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CSE 20 Lecture 13
Analysis: Counting with
Pigeonhole Principle
CK Cheng, May 17, 2011
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Pigeonhole Principle
5.6, 6.5 Schaum’s
Motivation:
The mapping of n objects to m buckets
E.g. Hashing.
The principle is used for proofs of certain
complexity derivation.
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Pigeonhole Principle
Pigeonhole Principle:
If n pigeonholes are
occupied by n + 1 or more pigeons, then
at least one pigeonhole is occupied by
more than one pigeon.
Remark:
The principle is obvious. No simpler fact or rule to
support or prove it.
Generalized Pigeonhole Principle:
If n pigeonholes are
occupied by kn + 1 pigeons, then at least one pigeonhole
is occupied by k + 1 or more pigeons.
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Example 1: Birthmonth
In a group of 13 people, we have 2 or more who are born
in the same month.
# pigeons
# holes
At least # born in
the same month
13
12
2 or more
20
12
2 or more
121
12
11 or more
65
12
6 or more
111
12
10 or more
≥
k
n
+1
n
k
+1 or more
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Example 2: Handshaking
Given a group of n people (n>1), each shakes
hands with some (a nonzero number of) people in
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This note was uploaded on 12/11/2011 for the course CSE 20 taught by Professor Foster during the Fall '08 term at UCSD.
 Fall '08
 Foster

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