In this presentation, Denise weaves together technical details from graph theory, computational complexity, and cryptography to ultimately discuss if the internet is secure. The discussion centers around whether or not N=NP. The video of the event will be available on YouTube here: http://youtu.be/jCvROOijhoY

1. Confidential 1
Can you connect the
dots, like the the graph
below, without lifting up
your pencil?
Can you connect each
dot on the top to each
dot on the bottom, like
above, without crossing
lines?
There are pens at the sign-in table if you want to try these out

2. Silicon Valley | Silicon Harbor
Can You Trust The Internet?
An intro to graph invariants, computational
complexity, and graph algorithms
(with a little bit of cryptography)
Denise K. Gosnell, Ph.D.
Data Scientist, PokitDok

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Graph Theory?
G = (V,E)

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The Plan
1. Graph Properties and Reddit
2. Computational Complexity
P: Bipartite Graph Matching
NP: Graph Coloring
3. A little bit of RSA

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Why should you care?
1. Everything is graph theoretic in nature.
2. Much easier to code.
1. or… you just want to be here to look cool.

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The Seven Bridges of Königsberg
"Konigsberg bridges" by Bogdan Giuşcă - Public domain (PD)
http://commons.wikimedia.org/wiki/File:Konigsberg_bridges.png#/media/File:Konigsberg_bridges.png

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The Seven Bridges of Königsberg:
Solved by Euler in 1735
Thank you Wikipedia: http://en.wikipedia.org/wiki/Seven_Bridges_of_Konigsberg

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Eulerian Paths
vs.
3
3
5 3
2
4 4
3 3

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screen shot from vizit: http://redditstuff.github.io/sna/vizit/

10. Silicon Valley | Silicon Harbor
P

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P
The set of all decision problems
that can be solved by a deterministic
Turing machine using a polynomial
amount of computational time

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Classes of Algorithm Complexity
O(1)
• Accessing an element of an array
• Determining if a number is even or odd
O(log(n))
• Binary search
[eg: intelligently searching through a dictionary]
O(n)
• Linear search: finding a min or max in an unsorted list
• Graph Search
O(n2)
• looking for a word in a word search
• Bad sorting algorithms like bubble sort, insertion sort
…

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Graph Search: O(|V| + |E|)
Depth vs. Breadth

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Graph Matching
Given a bipartite graph G = (V,E),
a matching M in G is a set
of edges in which no two
edges share a common vertex

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The User Preferences Problem
Candidates à Jobs

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The User Preferences Problem
Goal: assign candidates to jobs to fill as many jobs as possible

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The User Preferences Problem
Greedy Algorithm:
Keep adding edges until no more edges
can be added

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The User Preferences Problem
Greedy Approach

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The User Preferences Problem
A better solution: via augmenting paths

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Augmenting Paths
• A path P is M-alternating if the edges of the path alternate
between being in the matching M and not in the matching M.
• A path P is M-augmenting if it is m-alternating
and the first and last edges are not in the matching M.

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Augmenting Paths
You can improve the matching M by:
1. Remove from the matching M the edges of the path P
that are in the matching M
2. Add to the matching M the edges of the path P that are
not in the matching M
3. This will have one more edge than in the matching

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The User Preferences Problem
Augmenting Path Example:

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The User Preferences Problem
Augmenting Path Example:

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Berge’s Theorem:
A matching in a graph is maximum
if and only if there does not exist an
augmenting path in a graph

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Maximal Matching by
Constructing the Auxiliary Graph

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Maximal Matching by
Constructing the Auxiliary Graph:
Theorem:
G has an augmenting path if and only if it
has a directed path from the source to
the sink in the auxiliary graph

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Maximal Matching by
Constructing the Auxiliary Graph:

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Maximal Matching Final Solution:
EdmondsAlgorithm(G):
M = empty matching!
while there is an augmenting path P for M!
!M = M +- P!
output M!
!
AugmentingPath(G,M):
G’ = Auxiliary graph for G, M!
P = Path from source to sink (via BFS)!
if P is null:!
!return false!
else:!
!delete s and t from P and return P!

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Graph Algorithms in P
• Maximum (minimum) degree
• Finding connected components
[BFS, DFS]
• Pairwise shortest path algorithms
[Dijkstra’s, Bellman-Ford, Floyd-Warshall]
• Diameter
• Girth [shortest cycle]
• Edge Covering Number
• …

30. Silicon Valley | Silicon Harbor
NP

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NP
Nondeterministic Polynomial: the set
of all decision problems where a
“yes” instance can be verified by
a non-deterministic Turing machine
in polynomial time

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P à solvable in polynomial time
NP à verifiable in polynomial time

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Classic NP Problems:
Integer (prime?) Factorization
Graph Coloring (this isn’t what you think)
The Knapsack Problem
The Traveling Salesman

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Integer (prime?) Factorization

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Integer (prime?) Factorization
Factoring a 232 digit number
took over two years and utilized
hundreds of machines.
Paper: “Factorization of a 768-
bit RSA modulus”.
Kleinjung, et al. 2010.

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Graph Coloring
Minimum number of colors required
to color the vertices of G such that
no two adjacent vertices are the
same color

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Graph Coloring: Greedy approach

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Graph Coloring: Greedy approach

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Greedy Graph Coloring via a BFS:
ColorGraph(G,v):!
!colors = []; !
!let Q be a queue!
!Q.enqueue(v)!
!v.color = new color!
!while Q is not empty:!
! v ß Q.dequeue()!
! for all edges from v to w:!
! if w is not labeled as discovered!
! ! Q.enqueue(w)!
! ! label w as discovered!
! ! neighbor_colors = set of color assignments
! ! !of the neighbors of w!
! ! if neighbor_colors == colors:!
! ! w.color = new color, update colors!
! ! else: w.color = a color from !
! ! ! colors - neighbor_colors !
!!
! ! ! ! ! !!
! ! ! ! !!

40. Confidential 40
The Four Color Theorem
Every planar graph is four colorable.
1997: Thomas

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Non-Planar Graphs
Can you connect each
dot on the top to each
dot on the bottom (like
above), without
crossing lines?

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Finding non-planar graphs
K5 K3,3

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Graph Coloring
The chromatic number of a graph has a
constrained optimization version that is
impossible to approximate within any
constant factor unless P = NP.
-1996: Zuckerman

44. Silicon Valley | Silicon Harbor
About this P =? NP.

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The P versus NP Problem:
Essentially: can every problem whose
solution can be checked by a computer in
polynomial time also be solved by a
computer in polynomial time?
Formal Conjecture:
1971 by Stephen Cook

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Why should you care?
Integer (prime?) Factorization
Graph Coloring (this isn’t what you think)
The Knapsack Problem
The Traveling Salesman

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RSA:
1. Pick two extremely large prime numbers p and q
2. Public Key: (e,n) where:
n = p Ÿ q
e in [3, (p – 1)(q – 1)]
3. Private key: (d,n) where:
n = p Ÿ q
e Ÿ d = 1 mod ((p – 1)(q – 1))

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RSA:
The foundation of RSA’s security relies
upon the fact that given a composite
number, it is considered a hard problem
to determine it’s prime factors.
An NP problem, in fact.

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What just happened?
1. Graph Properties and Reddit
2. Computational Complexity
P: Bipartite Graph Matching
NP: Graph Coloring
3. A little bit of RSA

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Graph Theory Resources:
Introduction to Graph Theory 2nd Ed (West)
Introduction to Graph Theory (Chartrand)
Social Network Analysis (Wasserman)
Introduction to Algorithms 3rd Edition (Cormen, …, Stein)
… the Wikipedia pages aren’t too shabby.

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Links in the Presentation Notes:
Reddit Viz:
http://redditstuff.github.io/sna/vizit/#
YouTube Lecture on Graph Matching:
https://www.youtube.com/watch?v=NlQqmEXuiC8
Graph Matching Code:
http://www.geeksforgeeks.org/maximum-bipartite-matching/
RSA Detailed Example:
http://doctrina.org/How-RSA-Works-With-Examples.html
Factorization of a 768-bit RSA modulus:
http://eprint.iacr.org/2010/006.pdf

52. Confidential 52
Graph Tech Stack:
Databases:
Titan
Neo4J
OrientDB
Visualization:
Gephi
sigma.js
GraphViz
Graph Tech Stack:
Algorithm Libraries:
Spark
Gremlin (Gremthon)
Boost (c++)
JGraphT (java)
NetworkX
Python-Graph
… there are plenty more to
dive into. This is just a start.

53. Silicon Valley | Silicon Harbor
Can You Trust The Internet?
An intro to graph invariants, computational
complexity, and graph algorithms
(with a little bit of cryptography)
Denise K. Gosnell, Ph.D.
Data Scientist, PokitDok
T: @DeniseKGosnell

54. Confidential 54
Can you connect the
dots like the the graph
below without lifting up
your pencil?
yes.
Can you connect each
dot on the top to each
dot on the bottom (like
above), without
crossing lines?
no.