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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

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/

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 •  …

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 !!!! ! ! ! ! !!! ! ! ! !!

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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

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 � q e in [3, (p – 1)(q – 1)]

3. Private key: (d,n) where:

n = p � q e � 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

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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.

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

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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.