all pair shortest path--sdn

A Novel APSP and its Application in Multi Domain SDN

Sarat Chandra Prasad Gingupalli(12IS21F)

Under the guidance of

Mrs. Saumya Hegde

DEPARTMENT OF COMPUTER SCIENCE ENGINEERING

NITK Surathkal

June 27, 2014

Sarat Chandra Prasad Gingupalli (NITK Surathkal) Project Final Seminar June 27, 2014 1 / 37

Agenda

IntroductionLiterature SurveyProblem Statement and objectivesMethodologyAnalysisResults and DiscussionsConclusion and Future WorkReferences


Introduction: Ossification of Internet

The ossification of the Internet is a natural evolutionary stage in thedevelopment of any highly successful technology.The competition between the vendors brought lot of heterogeneityinto the Internet and now which is becoming a hindrance in deployingnew network technologies.To address these issues the concept of programmable networking wasintroduced.Software Defined Networking(SDN) is one such programmablenetworking technology which aims to provide network as a service likeany other resources such as computing and storage.


Introduction: Agile Networking

Agile Network and SDN offers an unprecedented experience in qualitywhile meeting the requirements of mobile applications, cloud computing,and social media.

Reduced Labor and Efficient Operation and ManagementQuick Service Innovation


Introduction: Shortest Path Problem

In graph theory, the shortest path problem is the problem of finding a pathbetween two vertices (or nodes) in a graph such that the sum of theweights of its constituent edges is minimized.The different types of shortest path problems are listed as follows:

single-source shortest path problemsingle-destination shortest path problemall pair shortest path problem


Introduction: Shortest Path Problem

The most important algorithms for solving this problem are:Dijkstra’s algorithm solves the single-source shortest path problem.BellmanFord algorithm solves the single-source problem if edgeweights may be negative.A* search algorithm solves for single pair shortest path usingheuristics to try to speed up the search.FloydWarshall algorithm solves all pairs shortest paths.Johnson’s algorithm solves all pairs shortest paths, and may be fasterthan Floyd- Warshall on sparse graphs.Viterbi algorithm solves the shortest stochastic path problem with anadditional probabilistic weight on each node.


Literature Survey

The brief study and analysis is made in following concepts:Overview of All Pair Shortest Path ProblemGraph Decomposition a NP Complete ProblemOverview of Security Issues in SDN


Overview of All Pair Shortest Path Problem

Given a graph G = (V ,E )Floyd Warshall algorithm takes O(n3) to compute the shortest pathsbetween the all pair of vertices in the worst case.(Bloniarz P.A.,1983) proposed an all pair shortest path algorithm withthe running time O(n2lognlogn) further improvement to this wasproposed by (Alistair and Takaoka,1985)with the expected runningtime O(n2logn).(Wei F.,2010) proposed TEDI(TreE Decomposition by Indexing)

In the case of sparse graphs the time complexity for constructing theindex is O(n2) and the query time is O(th), where t is the width of thetree and h is the height of the tree.

(Raghavendra, Jorge and Lee, 2012) proposed Dynamic TEDIDynamic TEDI was proposed based on TEDI and the limitations inTEDI are continued to exist in Dynamic TEDI.


Graph Decomposition a NP Complete Problem

Graph decomposition is basically a partition of a set into disjointsubsets taken from a given collection.Till today there exists no algorithm which can decompose the givengraph into components of equal size in a polynomial time.In 1980, Holyer conjectured that Graph decomposition isNP-complete whenever G is connected and has three edges or more.Many heuristic approaches are proposed to decomposed the parsegraphs in a polynomial time.In computer science, artificial intelligence, and mathematicaloptimization, a heuristic is a technique designed for solving a problemmore quickly when classic methods are too slow, or for finding anapproximate solution when classic methods fail to find any exactsolution.This is achieved by trading optimality, completeness, accuracy, orprecision for speed.


Overview of Security Issues in SDN

As SDN is a new design paradigm so the security challenges of SDNhave to be addressed from the scratch.As the control plane is separated from the data plane in SDN,it iscreating a clear centralized point of attack.Because of heterogeneity in the traditional networks an attacker haveto follow different attacking strategies in order to attack the entirenetwork but where as in the case of SDN, entire network can becompromised by compromising the controller.(Seungwon and Guofei, 2013) demonstrated an effective and efficientattack against SDN with the knowledge of some basic characteristicsof the SDN technology.


Problem Statement

Finding the novel All Pair Shortest Path algorithm which caneffectively decompose the graph and can efficiently finds the shortestpath in a source routing Multi Domain SDN interms of computationtime and memory requirements.As securing the network topology is a key criteria in a Multi DomainSDN, algorithms are proposed for securing the network topology.


Objectives

To Design a Novel Graph Decomposition Technique.To Design Algorithms for Securing the Network Topology.Performance comparison.


Methodology: Novel APSP

Finding the articulation points.Decompose the graph into number of components are of uniform size.In each component do the following.

Identify the peripheral vertices in the component.Apply the Dijkstra’s algorithm to find the shortest path from every nonperipheral vertex to every other peripheral vertex in the component.Apply the Dijkstra’s algorithm to find the shortest path betweenperipheral vertices in the component.

Construct a graph by taking the peripheral vertices from all thecomponents and apply Floyd Warshall algorithm to find the shortestpaths between all the pairs.


Methodology: Graph DecompositionThe algorithm for decomposing a graph G = (V ,E ) by applying

√|V |

reduction is as follows1 Start with any random vertex v ∈ V , which is not visited, create an

empty vertex set.2 Add the vertex v to the vertex set. Mark the status of the vertex v as

visited, and identify all its neighboring vertices.3 Visit any of the visited vertices neighboring vertex such that the

number of outgoing edges should be of minimum.4 If tie occurs, it will be resolved by giving priority to the vertex whose

parent vertex is visited first and if tie occurs among the verticeswhose parent vertex is same in that case it will be resolved byrandomly choosing any of the vertex.

5 Repeat steps 2-4 until the number of vertices visited is√|V | or there

exists no vertex to visit further.6 Identify the peripheral vertices of the vertex set and then identify the

neighbors of these peripheral vertices which are not included in thevertex set.


Methodology : Graph Decomposition

7 Scan the neighboring vertex set and include any vertex from it into aset of visited vertex set on the basis of below criteria.

If the vertex is not an articulation point, then the summation ofperipheral vertices and the leaving edges from the component shouldnot be increased.If the neighbor vertex is an articulation point, then exclude the edgefrom the corresponding parent vertex which is already visited and applyBFS on it to find out the number of non visited vertices which can bevisited. If the number of such vertices are of the order Ω(

√|V |), then

include it into the visited vertex set along with all the traversed vertices.8 Repeat step 7 until there exists no vertex to add further.9 Repeat the above process until all the vertices in the graph are visited.


Methodology : Securing Network Topology

Two approaches are proposed in this paper for generation of encryptedcomplete path in this paper.

1 Encrypting the sub paths of a domain using it’s corresponding sharedsecret keys

2 Encrypting the complete path using the all shared secret keys


Methodology : Encrypting the sub paths of a domain usingit’s corresponding shared secret keys

The algorithm to retrieve the complete path from P and encrypting it isdone as follows:while not at end of P do

Read current_switchRead next_switchRetrieve path from the corresponding slave controllerEncrypt the retrieved path using the secret key shared between themaster and the corresponding slave controller of the domain to whichcurrent_switch and next_switch belongs to by a master controllerAdd it to CPGo back to the beginning of current section

end


Methodology : Encrypting the complete path using the allshared secret keys

while not at end of P doRead current_switchRead next_switchRetrieve path from the corresponding slave controllerAdd it to CPEncrypt CP excluding the first switch using the secret key sharedbetween the master and the corresponding slave controller of thedomain to which current_switch and next_switch belongs to by amaster controllerGo back to the beginning of current section

end


Methodology : Analysis

The theoretical asymptotic bounds of the algorithms in terms of worst caseupdate time, query time and space requirements are shown in the belowtable. Here, V and E are the number of vertices and edges respectively.

Algorithm Update Time Query Time SpaceS-DIJ O (|E ||V |+

|V |2 log |V |) O (1) O

(|V |2

)D-PUP O (1) O

(|V |2

)O (|V ||E |)

D-TEDI O(|V |2

)O(k2h

)O(l |R2|

)APSP O

(|V |2 log |V |

log |V |)O (1) O

(|V |2

)Modified APSP O

(|V |2 log |V |

)O (1) O

(|V |2

)Novel APSP O

((|V |δ)2

)O(δ2) O (|V |l+

|V |δ2l ′)


Correctness of the Proposed Approach

The proposed algorithm is making use of the existing algorithms such asDijsktra’s and Floyd Warshall to compute the shortest paths between thevertices. Let G = (V ,E ) be a graph, s and d be the source anddestination vertices.

Proof.The shortest path between s and d which is computed by applying theDijsktra’s algorithm is defined as SP (s,d).Let v be any vertex in the shortest path, then according to Dijsktra’salgorithm the following expression holds true

SP (s,d) = SP (s,v) + SP (v ,d) (1)

Let s ∈ Ci = (Vi ,Ei ) and d ∈ Cj = (Vj ,Ej), Pi and Pj are thecorresponding peripheral vertex sets of s and d respectively.


Correctness of the Proposed Approach

Proof.There exists a vertex say s ′ ∈ Pi through which the shortest path leavesfrom the component Ci and there exists a vertex say d ′ ∈ Pj throughwhich the shortest path enters into the component Cj .According to the proposed approach the shortest path between s and dcan be expressed as

Shortest Path = SP(s,s ′)+ SP

(s ′,d ′)+ SP

(d ′,d)

= SP(s,d ′)+ SP

(d ′,d)

= SP (s,d) (2)

Hence Proved.


Why√|V | reduction

Let G = (V ,E ) be a given graph and is decomposed into Kcomponents.The time complexity of the initial setup is

K∑i=1

|Vi |∑j=1

j∑z=1

deg(vz) |vz ∈ Vi +(|Vi |−

|V |K

) |V |K +2|E |

+K∑

i=1(|Ei |+ |Vi | log |Vi |)|NPi ||Pi |+

( K∑i=1

Pi

)3

As the graph is decomposed into K components of equal size, then theaverage number of vertices in each component will be of |V |

K . Bysubstituting δ in place of a degree of a vertex, |V |δ in place of |E |, theaverage number of peripheral nodes in each component with δ, theaverage number of non peripheral nodes with |V |

K − δ in the aboveequation it equates to

K∑i=1

|V |K∑

j=1

j∑z=1

δ+2|V |δ

+K∑

i=1

( |V |δK + |V |K logVi

)δ

( |V |K − δ

)+( K∑

i=1δ

)3

= K |V |K

( |V |K +1

)δ+2|V |δ

+ K( |V |δ

K + |V |K log |V |)δ

( |V |K − δ

)+ (Kδ)3

= |V |( |V |

K +1)δ+2|V |δ

+ (|V |δ+ |V | log |V |)δ( |V |

K − δ)

+ (Kδ)3

(3)


Why√|V | reduction

The second order derivative of the above equation w.r.t K is positive, soby equating the first order derivative of it w.r.t K to zero, we’ll get a valueof K at which the above equation produces a minimum value.

ddx (3) =−

(|V |2δK 2

)(1+ |V |δ+ |V | log |V |) +3δ3K 2

By equating the above equation to zero we’ll get

K 4 = 13

((|V |2δ2

)(1+ δ+ log |V |)

)

K =√|V |

( log |V |√δ

) 14

K = O(√|V |)∵

( log |V |δ2

) 14√|V |

(4)


Results and Discussions

The proposed shortest path algorithm is tested based on the followingcriteria

1 Varying network topologies2 Varying number of controllers

We consider the basic, hybrid and randomly generated network topologiesto test our approach against Floyd Warshall algorithm under varyingnumber of controllers. The results are as follows



Figure: Floyd Warshall vs proposed approach in the case of random topology



Figure: Floyd Warshall vs proposed approach in the case of line topology



Figure: Floyd Warshall vs proposed approach in the case of ring topology



Figure: Floyd Warshall vs proposed approach in the case of mesh topology



Figure: Performance of proposed approach under various networking topologies


Conclusion and Future Work

Conclusion :In this project we presented a novel approach for finding the shortest pathsin a given network graph, and it was mathematically proven that for agiven graph whose average vertex degree δ is Ω(n) the proposed algorithmis efficient than any of the existing algorithms in terms of computationtime and storage.The proposed algorithm is applied in Multi Domain SDN to assign thehosts to a controller and to find the shortest path between the hosts. Itwas tested against the basic topologies like star, mesh, tree, chain andsome randomly generated graphs and in all the cases the proposedapproach producing promising results.We also proposed algorithms for hiding the network topology in a sourcerouting multi domain SDN, which is highly significant in a multi tenantcloud environment. The proposed approach can effectively countermeasureany kind of attacks which reveals the network topology.


Conclusion and Future Work

Future Work :As the proposed approach works effectively for the graphs which are ofparse nature, the future scope is to design an algorithm which is capableof decomposing the dense graphs having articulation points intocomponents of uniform size.


Published Papers

Paper titled Securing the Network Topology in a Source RoutingMulti Domain SDN was presented at International Conference OnAdvances in Computer Engineering and Application on Feb 15th. Itgot best paper award and published in International Journal ofComputer Applications.Paper titled A novel APSP algorithm and its application in MultiDomain SDN was presented at IEEE International Conference onRecent Advances and Innovations in Engineering on May 11th.Further, this paper will be published in IEEE Conference Proceedings.


References I

M. Alistair and T. Tadao.An all pairs shortest path algorithm with expexted running timeo(n2 logn

).

Technical report, IEEE, 1985.

P.A. Bloniarz.A shortest path algorithm with expected time o

(n2 logn logn

).

Technical report, SIAM J. Comput., 1983.

S. Chaudhuri and C. D. Zaroliagis.Shortest paths in digraphs of small treewidth. part i: Sequentialalgorithms.Technical report, Algorithmica, 2000.

C. Demetrescu and G. F. Italiano.Experimental analysis of dynamic all pairs shortest path algorithms.Technical report, ACM Trans.Algorithms, 2006.


References II

Fernando M.V. Diego, K. and P. Ramos.Towards secure and dependable software-defined networks.Technical report, HotSDN, 2013.

E. W. Dijkstra.A note on two problems in connexion with graphs.Technical report, Numerische Mathematic, 1959.

D. Dorit and T. Michael.Graph decomposition is np-complete: A complete proof of holyerâĂŹsconjecture.Technical report, SIAM J. COMPUT, 1997.

Astuto A. Mendonca, M. and T. Thierry.A survey of software-defined networking: Past, present, and future ofprogrammable networks.Technical report, HotSDN, 2013.


References III

Jorge L. Raghavendra, R. and Kang Won L.Dynamic graph query primitives for sdn-based cloud networkmanagement.Technical report, SIGMOD, 2012.

S. Seungwon and G. Guofei.Attacking software-defined networks: A first feasibility study.Technical report, HotSDN, 2013.

S. Stefan and S. Jukka.Exploiting locality in distributed sdn control.Technical report, HotSDN, 2013.

F. Wei.Tedi: efficient shortest path query answering on graphs.Technical report, SIGMOD, 2000.


References IV

Yan C. Xitao, W. and H. Chengchen.Towards a secure controller platform for openflow applications.Technical report, HotSDN, 2013.


End of Presentation

Thank You !!


all pair shortest path--sdn

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