplese(amples

Among %00 people# there are at leastAmong %00 people# there are at least

%00/%2%00/%21 $orn on the same month1 $orn on the same month

3o) man! students in a class must there3o) man! students in a class must there

$e to ensure that + students get the same$e to ensure that + students get the same

grade one of A# B# C# 5# or 678grade one of A# B# C# 5# or 678 The 9$o(es: are the grades. Thus#The 9$o(es: are the grades. Thus# kk1 1

Thus# )e setThus# )e set NN//1 +1 + ;o)est possi$le


6/14

+

Rosen# section 4.2# =uestion 4Rosen# section 4.2# =uestion 4

A $o)l contains %0 red and %0 !ello) $allsA $o)l contains %0 red and %0 !ello) $alls

a7a7 3o) man! $alls must $e selected to ensure * $alls of3o) man! $alls must $e selected to ensure * $alls ofthe same color8the same color8

>ne solution? consider the 9)orst: case>ne solution? consider the 9)orst: case

Consider 2 $alls of each colorConsider 2 $alls of each color@ou cant tae another $all )ithout hitting *@ou cant tae another $all )ithout hitting *

Thus# the ans)er is Thus# the ans)er is

ia generalied pigeonhole principleia generalied pigeonhole principle

3o) man! $alls are re=uired if there are 2 colors# and one color3o) man! $alls are re=uired if there are 2 colors# and one color

must ha


7/14

,

Rosen# section 4.2# =uestion 4Rosen# section 4.2# =uestion 4

A $o)l contains %0 red and %0 !ello)A $o)l contains %0 red and %0 !ello)

$alls$alls

$7$7 3o) man! $alls must $e selected to3o) man! $alls must $e selected toensure * !ello) $alls8ensure * !ello) $alls8 Consider the 9)orst: caseConsider the 9)orst: case

Consider %0 red $alls and 2 !ello) $allsConsider %0 red $alls and 2 !ello) $alls

@ou cant tae another $all )ithout hitting *@ou cant tae another $all )ithout hitting *

!ello) $alls!ello) $alls

Thus# the ans)er is %*Thus# the ans)er is %*


8/14

D

Rosen# section 4.2# =uestion *2Rosen# section 4.2# =uestion *2

+ computers on a net)or are connected to at least %+ computers on a net)or are connected to at least %other computerother computer

Sho) there are at least t)o computers that are ha


9/14

Rosen# section 4.2# =uestion %0Rosen# section 4.2# =uestion %0

Consider distinct points Consider distinct points xxii## yyii7 )ith integer


10/14

%0

More elegant applicationsMore elegant applications

Eot going o


11/14

11

Quick surveyQuick survey

I felt I understood the material in thisI felt I understood the material in this

slide setslide set

a)a)

Very wellVery well

b)b) With some review, Ill be goodWith some review, Ill be good

c)c) Not reallyNot really

d)d) Not at allNot at all


12/14

12


The pace of the lecture for thisThe pace of the lecture for this

slide set wasslide set was

a)a)

astast

b)b) !bout right!bout right

c)c) ! little slow! little slow

d)d) Too slowToo slow


13/14

13


"ow interesting was the material in"ow interesting was the material in

this slide set# $e honest%this slide set# $e honest%

a)a)

Wow% That was &'''''' cool%Wow% That was &'''''' cool%

b)b) &omewhat interesting&omewhat interesting

c)c) (ather borting(ather borting

d)d) ********************


14/14

1414

Todays demotivatorsTodays demotivators

2x4a64 pigeonhole principle

Documents