View by Category

Loading...

PPT – Virtual Private Network Layout PowerPoint presentation | free to download

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Virtual Private Network Layout

- A proof of the tree conjecture on a ring network
- Leen Stougie
- Eindhoven University of Technology (TUE)
- CWI, Amsterdam
- http//www.win.tue.nl/math/bs/spor/2004-15.pdf

Input to the VPN problem

- Undirected graph G(V,E)
- Subset of the vertices Wµ V (terminals)
- Communication bounds on the terminals b(i) for

all i2 W - Unit capacity costs on the edges c(e) for all e2

E

Communication bounds and scenarios

- b(i) is bound on total of incoming and outgoing

communication of node v (symmetric VPN) - A valid demand scenario is symmetric matrix

D(dik)ik2 W with dii0 satisfying - dik 0 8 i,k 2 W and ?k2 W dik b(i) 8

i2 W - D is the set of all valid scenarios

VPN Robust optimization

- Select for each pair i,k2 W a path for

communication - Reserve enough capacity on the edges E
- All demand in every valid communication scenario

D2D can be routed on the selected paths - The total cost of reserving capacity is minimum
- The paths are to be selected before seeing any

communication scenario

Routing variations of VPN

- SPR (Single path routing)
- For each pair i,k2 W exactly one path Pikµ E
- TTR (Terminal tree routing)
- SPR with for each i2 W, k2 WPik is a tree in G
- TR (Tree routing)
- SPR with i,k2 W Pik is a tree in G
- MPR (Multi-path routing)
- For each pair i,k2 W for each path P between i

and k, specify fraction of communication using P

Relation between the variations

- Lemma
- OPT(MPR) OPT(SPR) OPT(TTR) OPT(TR)
- Proof
- SPR is the MPR problem with the extra restriction

that all fractions must be 0 or 1. - The other inequalities are similarly trivial.

The open VPN-problem

- Conjecture 1
- SPR 2 P (polynomially solvable)
- Conjecture 2
- OPT(SPR)OPT(TR)
- Conjecture 3
- OPT(MPR)OPT(TR)

What do we know about VPN?

- TR 2 P
- Kumar et al. 2002
- OPT(TR) OPT(TTR)
- Gupta et al. 2001
- OPT(TR) 2OPT(MPR)
- Gupta et al. 2001
- MPR 2 P
- Erlebach and Ruegg 2004, Altin et al. 2004,

Hurkens et al. 2004

The asymmetric VPN

- b(v) outgoing communication bound
- b-(v) incoming communication bound
- TR is NP-hard
- Gupta et al. 2001
- TR 2 P if ?v2 Wb-(v) ?v2 Wb(v)
- Italiano et al. 2002
- MPR 2 P
- Erlebach and Ruegg 2004, Altin et al. 2004,

Hurkens et al. 2004 - Constant Aprroximation ratios for SPR
- Gupta et al. 2001, Eisenbrandt et al. 2005

(randomized)

Conjecture 3 is true

- If G is a tree (trivial)
- If G is K4
- If G is a cycle !!!!
- If G is a 1-sum of graphs for which Conjecture 3

is true

Path-formulation of VPN

- Pik set of paths in G between i and k
- P set of all paths in G
- For each path p in G we define xp
- For all i and k 2 W, ?p2 Pikxp1
- SPR xp2 0,1 8p2 P
- MPR 0 xp 1 8p2 P

The capacity problem

- Given selected paths given values for x(p)
- Problem find capacities on edges z(e) 8 e2 E
- ?ep1 if e2P and 0 otherwise

Dual of the capacity finding problem

Path-formulation of SPR

- SPR Find x(p) minimizing ?e2 Eceze

Path-formulation of MPR

- MPR SPR with x(p) 0 i.o. x(p)2 0,1

Dual of the Path-formulation of MPR

- Dual-MPR

MPR and TR

- OPT(MPR) OPT(TR)
- Weak duality any feasible (?, ?) has ?

?ik OPT(MPR) - Conjecture 3 OPT(MPR)OPT(TR)
- Conjecture 3 OPT(TR)Optimal solution value of

the dual of MPR

Optimal solution of TR (1)

- Notation b(U)?v2 U b(v)
- Take tree T
- Each e2 T is cut in T splitting V in L(e) and

R(e) - Direct e to minimum of b(L(e)) and b(R(e))
- There is a unique vertex r with indegree 0, root
- Cost of T ?eminb(Le),b(Re) c(e)
- The minimum cost tree with r as the root is the

shortest path tree from r in G w.r.t. length

function c - OPT(TR) can be found in polynomial time

Optimal solution of TR (2)

- Let dG(u,v) the distance between u and v in G

w.r.t. length function c - The cost of optimal tree T is given by
- ?v b(v) dT(r,v)
- for some root vertex r.
- Moreover, it is bounded from below by
- ?v b(v) dG(r,v).
- Clearly it is bounded from above by
- ?v b(v) dT(u,v) forall u2V
- Compute shortest path tree rooted at u for all

u2V and select - the one with minumum cost solves OPT(TR) in

polynomial time

Conjecture 3 true for the cycle

- Lemma If Conjecture 3 is true for any cycle

with - - WV
- b(v)1 8 v2 V
- V is even
- Then Conjecture 3 is true for any cycle
- Theorem Conjecture 3 is true for any even cycle

with the above three properties

The even cycle (1)

- Vertices 0,1,2,...,2n-1
- Edges e1,e2,...,e2n
- Cost of tree by deleting edge ek
- (using ?eminb(Le),b(Re) c(e))
- We show there exist a dual solution with value

equal to - minek

The even cycle (2)

- MPR-dual restricitions for even cycle with b(v)1
- Only two possible paths between each pair of

vertices

The even cycle (3)The Tool Lemma

- The Tool Lemma
- - Let G(V,E) even circuit
- - b 1.
- - F µ E, F?
- Then there exist ?E! R, ? not equal 0, and K

such that - support(?)µF
- 8 f2 F KC(f?)mine2 E C(e ?)
- There is a dual solution (?, ?) with value K for

the MPR-dual problem with cost function ?

The even cycle (4)Part of Proof of Tool Lemma

- Proof By induction on F
- F1 (easy) Fek
- Take ?k1 and ?i0 8i? k
- Clearly, mine2 EC(e ?)C(ek ?)0
- A feasible dual solution with value 0 is ?eih0,

?ih0 8e2 E 8i,h 2 V

The even cycle (5)Part of Proof of Tool Lemma

- Proof (continued) Fgt1
- Case (i) There is a k such that ek2 F and its

opposite edge ekn2 F - (in figure read eka and eknb)

- Choose ?k?kn1 and ?i0 8 i? k,nk
- ) C(e?)n 8e2 E
- Choose
- Verify that ? ?ijn

The even cycle (6)

- Theorem Let G(V,E) be an even circuit, c E!R

and b(v)1 8v2 V. Then the cost of an optimal

tree solution equals the value of an optimal dual

solution. - Proof An inductive primal-dual argument using

the Tool Lemma. - (By request on the blackboard)

Postlude

- OPT(MPR)OPT(TR) for any graph?
- SPR polynomially solvable for any graph?
- Proof for the cycle is complicated!
- Is there an easier proof for the cycle?
- The crucial insight?
- Complexity of the non-robust MPR-problem is also

open!

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

disablevideo

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2015 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2015 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Virtual Private Network Layout" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Cmu University and other schools with their online training by sharing educational presentations for free