1. Heuristic Algorithms for Multiconstrained Quality-of-Service Routing Xin Yuan, Member, IEEE IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 10, VO. 2, APRIL 2002

2. Outline Introduction Extended Bellman-Ford Algorithm Limited Granularity Heuristic Limited Path Heuristic Simulation Conclusion

3. Introduction QoS constraint ： Link-constraint (bandwidth) Path-constraint (delay, cost,..) k-constrained routing ： Refers to multiconstrained QoS routing problems with exactly k path constraints. Is known to be NP-hard.

4. Assumptions and Notations Directed graph G(N,E), N : nodes, E : edges For each edge e=u → v, w l (e)єR + and w l (e)>0 for all 1 ≦ l ≦ k w(e)=w(u → v)=(w 1 (e), w 2 (e), …, w k (e)) Assume for a path p=v 0 → v 1 → … → v n, w(p) ≦ w(q) ： w l (p) ≦ w l (q) for all 1 ≦ l ≦ k

5. Multiconstrained QoS Routing Multiconstrained QoS routing is to find a path p from src to dst such that w(p) ≦ c, that is w 1 (p) ≦ c 1, w 2 (p) ≦ c 2, …, w k (p) ≦ c k where k ≧ 2. A path p=src → v 1 → v 2 → … → dst is said to be an optimal QoS path from src to dst if there does not exist another path q form src to dst such that w(q)<w(p).

6. Example Non-optimal Optimal The number of optimal QoS paths from node scr=0 to dst=3k is equal to 2 k.

7. Extended Bellman-Ford Algorithm Extended Bellman-Ford algorithm (EBFA) for multiconstrained QoS routing problems. Executes the RELAX operation O(|N||E|) times Depends on the sizes of PATH(u) and PATH(v)

8. Limited Granularity Heuristic Basic idea ： Use bounded finite ranges to approximate QoS metrics. Reduce NP-hard problem to be solved in polynomial time.

9. Limited Granularity Heuristic This heuristic approximates k-1 metrics with k-1 bounded finite values. For 2 ≦ i ≦ k, the range (0,c i ] is mapped into X i elements, r i 1,r i 2, …,r i X i, where 0<r i 1 <r i 2 < … <r i X i =c i. The w i (e)є(0,c i ] is approximated by r i j if and only if r i j-1 <w i (e) ≦ r i j. aw i (p) ： denote the approximated w i (p)

10. Limited Granularity Heuristic Each node u maintains a table d u [1:X 2,1:X 3, …,1:X k ] with X=X 2 X 3..X k elements. An entry d u [i 2,i 3, …,i k ] in the table records the path that has the smallest w 1 weight among all paths p from the source to node u that satisfy w l (p) ≦ r l i l for 2 ≦ l ≦ k.

11. Limited Granularity Heuristic Time complexity ： O(X 2 X 3 … X k ) Time complexity ： O(X|N||E|) X=X 2 X 3 … X k

12. Limited Granularity Heuristic Lemma I ： In order for the limited granularity heuristic to find any path of length L that satisfies the QoS constraints, the size of the table in each node must be at least L k-1. That is X=X 2 X 3 … X k ≧ L k-1. (by using aw i (p(n)) ≧ r i n ) For a N-node network, paths can potentially be of length N. Thus, each node should at least maintains a table of size O(|N| k-1 ). It is quite sensitive to the number of constraints k.

13. Limited Granularity Heuristic Lemma II ： Let n be a constant, X 2 =X 3 =…=X k =nL so that X=X 2 X 3 … X k = n k-1 L k-1. For all 2 ≦ i ≦ k, let the range (0, c i ] be approximated with equal spaced values {r i l =(c i /X i )*l}. The limited granularity heuristic guarantees finding a path q that satisfies w(q) ≦ c if there exists a path p of length L that satisfies w 1 (p) ≦ c 1 and w i (p) ≦ c i -(c i /n), for 2 ≦ i ≦ k. When each node maintains a table of size n k-1 |N| k- 1 =O(|N| k-1 ) and when n is a reasonably large constant, the heuristic can find most of the paths that satisfy the QoS constraints.

14. Limited Path Heuristic Basic idea ： Maintain a limited number of optimal QoS paths, say X optimal QoS paths, in each node. X corresponds to the size of the table maintained in each node in the limited granularity heuristic.

15. Limited Path Heuristic We check the size of PATH(v), which is X, before a path is inserted into. We prove that X=O(|N| 2 lg(|N|)) is sufficient to supply high probability to solve general k-constrained problems.

16. Limited Path Heuristic For a set S of |S| paths of the same length, we derive the probability prob i that set S contains i optimal QoS paths. When X=O(|N| 2 lg(|N|)), Σ X i=1 prob i is very large (or Σ |S| i=X+1 prob i is very small), which indicates the heuristic have very high probability to record all optimal QoS paths in each node.

17. Limited Path Heuristic Process ： 1. Choose path p with the smallest w1 weight from set S 2. Let set T include all non-optimal QoS paths q which w j (p) ≦ w j (q) for 2 ≦ j ≦ k. 3. Go to 1 with set S ’ = S-T P k i,j ： the probability of the remaining set size equal to j when the process is applied to a set of i paths and the number of QoS metrics is k. (0 ≦ j ≦ i-1)

18. Limited Path Heuristic A m k (|S|,0) ： The probability that the set S contains exactly m optimal QoS paths.

19. Limited Path Heuristic To determine the value X such that Σ |s| m=X A m k (|S|,0) is very small. Theorem ： Given a N-node graph with k independent constraints, the limited path heuristic has very high probability to record all optimal QoS paths and thus has very high probability to find a path that satisfies the QoS constraints when one exists, when each node maintains O(|N| 2 lg(|N|)) paths. (insensitive to k)

20. Simulation Network topologies (a) A 4*4 mesh (b) MCI backbone

21. Simulation Existence percentage ： The ratio of the total number of requests satisfied using the exhaustive algorithm and the total number of requests generated. Competitive ratio ： The ratio of the number of requests satisfied using a heuristic algorithm and the number of requests satisfied using the exhaustive algorithm.

22. Simulation 2-constrained problems on (a) 8*8 meshes (b) 16*16 meshes by limited granularity heuristic. Degradation

23. Simulation 2-constrained problems on (a) 8*8 meshes (b) 16*16 meshes by limited path heuristic. Almost the same

24. Simulation 3-constrained problems on 8*8 meshes. (a) limited granularity heuristic (b) limited path heuristic. Increase dramatically Increase slightly

25. Simulation 3-constrained problems on MCI backbone (a) limited granularity heuristic (b) limited path heuristic.

26. Simulation

27. Conclusion The limited granularity heuristics must maintain a table of size O(|N| k-1 ) in each node to achieve good performance, which results in a time complexity of O(|N| k |E|). The limited path heuristic only needs to maintain O(|N| 2 lg(|N|)) entries in each node. Both heuristics can solve k-constrained QoS routing problems with high probability in polynomial time.