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 M k of maximal failures has to be calculated
A key observation
Additional definitions ● M k can be computed if sets M k,{u,v} can be computed ● Let E k be the set of node pairs {u,v} for which C k,{u,v} is not empty ● Lists M k,{u,v} are needed to be considered only for edges from E k .
Apples ● For ease of formulation
Framework for algorithms ● Processing apple A k{u,v} means determining M k{u,v} known A k{u,v} ● Framework for determining M k : – 1. Determine E k – 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 M k{u,v} → locally maximal cardinality while sweeping → gather theese → O(θ k ) elements
Trivial complexity bounds ● A naive algorithm for computing M k has the following complexity:
Improvements... ● Any M k,{u,v} can be stored in O(θ k )
Better estimations... ● The number of edges m=O(nθ 0 ) – E 0 → Delaunay triangulation → <2n triangles – Each triangle covers < θ 0 edges ● θ 0 is a graph density parameter
Improved complexity bounds ● Let θ k be the maximal number of covered edges by a circle from C k . Parameter θ k is considered to be ‘small’. ● A naive and an improved algorithm for computing M k has the following complexity:
Improved complexity bounds on M k ● Since θ k is small, it can be deducted that M k is relatively short ● For small k, |E k | is considered to be ‘not big’, thus M k can be computed in O(n 3 )
Speedup for k=0 and k=1
Speedup for k=0 and k=1 ● Further improvements can be made for small k values ● Graph T k and parameter τ i is to be defined later
Case of k=0 ● Graph D 0 induced by E 0 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 D 0 Delaunay triangulation and triangle graph T 0 :
Note ● T 0 is almost the Voronoi diagram of node set V, which is the dual graph of the D 0 Delaunay triangulation ● Both T 0 and D 0 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 nθ 0 ) ● Should be mapped: – Incident edges for each triangle circumcircle ● O(n) triangle, O(n θ 0 ) edge → checking each pair not good enough ● For each edge – connected subgraph of T 0 – Search on graph – max. degree is 3 ● All together: O(n log n θ 0 )
Merging lists M k,{u,v} ● It is enough to do the following for all edge e : – Compare all lists M k,{u,v} which contain e ● For every edge e let e i be the number of apples containing e ● Let τ 0 be the square mean of the e i values. ● Claim: merging can be done in O(n θ 03 τ 0 )
Total complexity for k=0 ● According to the corollary, M 0 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
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