PIGEON HOLE PRINCIPAL
By: Palash Shah
Roll No: 16014123035
INTRODUCTION TO THE
PIGEONHOLE PRINCIPLE
Definition:
The Pigeon hole Principle states that
if n items are put into m containers,
and if n > m, then at least one container
must hold more than one item.
Basic Example:
If you have 10 pairs of socks and only 9
drawers, at least one drawer will contain
more than one pair of socks.
MATHEMATICAL FORMULATION
AND EXAMPLES
Mathematical Representation:
If n items are distributed among m containers, then at least one container has at
least ⌈ n / m⌉ items.
Additional Examples:
• Example 1: In a group of 13 people, at least two people must share a birth month
(12 months).
• Example 2: In a classroom of 30 students, at least two students must have the same
shoe size if there are only 10 different sizes available.
APPLICATIONS OF THE PIGEON
HOLE PRINCIPLE IN ENGINEERING
Computer Networks:
Data Structures:
Cryptography:
IP Addressing Schemes:
Hash Collisions:
1. If the number of devices (n) exceeds
the number of available IP
addresses (m), some devices must
share an IP, causing address
conflicts.
2. Example: In IPv4, with 4.3 billion
possible addresses, the increasing
number of devices leads to overlaps,
pushing the development of IPv6.
3. Impact: Efficient network design
ensures optimal use of IP space, and
the Pigeonhole Principle highlights
why new protocols like IPv6 were
necessary.
1. When more inputs (n) are mapped
to fewer outputs (m), collisions are
inevitable in hash tables.
2. Example: If 1000 inputs are hashed
into 100 buckets, at least one bucket
must contain multiple inputs.
3. Impact: Understanding hash
collisions helps in designing efficient
data structures like hash tables used
in databases and memory
optimization in software
development.
Vulnerabilities in Cryptographic
Systems:
1. The Pigeonhole Principle explains
potential weaknesses when
encrypting large datasets into
smaller encrypted outputs.
2. Example: Birthday Attack exploits
hash collisions to find two distinct
inputs that produce the same hash
output.
3. Impact: Enhances cryptographic
security by forcing developers to
design encryption algorithms that
minimize collision vulnerabilities.
REFERENCES
1. K. H. Rosen, Discrete Mathematics and Its Applications, 7th ed. New York:
McGraw-Hill, 2012.
2. "Discrete Mathematics - The Pigeonhole Principle," GeeksforGeeks. Available:
https://www.geeksforgeeks.org/discrete-mathematics-the-pigeonhole-principle/.
[Accessed: 11-Oct-2024].
3. u/digital_deviant, "Where is the Pigeon Hole Principle applied in computer
science?", Reddit. Available:
https://www.reddit.com/r/compsci/comments/2wlynw/where_is_the_pigeon_hole
_principle_applied_in/. [Accessed: 11-Oct-2024].
4. Ted-Ed, "How the Pigeonhole Principle proves you’re more creative than you
think," YouTube. Available: https://www.youtube.com/watch?v=B2A2pGrDG8I.
[Accessed: 11-Oct-2024]