Establishing Alternate Routes in Networks and Routes Between Polygons in 2D Space

1. Establishing Alternate Routes in Networks and Routes Between Polygons in 2D Space Amit M Bhosle Department of Computer Science University of California, Santa Barbara

2. Overview • Bridges connecting polygons • Euclidean bridges: Definition and Results • Geodesic bridges: Definition and Algorithm • Transient failures and proactive recovery schemes • Single link/ node failure recovery (SLFR/SNFR): Problem definitions and our results • Distributed Algorithm: SNFR, SLFR and others • Open Problems

3. Bridges Connecting Simple Polygons

4. Bridge Connecting Simple Polygons • Two islands, to be connected by a bridge • Flyover-like bridges, and end points need not be visible from each other • Instead of a bridge, we may want to find the route for a ferry that connects the two islands • The (non-amphibian) ferry needs to stay in the water !

5. Bridge Connecting Simple Polygons • P & Q are two simple, disjoint polygons • Bridge (p, q), with p єρ(P), qєρ(Q) • ρ() defines the region enclosed by the boundary of the polygon • Weight of the bridge (p,q)is: w (p,q) = gd(p, P) + Φ(p,q) + gd(q, Q) • gd(x, X): Geodesic distance between x and its geodesic furthest neighbor in X • Φ(p,q): d(a,b) or gde(p,q), depending on the problem. • Problem asks to find a bridge connecting P and Q which has minimum weight

6. Simple Polygons: Euclidean Bridges P Q p q gd(p,P) = gd(p,p’) gd(q,Q) = gd(q,q’) p’ q’ w(p,q) = gd(p,P) + d(p,q) +gd(q,Q)

7. Simple Polygons: Geodesic Bridges P Q p q gd(p,P) = gd(p,p’) gd(q,Q) = gd(q,q’) p’ q’ w(p,q) = gd(p,P) + gde(p,q) + gd(q,Q)

8. Visible Bridges y y w(opt visible bridge) ~ 2x + 2y w(opt bridge) ~ x + 2y Q x P z An opt bridge’s end points may not be mutually visible: a restriction imposed/assumed in earlier results on the bridge problem Opt visible bridge can be found in O(n log3 n) time. Tan [IJCGA ’02] x >> y >> z

9. Opt Bridge has End Points on the Polygon Boundaries p q P r Q r’ q’ gd(q,Q) = gd(q,q’) gd(r,Q) = gd(r,r’) • Bridge (p,r)can be shown to be no worse than the bridge (p,q) • gd(r,r’)≤ d(r,q) + gd(q,q’) • Proof holds for geodesic bridges as well

10. Optimal Bridges: Simple Polygons • We narrow the set of points that can support an optimal bridge to O(n2) anchors on each polygon’s boundary • These O(n2) anchors can be found in O(n2 log n) time • Geodesic furthest point Voronoi Diagram answers the gd(x,X) queries in O(log n) time (can also use Suri’s algorithm [JCCS ’89] to precompute these)

11. Optimal Euclidean Bridges: Results • An O(n2 log n) time algorithm for the Optimal Euclidean bridge problem • The point to Polygon version of the Euclidean bridge problem can be solved in O(n log n) time • Approximately optimal bridge, within a factor of 1+2/(k+1) of the optimal, in O(kn log n) time • An FPTAS for the Euclidean bridge problem

12. Geodesic Bridges: Point to Polygon p Q • O(n2) candidate bridge end points on Q • Build the SP tree of p to all the O(n2) points in O(n2 log n)time using Hershberger and Suri’s algorithm [SIAM J. ’99] • Precompute geodesic furthest neighbors of each of the O(n2) points in O(n2 log n) time q’ q Works. But for the 2 polygon case, will need ~ O(n4)time (since we have O(n2) anchors on each polygon)

13. Geodesic Bridges: Point to Polygon p q Q • Either (p,q) is a visible bridge • Can be found in O(n logn) time q0 q • OR • The 2nd last vertex of gd(p,q) (q0in fig) is a • vertex of Q. • And, (q0 ,q) is optimal visible bridge from q0to Q (sub-path optimality)

14. Optimal Geodesic Bridge: Point to Polygon • From p and each vertex of Q, precompute the optimal visible bridge to Q in O(n2 log n) total time • From p, find the shortest paths to all vertices of Q: O(n log n) time • Select best among vis(p,Q) and MIN q єV(Q){gd(p,q) + vis(q,Q)} • Total of O(n2 log n) time

15. Geodesic Bridges: Simple Polygons P Q p q Case I: (p,q) is a one-link bridge => (p,q) is a visible bridge. An optimal visible bridge between P and Q can be found in O(n log3 n) time using Tan’s algorithm.

16. Geodesic Bridges: Simple Polygons P Q q p x • Case II: (p,q) is a two-link bridge, pivoted at a vertex, say x • (x, p) is an optimal visible bridge from x to P • (x, q) is an optimal visible bridge from x to Q • There are a total of 2n such bridges, that can be found in O(n2 log n) total time

17. Geodesic Bridges: Simple Polygons q0 P Q q p p0 • Case III: (p,q) is a multi-link bridge • the second and second-last vertices of gde(p,q), p0and q0, are vertices of P and/or Q • (p0 , p) is an optimal visible bridge from p0to P • (q0 , q) is an optimal visible bridge from q0to Q • There are a total of O(n2)such bridges, that can be found in O(n2 log n) total time

18. Optimal Geodesic Bridges: Simple Polygons • Each of the three types of bridges, one-link, two-link and multi-link, can be found in O(n2 log n) time or less. The best among them is an optimal geodesic bridge between the two polygons. • Therefore, we can find an optimal geodesic bridge in O(n2 log n) time • The time bound matches our bound for finding an optimal Euclidean bridge between two simple polygons • The point to Polygon version of the geodesic bridge problem can be approximated to a factor of 1+2/(k+1) in O(kn log n) time

19. Some Open Problems Related to Bridges • Faster exact algorithms for the optimal Euclidean bridge and optimal geodesic bridge problems? • o(n2)-time approximation algorithms for the geodesic bridge problem?

20. Finding Alternate Paths in Networks

21. Failures in IP Backbones • Five 9s availability (99.999% uptime) is highly desirable • VOIP, VPN, Banking, Multimedia Conferencing, Shopping, … • Failures occur quite frequently • Faulty interfaces, router/link failures, maintenance activity • Markopolu, et al. (INFOCOM 2004) characterized the failures in IP backbones • About 20% failures due to planned activity (e.g. maintenance) • Most unplanned failures in communication networks are transient and affect a single element • 46% last less than 1 minute • 86% last less than 10 minutes • 85% of all failures affect a single link or node • Handling transient single node or link failures is key to ensuring high availability

22. Proactive Recovery Schemes • Popular technique of handling transient failures • Precompute alternate paths that avoid the failed element • When a failure actually occurs • Suppress the failure, instead of the traditional approach of advertising it across the network (and hope that it is a transient issue) • Use the precomputed alternate paths to reroute traffic (and survive the failure)

23. Single Link Failure Recovery (SLFR) • Given an edge weighted graph G (V,E), and the shortest paths tree Ts = {e1 , e2 ,…, en-1}of a node s, where ei= {xi , yi} and xi is the parent of yiin Ts • For 1 ≤ i ≤ n-1, find the shortest path from yi to s in the graph G-i = G \ ei • G-i is obtained from G by deleting the edge ei • G assumed to be 2-edge-connected, and undirected

24. Single Node Failure Recovery (SNFR) • Given an edge weighted graph G(V,E), and the shortest paths tree Tsof a node s, where Cx= {x1 , x2 , x3 ,…, xk}denote the children of x in Ts • For each node x є G, and x ≠ s, find a path from each xiєCxto s in the graph G-x = (V \ x, E \ Ex) • G-x is obtained from G by deleting the node x and all its edges • G assumed to be 2-node-connected, and undirected

25. SLFR SNFR s s x x y xi xj xk

26. SLFR, SNFR: Naïve Algorithm • Recompute the shortest paths tree of s separately in each of the n-1 failure graphs (G-igraphs for SLFR problem, and G-xgraphs for SNFR problem) • Time Complexity: Since we have O(n)failure graphs, the naïve algorithm requires O(n(m+nlog n))time • Simple, but inefficient (at least for undirected graphs)

27. SLFR, SNFR: Our Results (Centralized Algorithms) • SLFR: O(m + nlog n)time for computing optimal alternate paths • Undirected, planar graphs: linear time • Undirected, integer edge weights: O(m + Tsort(n)) • Some improvements for replacement-paths problem • Given an s-t shortest path, find alternate s-t paths if each edge of the original s-t shortest path fails, one at a time • SNFR: O(m log n)time algorithm • Computes good alternate paths – within 15% of optimal for randomly generated graph with 100 – 1000 nodes, and an average degree of 10 – 30 • Protocols: A new proactive recovery scheme that uses the paths computed by these algorithms to recover from transient single element failures

28. SNFR Algorithm: Empirical Evaluation

29. Single Node Failure Recovery:A Distributed Algorithm? • All previous algorithms, including our earlier algorithms, are centralized algorithm that need complete information of the Network graph as input to the algorithms • Resource intensive – need a single computer with enough processing power, and memory to deal with potentially huge network graphs • Manually intensive • Need a consistent snapshot of the network • Results of the computation need to be deployed to the corresponding hosts • Especially desirable if the other steps performed at the Network setup are distributed algorithms • Thesis proposal question – can our ideas from the centralized algorithms be used for designing a distributed algorithm for computing alternate paths?

30. SNFR: Key Properties s G x xj xi x1 gj bq bp gi • Green edges (e.g. gi, gj) have one end point outside the subtree of x. • giand gjcan be used by xiand xj respectively to reach s. • Blue edges (e.g. bp, bq) run between subtrees of siblings. • Together with green edges, they can offer a valid alternate paths. • E.g. bpoffers an alternate path to x1(either bq gj, or directly using gi).

31. SNFR: Constructing Rx s G x xj x1 xi gj bq bp gi sx sx yi yj y1 yi yj y1 Rx Tsx

32. SNFR • Each node needs the following information • Cheapest green edge for each child node • Blue edges among the subtrees of the child nodes • An edge is green for a node if it has one end point in the node’s subtree, and the other outside the node’s parent’s subtree • An edge is blue in Rx if x is the nearest-common-ancestor of the two end points of the edge

33. DFS Start / End Labels (1,14) (12,13) (2,11) u w (5,10) (3,4) (8,9) (6,7) v • Assign start and end labels to nodes as a depth-first-search traversal of the tree starts or ends at the node • A node v belongs to the subtree of u iff dfsStart(u) < dfsStart(v) < dfsEnd(v) < dfsEnd(u) • Green edges have exactly one end point in the node’s subtree

34. Distributed SNFR: Gathering Information to Construct Rx (1,200) s (2,101) x (102, 199) xi (3,50) xj (51,100) b v u (80,91) (10,21) • M(u,v): • (10,21) – (80,91) • loop weight: d(s,u) + cost(u,v) + d(v, s)

35. Distributed SNFR: Gathering Information to Construct Rx (1,200) s (2,101) x (102, 199) xi (3,50) xj (51,100) z b v g (110,151) u y (60,71) • M(y,z): • (60,71) – (110,151) • loop weight: d(s,y) + cost(y,z) + d(z, s) Forward a child node’s edge to your parent if and only if it is a green edge

36. Distributed SNFR Algorithm • DFS labels computed using the algorithm by Fraigniaud, et. al. [PODC ’00] • The root node, s, initiates the SNFR algorithm by sending a control message to all nodes – asking them to start forwarding their non-tree edges • After a leaf node has forwarded all its non-tree edges, it sends a special DONE message to its parent to indicate that it has finished fwd’ing all non-tree edges • An intermediate node sends a DONE message to its parent after it is done, and it has received DONE from all its children • Once a node x has received DONE from all its children, it has enough information to construct Rxlocally, and finish the remaining steps of the alternate path computation algorithm

37. Distributed SNFR Algorithm: Summary • First distributed algorithm for computing alternate paths • Based on request-response model – does not need any global coordination among the nodes • Requires O(m+n)messages to be transmitted among the nodes, and has a space complexity of O(m+n)across all nodes (asymptotically optimal) • Can be generalized to the Single Link Failure Recovery and a few other related problems.

38. Some Open Problems Related to SLFR/SNFR • Faster algorithm and/or better alternate paths for SNFR problem • Our solution runs in O(m log n) time, and does not compute optimal alternate paths • Naïve upper bound for computing optimal alt paths: O(n(m + n log n)) • Directed graphs • Our upper bounds are strictly for undirected graphs •  (m n) lower bound holds for directed graphs in path comparison model (as long as m = O(n n) ) • Recently, replacement paths problem solved in O(n log3n) time for directed planar graphs [EPR ’08] • Multiple failures • Failure of any two links or nodes in the network? We presented an O(n3) algorithm for computing alternate s-t paths when the 2 failed links or nodes are on the original s-t path • Trivial lower bound of  (n2)

39. Thank You

40. O(n log n)Algorithm for Optimal Vertex Bridge • Vertex bridge is between the vertices of the 2 polygons • Additively weighted nearest-point Voronoi diagram • Each Voronoi point u has an associated weight wv • Query for point p returns a point v such that d(p, v) + wv = MIN u є V{d(p,u) + wu} • Can be built in O(n log n) time [Aurenhammer-Imai ’88] • Query can be answered in O(log n) time • Geodesic furthest neighbors of all vertices of a simple polygon can be found in O(n log n) time [Suri ’89]

41. Optimal Vertex Bridge (Contd.) • For each vertex q є Q, assign weight wq = gd(q, Q) • O(n log n) total time • For each vertex p є P, find the vertex q є Q with minimum d(p, q) + wq • This would be the best vertex bridge with p as one end point • Each query in O(log n) time • Check all vertices of P for the optimal vertex bridge • Total time O(n log n)