~1~ Infocom’04 Mar. 10th On Finding Disjoint Paths in Single and Dual Link Cost Networks Chunming Qiao* LANDER, CSE Department SUNY at Buffalo *Collaborators: Dahai Xu, Yang Chen, Yizhi Xiong and Xin He

~2~ Infocom’04 Mar. 10th Outline  The Min-Min Problem  Motivation and Definition  Existing and Proposed Heuristics  Application and Performance Evaluation  Summary

~3~ Infocom’04 Mar. 10th Finding Disjoint Path Pairs  Basic and important problem in survivable routing  The Min-Min Problem  Definition: Finding a link (node) disjoint path pair such that the length of the shorter path is minimized.  Applications  Encrypted data on the shorter path, and decryption key on the longer path  Shared Path Protection (use the shorter path as AP)  Counterpart problems  Min-Max  Min-Sum

~4~ Infocom’04 Mar. 10th Computational Complexities  Min-Sum (P) [Suurballe-74]  Min-Max (NP Complete) [Li-90]  Min-Min (P or NP Hard?)  NP Complete! proved by Xu et. al. in INFOCOM’04]  Reduction from a well-known NPC problem 3SAT  We also proved that it is NP-hard to obtain a k- approximation to the optimal solution for any k > 1

~5~ Infocom’04 Mar. 10th Solving The Min-Min Problem  Active Path First (APF) Heuristic  Finds a shortest path for use as AP, followed by searching a disjoint BP.  It may fail to find such a BP even though a disjoint path pair does exist.  K Shortest Path (KSP) Heuristic  First K shortest paths are found and tested in the increasing order of their costs (path lengths) to see if a disjoint BP exists.  Could be time-consuming

~6~ Infocom’04 Mar. 10th Inefficiency of KSP  Any path from s to d consists of two sub-paths in domain E1 and E2 respectively.  Links in E1 is much shorter than those in E2.  The number of all possible sub-paths in E1 is very large 1st 2nd

~7~ Infocom’04 Mar. 10th Proposed Approach  Find a shortest AP first (as in APF)  If the AP doesn’t have a disjoint BP, determine the “conflicting link set” that are causing the problem  Try another AP without using these problematic links

~8~ Infocom’04 Mar. 10th Conflicting Link Set  Definition  A minimal subset of the links on AP such that no path using ALL these “problematic” links can find a disjoint counterpart, e.g., e1 and e2 in the following example.  The Min-Min problem can be solved much faster by avoiding using at least one link in the conflicting link set for the next shortest AP.

~9~ Infocom’04 Mar. 10th Divide and Conquer  Let P(I, O) be the problem of finding a disjoint path pair where AP must use the links in set I (Inclusion) but not the links in O (the Exclusion set).  Denote the original Min-Min problem by P( ,  )  Find a shortest AP; If no disjoint BP, find the Conflicting Link Set  Divide P( ,  ) into sub-problems based on the conflicting link set, e.g., P( , {e1}) and P({e1},  ) in the previous example.  The same procedure may be applied recursively on these sub-problems, e.g., P({e1},  ) can be further divided into P({e1},{e2}) and P({e1,e2},  ).  The definition of conflicting link set means that we do not need to try to solve P({e1,e2},  ).

~10~ Infocom’04 Mar. 10th The Proposed Conflicting Link Exclusion (COLE) Heuristic  An algorithm to find the conflicting link set (to be discussed)  Usually has fewer links than the half of the links on AP  Fewer sub-problems than KSP  “Divide and Conquer” based on the conflicting link set (rather than all the links on AP as in KSP)  Then pick a best solution (with a shorter AP) among those for the sub-problems.  Find a optimal or near-optimal result for each sub- problem  Each sub-problem may be solved recursively using Divide-and-Conquer

~11~ Infocom’04 Mar. 10th Solving the Sub-problem  Finding a shortest path consisting of certain links (e.g. set I) is itself NP-Hard  Approximation method to speed up the computation. [Xu et al. OFC’04]

~12~ Infocom’04 Mar. 10th Finding Conflicting Link Set  Finding a link-disjoint path pair between nodes s and d in graph G=(V, E) = Finding two unit-flows in a flow network where each link's capacity is set to 1 unit  Assume that the network is symmetrical  For the chosen AP, construct a new graph G 0  G 0 uses the same V and E of G  The capacity of the links in AP is set to 1  The capacity of the reverse links in AP is set to 0.  The capacity of all other links with non-zero capacity in G (except those in AP) is set to |AP|+1 (or a larger value).

~13~ Infocom’04 Mar. 10th Finding Conflicting Link Set (II)  Let Φ 0 =(S, D) be a min-cut of G 0, S={s, 3, 7} D={1, 2, 4, 5, 6, d}  The set of negative links (from D to S) on AP of Φ 0 is a Conflicting Link Set {e1, e2}

~14~ Infocom’04 Mar. 10th Reason for Not Using an Ordinary Min-Cut  Divide and conquer based on Ordinary Min-Cut might not help reducing the computational complexity.  AP 0 is the shortest path (and no link-disjoint BP exists)  AP opt is the shortest path with a link-disjoint BP  The min-cut: The partition S = {s}, positive links are a and b  Divide the original Min-Min problem into P( , {a}) and P({a}, {b}) (no solution in P({a, b},  ) )  Solving P( , {a}) leads to a non-optimal solution, and trying to solve P({a}, {b}) will again yield AP 0

~15~ Infocom’04 Mar. 10th Shared Path Protection  Two BPs can share backup bandwidth on a common link as long as their APs are disjoint (with a single failure)

~16~ Infocom’04 Mar. 10th Performance Evaluation  Solution to Min-Min problem (Single Link Cost Networks)  COLE will stop iteration after finding optimal result.  KSP can find the optimal result with a large enough K but has a longer running time than COLE  In both algorithms, the time for each invocation of the Dijkstra Algorithm to find the (next) shortest path dominates  Application to shared path protection (Dual Link Cost Networks)  COLE is compared with the optimal shared Min-Sum and optimal shared Min-Max solutions (based on ILP)  Tradeoffs between bandwidth overhead and recovery time.

~17~ Infocom’04 Mar. 10th Number of Dijkstra Invocations (Min-Min)  Net 1 (46 Nodes, 76 edges), Net 2 (79 Nodes, 108 edges), Net 3 (119Nodes, 190 edges)  KSP calls the sub-routine significantly more times than COLE, especially for large networks

~18~ Infocom’04 Mar. 10th Performance in Shared Path Protection Min-Sum Min-Min Min-Max  Bandwidth Overhead: Percentage increase in the total bandwidth (active + backup) required over the standard active bandwidth

~19~ Infocom’04 Mar. 10th Summary  The Min-Min problem is formulated and applied to shared path protection  The concept of Conflict Link Set is defined, which helps to solve the Min-Min problem fast  A novel heuristic algorithm COLE capable of solving the Min-Min problem faster than KSP is proposed  COLE is also found to be competitive in providing shared path protection.