Shared Risk Link Groups of Regional Failures Covering k Nodes Balzs - - PowerPoint PPT Presentation

Shared Risk Link Groups of Regional Failures Covering k Nodes

Balázs Vass BME TMIT vb@tmit.bme.hu

How to make good SRLG lists?

• Network

– Considered as a geometric graph

G(V,E) in the Euclidean plane

• Links are considered as line

segments

• Last week:

– List of maximal SRLGs which can

be covered by a disk with radius at most r

– Problem: cannot control: # of

covered nodes → traffic matrix change

How to make good SRLG lists?

• Last week:

– List of maximal SRLGs which

can be covered by a disk with radius at most r

– Problem: does not control: # of

covered nodes → traffic matrix change

– A solution:

• List of maximal SRLGs which can

be covered by a disk with radius at most r and covering k nodes

• small algorithm modification

How to make good SRLG lists?

• This week:

– List of maximal SRLGs which can be covered by a disk

with radius at most r and covering k nodes

– k is considered to be ‘small’

Definitions

• Parameter k is given
• Set Mk of maximal failures has to be

calculated

A key observation

Additional definitions

• Mk can be computed if sets Mk,{u,v} can be computed
• Let Ek be the set of node pairs {u,v} for which Ck,{u,v} is not empty
• Lists Mk,{u,v} are needed to be considered only for edges from Ek.

Apples

• For ease of formulation

Framework for algorithms

• Processing apple Ak{u,v} means determining

Mk{u,v} known Ak{u,v}

• Framework for determining Mk:

– 1. Determine Ek – 2. Determine nonempty apples – 3. Process apples – 4. Merge lists

How to process apples

• By sweeping (transforming) a circle from c+ to c-
• Its continuous motion is discretized.
• Failure is in Mk{u,v} →

locally maximal cardinality while sweeping → gather theese→ O(θk) elements

Trivial complexity bounds

• A naive algorithm for computing Mk has the

following complexity:

Improvements...

• Any Mk,{u,v} can be stored in O(θk)

Better estimations...

• The number of edges m=O(nθ0)

– E0 → Delaunay triangulation → <2n triangles – Each triangle covers <θ0 edges

• θ0 is a graph

density parameter

Improved complexity bounds

• Let θk be the maximal number of covered edges by a circle from
• Ck. Parameter θk is considered to be ‘small’.
• A naive and an improved algorithm for computing Mk has the

following complexity:

Improved complexity bounds on Mk

• Since θk is small, it can be deducted that Mk is relatively short
• For small k, |Ek| is considered to be ‘not big’, thus Mk can be

computed in O(n3)

Speedup for k=0 and k=1

Speedup for k=0 and k=1

• Further improvements can be made for small k

values

• Graph Tk and parameter τi is to be defined later

Case of k=0

• Graph D0 induced by E0 is the

Delaunay triangulation on node set V.

• Delaunay triangulation:

– All triangles: empty circle

property

– Maximizes the minimal angle – Unique if nodes are in general

position

– Plane graph – Can be deternined in O(n log n)

Example

• A German optic network:

Example

• The network and its Delaunay triangulation:

– Can be very different from each other

Example

• The network’s Delaunay triangulation with

circumcircles:

Example

• The network’s D0 Delaunay triangulation and

triangle graph T0:

Note

• T0 is almost the Voronoi diagram of node set V,

which is the dual graph of the D0 Delaunay triangulation

• Both T0 and D0 can be determined in O(n log n)

More about Delaunay triangulation

• Finding Delaunay triangulation & more

computational geometry

– www.cs.uu.nl/geobook/

Determining apples in O(n log nθ0)

• Should be mapped:

– Incident edges for each

triangle circumcircle

• O(n) triangle, O(nθ0) edge

→ checking each pair not good enough

• For each edge

– connected subgraph of T0 – Search on graph – max. degree is 3

• All together: O(n log n θ0)

Merging lists Mk,{u,v}

• It is enough to do the following for

all edge e:

– Compare all lists Mk,{u,v} which contain e

• For every edge e let ei be the

number of apples containing e

• Let τ0 be the square mean of the ei

values.

• Claim: merging can be done in O(n

θ03 τ0)

Total complexity for k=0

• According to the corollary, M0 has a linear size,

and can be computed in almost linear time

Complexities again

• Case of k=1 preserves many useful

properties of case k=0.

• Case k=2, 3 , …<<n : fine properties,

further study

• Implementation ongoing

New topic

PSRLGs

• Paper: Diverse Routing in Network with

Probabilistic Failures (H.W. Lee and E. Modiano)

PSRLGs

• Correlated probabilistic link failures instead of

deterministic failure model

• A probabilistic SRLG (PSRLG) is a set of links

with positive failure probability in the event of an SRLG failure.

• Edges from an SRLG are correlated
• If failure probabilities in SRLGs are 1, we get

back the deterministic model

Motivation

• Multiple failures due to:

– Multiple communication links sharing the same fiber – Second link fails before first was repaired – Natural disasters / attacks destroy several links

Problem

• Single source-destination pair
• Independent link failure model / PSRLG based

correlated failure model

• Single path problem / Path pair problem with

disjointness constraint / Path pair problem without disjointness constraint

Approaches

• Independent link failure model + Single path

problem → Dijkstra

• Other cases: hard

– INLP formulations – Heuristics

• Heuristics are often faster and give better

solution

Thank you for attention