Addressing Network Survivability Issues by Finding the K-best Paths through a Trellis Graph

1. Addressing Network Survivability Issues by Finding the K-best Paths through a Trellis Graph Stavros D. Nikolopoulos, Andreas Pitsillides, and David Tipper Department of Computer Science, University of Cyprus, CY-1678 Nicosia, Cyprus. Presented by Chung-lien Tseng INFOCOM '97

2. Agenda • Introduction • Graph Modeling and K-best Paths Problems • Problem Transformation • Example • Conclusions INFOCOM '97

3. Introduction Survivability K-best (Disjoint) Paths Problems Trellis Graph INFOCOM '97

4. Survivability • Network design and management procedures to minimize the impact of failures on the network. • Events that cause failures: • Accidental cable cuts • Hardware malfunctions • Software errors • Natural disasters (e.g., fire) • Human error (e.g., incorrect maintenance) INFOCOM '97

5. 3 Categories of Survivability Techniques • Prevention • Focus on improving component and system reliability. • Network design • Mitigate the effects of system level failures such as link or node failures by placing sufficient diversity and capacity in the network topology. • Traffic management and restoration • Direct the network load such that a failure has minimum impact when it occurs and that connections affected by a failure are reconnected around the failure. INFOCOM '97

6. K-best Disjoint Paths Problems • Select k-best disjoint paths (or unmerged paths) between any OD pair. • Used for load balancing or rerouting. • “Best” paths are paths which are as diverse as possible (with minimal overlap between the paths in terms of nodes and links). • Maximize the chance of survivability • Ensure a graceful degradation INFOCOM '97

7. K-best Disjoint Paths Problems (Cont’d) • Factors of selecting K-best disjoint paths: • Shortest paths (minimum delay) • Minimization of the bandwidth allocation (given the bandwidth demanded by users) • Maximization of network throughput • Comparisons • K-best paths problems: find k disjoint paths with minimum total path costs at the same time. • K-shortest paths problems: find the shortest path for K times, i.e., remove the shortest path once it is found. INFOCOM '97

8. Trellis Graph • A structured graph useful for formulating problems of diverse fields such as radar, sonar. • Some past works which find K-best paths through a trellis: • J. K. Wolf, A. M. Viterbi, and G.S Dixon, “Finding the best set of K paths through a trellis with applications to multitarget tracking”, IEEE Trans. AES 25, 1989. • D. Castanon, “Efficient Algorithm for Finding the K Best Paths Through a Trellis”, IEEE Trans. AES 26, 1990. INFOCOM '97

9. Objectives • Focus on transforming a network topology into a trellis graph. • Use the algorithm proposed by Castanon to find the K-best paths. --------------------------------------------------------------- • Error pattern: random failure. • Error dependency: independent. • Recovery mechanism: without. INFOCOM '97

10. Graph Modeling and K-best Path Problems Defining a Trellis Graph Formulate the K-best Paths Problem INFOCOM '97

11. Defining a Trellis Graph • A walk in a trellis is an alternating sequence of nodes and links, i.e., . • The length L(P) of a walk is the number of links in it. • A path is a walk in which all nodes are distinct( ). • A directed graph G=(V, E) is a structure consisting of a finite set of nodes and a finite set of links and , where each link is an ordered pair. INFOCOM '97

12. Defining a Trellis Graph (Cont’d) • A trellis is a directed graph G=(V, E), with nodes and directed links that satisfies following conditions: • The node set V is partitioned into L (mutually disjoint) subsets , such that • Links connect nodes only of consecutive subsets and , i.e., if , then and • L is the depth of the trellis, i.e., there’re L levels in the trellis. L=4, H=3 INFOCOM '97

13. Defining a Trellis Graph (Cont’d) • A K-trellis is a trellis graph with 2 additional properties: • It has 2 more nodes and , such that , for every and , for every , . • The node of the set is connected (where possible) with K=2g+1 nodes of set , where , and . • The depth of a K-trellis graph will be equal to L+2 • Through this paper, a trellis graph is a K-trellis graph with K=H. INFOCOM '97

14. Defining a Trellis Graph (Cont’d) Depth=5+2=7 • Add 2 nodes s and t into the trellis, and connect them to the graph. • Numbers (K) of nodes every node links to are the same, and the form of connection is symmetric. A K-trellis graph with L=5, H=4 and K=3. INFOCOM '97

15. Formulate the K-best Paths Problem • Define link cost c(i, j) where Q is the node metric, and D is the link metric. • Define cost c(P) of a path P ( ) where is the cost of link . • The shortest path from the node to node is a path 111111111111111with minimum cost. INFOCOM '97

16. Formulate the K-best Paths Problem (Cont’d) • Let G=(V, E) be a trellis graph with L H nodes, i.e., |V|=LH. • Two paths and are said to be mutually exclusiveor unmergedif for all ; mergedotherwise. • Refer to unmerged paths asdisjointpaths. INFOCOM '97

17. Formulate the K-best Paths Problem (Cont’d) • Problem:Find K paths , through the trellis G=(V, E) which minimize the total cost. subject to are mutually disjoint. INFOCOM '97

18. Formulate the K-best Paths Problem (Cont’d) • Canstanon’s work • the problem of finding the K-best paths through a trellis graph can be defined asa Minimum Cost Network Flow (MCNF) problem. • Time complexity of worse case is . • Different from “finding K-successively shortest paths”. • If there’re m disjoint paths between an OD pair, they can be exactly computed. *From: D. Castanon, “Efficient Algorithm for Finding the K Best Paths Through a Trellis”. INFOCOM '97

19. Formulate the K-best Paths Problem (Cont’d) *From: D. Castanon, “Efficient Algorithm for Finding the K Best Paths Through a Trellis”. INFOCOM '97

20. Formulate the K-best Paths Problem (Cont’d) • Problem: Find , which minimize the total cost. , is the cost of arc(I, j) subject to: where is the variable representing the flow on arc (I, j) in *From: D. Castanon, “Efficient Algorithm for Finding the K Best Paths Through a Trellis”. INFOCOM '97

21. Problem Transformation 5 Main Steps Operation P1 & P2 An Example INFOCOM '97

22. 5 Steps for Transforming the Network Topology into a Trellis Graph • Partition • Disconnect • Apply (operation P1 & P2) • Merge • Complete INFOCOM '97

23. 1. Partition • Define L(G, v) where is the length of the partition , v is a node . • Define A(v, l) L(G,s)={AL(s,1), AL(s,2), AL(s,3)}, where AL(s,1)={A, B, C}, AL(s,2)={D, E, F}, AL(s,3)={J, K}. INFOCOM '97

24. 2. Disconnect • Disconnect the network into 2 sub-networks G’ and G”. N(t)=all the nodes linking to t G’ G” {t}+ N(t) V-{t} INFOCOM '97

25. 3. Apply Operation P1 & P2 • If there’re any “vertical” links in G’, operation P1 & P2 will be applied to eliminate “vertical” links. • The operations are based on the addition of dummy nodes and 0-cost links. INFOCOM '97

26. 3. Apply Operation P1 & P2 (Cont’d) • Definition for the shortest path from node s to a node y through a specific link: • Let x, y be 2 nodes of consecutive levels which are connected by a link. The s-cost of a link (x, y) is defined to be the minimum cost of the path from s to y through node x, i.e., the cost of the path P={s,…, x, y}, and is denoted byΦ(y, x). • Find the s-cost of every link until a “ vertical link ” is found. INFOCOM '97

27. Operation P1 • Let G(V, E, c) be a network partitioned into adjacency-levels and let be a link, where nodes x, y are on the same level l, i.e., . Let x be the node satisfying the following properties: INFOCOM '97

28. Operation P1 (Cont’d) • Replace node x with a dummy node x’, move node xinto level l+1 and update the following parameters: Min s-cost of x=5 Min s-cost of y=5 Sum of s-cost of x=14 Sum of s-cost of y=13 INFOCOM '97

29. Operation P2 • Let G(V, E, c) be a network and let x be a node at level I which, after operation P1, remains without neighborhoods in level I - 1 , i.e., • move node x into level I +1and update the following parameter: INFOCOM '97

30. 4. Merge • Merge the resulting graph G’ with graph G” by adding necessary dummy nodes and 0-cost links. INFOCOM '97

31. 5. Complete • If it’s necessary, add more dummy nodes with infinite link cost to make the trellis graph completed. INFOCOM '97

32. Example 4(B)>1(C) Step 1 Step 2 INFOCOM '97

33. Example (Cont’d) E has no neighbors on level 1 11(J)>7(E)>3(K) Step 3 INFOCOM '97

34. Example (Cont’d) 11(J)>7(E) 7(E)>3(K) Step 3 INFOCOM '97

35. Example (Cont’d) Step 4 Step 5 INFOCOM '97

36. Conclusions Cutpoint Problem Conclusions Future Works INFOCOM '97

37. Cutpoint Problem • If there’s a cutpoint (also a critical point) in a trellis graph, it’s impossible to find K-best disjoint paths. • Try to find the K-best paths which are as diverse as the network topology allows. • Break up the trellis graph at the cutpoint to form 2 trellis subgraphs, and find the K-best disjoint paths of each subgraph. • Concatenate 2 subgraphs to obtain the K-best paths for the complete graph. cutpoint INFOCOM '97

38. Conclusions • Use graph theoretic techniques to address network survivability issues by finding the K-best (disjoint) paths through a trellis graph. • Transform any network topology into a trellis graph, which can be used to find the K-best paths for any O-D pair in a given network topology. INFOCOM '97

39. Future Works • Enhance the efficiency of the algorithm. • Apply the knowledge of K-best paths and trellis graph transformation in a real network. INFOCOM '97

40. Personal Conclusions • There’ll be some more operations needed when transforming. • It can’t be applied to scale-free networks well, for the possibility of “cutpoint problem” is high. INFOCOM '97