Graph Coloring using Peer-to-Peer Networks

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1) The document proposes using a peer-to-peer network model called Content Addressable Network (CAN) to solve the graph coloring problem in a distributed manner to reduce computational time. 2) It describes how CAN works as a virtual coordinate space and how nodes are inserted. 3) It then explains the graph coloring problem and presents a recursive algorithm that divides the problem among peer nodes, combines partial solutions, and resolves conflicts to find the final solution.

Technology

Graph Coloring using Peer-to-Peer Networks Adrian Iftene, Cornelius Croitoru {adiftene,croitoru}@infoiasi.ro Faculty of Computer Science “ Al.I.Cuza” University, Iaşi, ROMANIA RoEduNet 2006, Sibiu

Content ,[object Object],[object Object],[object Object],[object Object]

Peer-to-Peer Systems ,[object Object],[object Object],(0,100) (100,100) (0,0) (100,0) C (0-50,50-100) D (50-75,50-100) E (75-100,50-100) A (0-50,0-50) B (50-100,0-50)

CAN - Movement ,[object Object],[object Object],6 2 3 1 5 4 k j Neighbours for 1 = {2, 3, 4, 5} Neighbours for 7 = {}

CAN - Insertion ,[object Object],[object Object],[object Object],[object Object]

CAN – Insertion – Sample 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 A node can refuse to split if it thinks it neighbors should be split instead, keeps the load balanced 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 4 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 4 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 4 Node 5 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 5 Node 4 Node 6

Graph Coloring ,[object Object],[object Object],[object Object]

Algorithm Phases ,[object Object],[object Object],[object Object],[object Object]

Recursive Graph Coloring A node from network will initiate a coloring problem All available neighbours will be involved in problem solving In the end all partial solutions are sended to the initiator All nodes will communicate with problem initiator

Phases I and II ,[object Object],[object Object],[object Object],[object Object],[object Object]

Phases III and IV ,[object Object],[object Object],[object Object],[object Object]

Tests and Results 78 s 127 s 38 36 35 25 17425 450 Le450-25d 90 s 145 s 39 37 36 25 17343 450 Le450-25c 85 s 129 s 32 32 30 15 16750 450 Le450-15d 79 s 110 s 32 33 30 15 16680 450 Le450-15c 26 s 44 s 18 18 18 5 9757 450 Le450-5d 13 s 20 s 14 15 13 5 5734 450 Le450-5b 14 s 21 s 13, 14 15 13 5 5714 450 Le450-5a 27 s 30 s 25 32 25 16 12640 256 Queen16-16 2 s 1 s 14, 15, 16 17 18 12 3216 128 Queen10-10 1 s 0 s 13, 14, 15 16 14 10 2112 81 Queen9-9 1 s 0 s 12, 13, 14 16 13 9 728 64 Queen8-8 1 s 0 s 15 , 16, 17 17 15 12 1368 96 Queen8-12 1 s 0 s 10, 11 , 12 13 12 7 476 49 Queen7-7 1 s 0 s 9 , 10, 11 13 9 7 290 36 Queen6-6 1 s 0 s 6, 7, 8 , 9 8 7 5 160 25 Queen5-5 1 s 1 s 9, 10 9 10 9 638 120 Games120 0 s 0 s 10 , 11 11 11 10 254 80 Jean 0 s 0 s 11 11 11 11 301 74 Huck 0 s 0 s 11, 12, 13 12 12 11 406 87 David 2 s 1 s 11, 12 12 12 11 493 138 Anna 0 s 0 s 7, 8 8 7 7 755 95 Myciel7 0 s 0 s 6 , 7 7 6 6 236 47 Myciel5 0 s 0 s 5 , 6 6 5 5 71 23 Myciel4 P2P Local Random Order Theorem  Edges Nodes ID

Conclusions and Future Work ,[object Object],[object Object],[object Object],[object Object]

Bibliography ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The graph colouring problem can be solved using methods based on Satisfiability (SAT). An instance of the problem is defined by a Boolean expression written using Boolean variables and the logical connectives AND, OR and NOT. It has to be determined whether there is an assignment of TRUE and FALSE values to the variables that makes the entire expression true.A SAT problem is syntactically and semantically quite simple. Many Constraint Satisfaction Problems (CSPs)in AI and OR can be formulated in SAT. These make use of two kinds of searchalgorithms: Deterministic and Randomized.It has been found that deterministic methods when run on hard CSP instances are frequently very slow in execution.A deterministic method always outputs a solution in the end, but it can take an enormous amount of time to do so.This has led to the development of randomized search algorithms like GSAT, which are typically based on local (i.e., neighbourhood) search. Such methodshave been applied very successfully to find good solutions to hard decision problems

RC4 Technique in Visual Cryptography

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In this study, we are doing a cryptography scheme which can modify the visualization of pictures. The protection of images is critical. This protection does not alter the value of the header and the metadata. Every image consists of three color layers. There are red, green and blue. Each layer has numbers which represent the color intensity. RC4 is used to change the color intensity in every layer. We can choose how many layers will be encrypted. The fuzziness of the encrypted image depends on how many layer are taking a role.

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Machine learning on graphs is an important and ubiquitous task with applications ranging from drug design to friendship recommendation in social networks. The primary challenge in this domain is finding a way to represent, or encode, graph structure so that it can be easily exploited by machine learning models. However, traditionally machine learning approaches relied on user-defined heuristics to extract features encoding structural information about a graph. In this talk I will discuss methods that automatically learn to encode graph structure into low-dimensional embeddings, using techniques based on deep learning and nonlinear dimensionality reduction. I will provide a conceptual review of key advancements in this area of representation learning on graphs, including random-walk based algorithms, and graph convolutional networks.

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The famous Lovász Local Lemma was derived in the paper of P. Erdős and Lovász to prove that any _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least _Δ(H) ≥ ¼ r_<sup>_n−1_</sup>. A long series of papers is devoted to the improvement of this classical result for different classesof uniform hypergraphs. In our work we deal with colorings of simple hypergraphs, i.e. hypergraphs in which everytwo distinct edges do not share more than one vertex. By using a multipass random recoloringwe show that any simple _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least _Δ(H) ≥ с · nr_<sup>_n−1_</sup> where _c_ > 0 is an absolute constant. We also give some applications of our probabilistic technique, we establish a new lower bound for the Van der Waerden number and extend the main result to the _b_-simple case. The work of the second author was supported by Russian Foundation of Fundamental Research (grant № 12-01-00683-a), by the program “Leading Scientific Schools” (grant no. NSh-2964.2014.1) and by the grant of the President of Russian Federation MK-692.2014.1

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Nowadays, more and more data is collected in large amounts, thus the need of studying it both efficiently and profitably is arising; we want to achieve new and significant informations that weren’t known before the analysis. At this time many graph mining algorithms have been developed, but an algebra that could systematically define how to generalize such operations is missing. In order to propel the development of a such automatic analysis of an algebra, We propose for the first time (to the best of my knowledge) some primitive operators that may be the prelude to the systematical definition of a hypergraph algebra in this regard.

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1. Graph Coloring using Peer-to-Peer Networks Adrian Iftene, Cornelius Croitoru {adiftene,croitoru}@infoiasi.ro Faculty of Computer Science “ Al.I.Cuza” University, Iaşi, ROMANIA RoEduNet 2006, Sibiu

6. CAN – Insertion – Sample 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 A node can refuse to split if it thinks it neighbors should be split instead, keeps the load balanced 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 4 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 4 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 4 Node 5 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 Node 1 Node 2 Node 3 Node 5 Node 4 Node 6

9. Recursive Graph Coloring A node from network will initiate a coloring problem All available neighbours will be involved in problem solving In the end all partial solutions are sended to the initiator All nodes will communicate with problem initiator

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12. Tests and Results 78 s 127 s 38 36 35 25 17425 450 Le450-25d 90 s 145 s 39 37 36 25 17343 450 Le450-25c 85 s 129 s 32 32 30 15 16750 450 Le450-15d 79 s 110 s 32 33 30 15 16680 450 Le450-15c 26 s 44 s 18 18 18 5 9757 450 Le450-5d 13 s 20 s 14 15 13 5 5734 450 Le450-5b 14 s 21 s 13, 14 15 13 5 5714 450 Le450-5a 27 s 30 s 25 32 25 16 12640 256 Queen16-16 2 s 1 s 14, 15, 16 17 18 12 3216 128 Queen10-10 1 s 0 s 13, 14, 15 16 14 10 2112 81 Queen9-9 1 s 0 s 12, 13, 14 16 13 9 728 64 Queen8-8 1 s 0 s 15 , 16, 17 17 15 12 1368 96 Queen8-12 1 s 0 s 10, 11 , 12 13 12 7 476 49 Queen7-7 1 s 0 s 9 , 10, 11 13 9 7 290 36 Queen6-6 1 s 0 s 6, 7, 8 , 9 8 7 5 160 25 Queen5-5 1 s 1 s 9, 10 9 10 9 638 120 Games120 0 s 0 s 10 , 11 11 11 10 254 80 Jean 0 s 0 s 11 11 11 11 301 74 Huck 0 s 0 s 11, 12, 13 12 12 11 406 87 David 2 s 1 s 11, 12 12 12 11 493 138 Anna 0 s 0 s 7, 8 8 7 7 755 95 Myciel7 0 s 0 s 6 , 7 7 6 6 236 47 Myciel5 0 s 0 s 5 , 6 6 5 5 71 23 Myciel4 P2P Local Random Order Theorem  Edges Nodes ID

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Graph Coloring using Peer-to-Peer Networks

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Graph Coloring using Peer-to-Peer Networks