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Heuristic Algorithms for Multiconstrained Quality-of-Service Routing

Heuristic Algorithms for Multiconstrained Quality-of-Service Routing. Xin Yuan, Member, IEEE IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 10, VO. 2, APRIL 2002. Outline. Introduction Extended Bellman-Ford Algorithm Limited Granularity Heuristic Limited Path Heuristic Simulation Conclusion.

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Heuristic Algorithms for Multiconstrained Quality-of-Service Routing

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  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, wl(e)єR+ and wl(e)>0 for all 1≦l≦k • w(e)=w(u→v)=(w1(e), w2(e),…, wk(e)) • Assume for a path p=v0→v1→…→vn, • w(p)≦w(q)：wl(p)≦wl(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 w1(p)≦c1, w2(p)≦c2, …, wk(p)≦ck where k≧2. • A path p=src→v1→v2→…→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 Optimal Non-optimal Optimal The number of optimal QoS paths from node scr=0 to dst=3k is equal to 2k.

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

  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,ci] is mapped into Xi elements, ri1,ri2,…,riXi, where 0<ri1<ri2<…<riXi=ci. • The wi(e)є(0,ci] is approximated by rij if and only if rij-1<wi(e)≦rij. • awi(p)：denote the approximated wi(p)

  10. Limited Granularity Heuristic • Each node u maintains a table du[1:X2,1:X3,…,1:Xk] with X=X2X3..Xk elements. • An entry du[i2,i3,…,ik] in the table records the path that has the smallest w1 weight among all paths p from the source to node u that satisfy wl(p)≦rlil for 2≦l≦k.

  11. Limited Granularity Heuristic Time complexity：O(X2X3 …Xk) Time complexity：O(X|N||E|) X=X2X3 …Xk

  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 Lk-1. That is X=X2X3…Xk≧Lk-1. (by using awi(p(n))≧rin) • 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, X2=X3=…=Xk=nL so that X=X2X3…Xk= nk-1Lk-1. For all 2≦i≦k, let the range (0, ci] be approximated with equal spaced values {ril=(ci/Xi)*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 w1(p)≦c1 and wi(p)≦ci-(ci /n), for 2≦i≦k. • When each node maintains a table of size nk-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|2lg(|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 probi that set S contains i optimal QoS paths. • When X=O(|N|2lg(|N|)), ΣXi=1probi is very large (or Σ|S|i=X+1probi 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 wj(p)≦wj(q) for 2≦j≦k. • 3. Go to 1 with set S’ = S-T • Pki,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. Amk(|S|,0)：The probability that the set S contains exactly m optimal QoS paths. Limited Path Heuristic

  19. Limited Path Heuristic • To determine the value X such that Σ|s|m=X Amk(|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|2lg(|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 Degradation 2-constrained problems on (a) 8*8 meshes (b) 16*16 meshes bylimited granularity heuristic.

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

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

  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|2lg(|N|)) entries in each node. • Both heuristics can solve k-constrained QoS routing problems with high probability in polynomial time.

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