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![Confidential 12
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
…](https://image.slidesharecdn.com/gosnell6315-150604210346-lva1-app6892/85/Can-you-trust-the-internet-An-introduction-to-graph-theory-computational-complexity-and-a-little-bit-of-RSA-12-320.jpg)
![Confidential 29
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
• …](https://image.slidesharecdn.com/gosnell6315-150604210346-lva1-app6892/85/Can-you-trust-the-internet-An-introduction-to-graph-theory-computational-complexity-and-a-little-bit-of-RSA-29-320.jpg)
![Confidential 39
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 !
!!
! ! ! ! ! !!
! ! ! ! !!](https://image.slidesharecdn.com/gosnell6315-150604210346-lva1-app6892/85/Can-you-trust-the-internet-An-introduction-to-graph-theory-computational-complexity-and-a-little-bit-of-RSA-39-320.jpg)
![Confidential 47
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))](https://image.slidesharecdn.com/gosnell6315-150604210346-lva1-app6892/85/Can-you-trust-the-internet-An-introduction-to-graph-theory-computational-complexity-and-a-little-bit-of-RSA-47-320.jpg)
Uploaded byDenise Gosnell, Ph.D.
Can you trust the internet? An introduction to graph theory, computational complexity, and a little bit of RSA.
The document discusses graph theory, focusing on concepts such as graph properties, computational complexity, and algorithms related to bipartite graph matching and graph coloring. It highlights the significance of these topics in computer science and presents examples like the Seven Bridges of Königsberg and various algorithms including augmenting paths and greedy approaches. Additionally, it touches on NP problems and introduces RSA cryptography, emphasizing its reliance on factoring large prime numbers.
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Can you trust the internet? An introduction to graph theory, computational complexity, and a little bit of RSA.
- 1. Confidential 1 Can youconnect 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
- 3.
- 4. Confidential 4 The Plan 1. Graph Properties and Reddit 2. Computational Complexity P: Bipartite Graph Matching NP: Graph Coloring 3. A little bit of RSA
- 5. Confidential 5 Why shouldyou care? 1. Everything is graph theoretic in nature. 2. Much easier to code. 1. or… you just want to be here to look cool.
- 6. Confidential 6 The SevenBridges of Königsberg "Konigsberg bridges" by Bogdan Giuşcă - Public domain (PD) http://commons.wikimedia.org/wiki/File:Konigsberg_bridges.png#/media/File:Konigsberg_bridges.png
- 7. Confidential 7 The SevenBridges of Königsberg: Solved by Euler in 1735 Thank you Wikipedia: http://en.wikipedia.org/wiki/Seven_Bridges_of_Konigsberg
- 8. Confidential 8 Eulerian Paths vs. 3 3 53 2 4 4 3 3
- 9. Confidential 9 screen shotfrom vizit: http://redditstuff.github.io/sna/vizit/
- 10. Silicon Valley |Silicon Harbor P
- 11. Confidential 11 P The setof all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computational time
- 12. Confidential 12 Classes ofAlgorithm 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 …
- 13. Confidential 13 Graph Search:O(|V| + |E|) Depth vs. Breadth
- 14. Confidential 14 Graph Matching Givena bipartite graph G = (V,E), a matching M in G is a set of edges in which no two edges share a common vertex
- 15. Confidential 15 The UserPreferences Problem Candidates à Jobs
- 16. Confidential 16 The UserPreferences Problem Goal: assign candidates to jobs to fill as many jobs as possible
- 17. Confidential 17 The UserPreferences Problem Greedy Algorithm: Keep adding edges until no more edges can be added
- 18. Confidential 18 The UserPreferences Problem Greedy Approach
- 19. Confidential 19 The UserPreferences Problem A better solution: via augmenting paths
- 20. Confidential 20 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.
- 21. Confidential 21 Augmenting Paths Youcan 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
- 22. Confidential 22 The UserPreferences Problem Augmenting Path Example:
- 23. Confidential 23 The UserPreferences Problem Augmenting Path Example:
- 24. Confidential 24 Berge’s Theorem: Amatching in a graph is maximum if and only if there does not exist an augmenting path in a graph
- 25. Confidential 25 Maximal Matchingby Constructing the Auxiliary Graph
- 26. Confidential 26 Maximal Matchingby 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
- 27. Confidential 27 Maximal Matchingby Constructing the Auxiliary Graph:
- 28. Confidential 28 Maximal MatchingFinal 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!
- 29. Confidential 29 Graph Algorithmsin 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
- 31. Confidential 31 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
- 32. Confidential 32 P àsolvable in polynomial time NP à verifiable in polynomial time
- 33. Confidential 33 Classic NPProblems: Integer (prime?) Factorization Graph Coloring (this isn’t what you think) The Knapsack Problem The Traveling Salesman
- 34. Confidential 34 Integer (prime?)Factorization
- 35. Confidential 35 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.
- 36. Confidential 36 Graph Coloring Minimumnumber of colors required to color the vertices of G such that no two adjacent vertices are the same color
- 37. Confidential 37 Graph Coloring:Greedy approach
- 38. Confidential 38 Graph Coloring:Greedy approach
- 39. Confidential 39 Greedy GraphColoring 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 FourColor Theorem Every planar graph is four colorable. 1997: Thomas
- 41. Confidential 41 Non-Planar Graphs Canyou connect each dot on the top to each dot on the bottom (like above), without crossing lines?
- 42. Confidential 42 Finding non-planargraphs K5 K3,3
- 43. Confidential 43 Graph Coloring Thechromatic 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.
- 45. Confidential 45 The Pversus 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
- 46. Confidential 46 Why shouldyou care? Integer (prime?) Factorization Graph Coloring (this isn’t what you think) The Knapsack Problem The Traveling Salesman
- 47. Confidential 47 RSA: 1. Picktwo 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))
- 48. Confidential 48 RSA: The foundationof 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.
- 49. Confidential 49 What justhappened? 1. Graph Properties and Reddit 2. Computational Complexity P: Bipartite Graph Matching NP: Graph Coloring 3. A little bit of RSA
- 50. Confidential 50 Graph TheoryResources: 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.
- 51. Confidential 51 Links inthe 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 TechStack: 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 youconnect 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.