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

Introduction QoS constraint ：

Link-constraint (bandwidth) Path-constraint (delay, cost, ..)

k-constrained routing ： Refers to multiconstrained QoS routing p

roblems with exactly k path constraints. Is known to be NP-hard.

Assumptions and Notations Directed graph G(N,E), N : nodes, E : edges For each edge e=u→v, wl(e) Rє + and wl(e)>0 f

or 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

n

i iill vvwpw1 1 )()(

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).

Example

Non-optimal

Optimal

Optimal

The number of optimal QoS paths from node scr=0 to dst=3k is equal to 2k.

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)

Limited Granularity Heuristic Basic idea ：

Use bounded finite ranges to approximate QoS metrics.

Reduce NP-hard problem to be solved in polynomial time.

Limited Granularity Heuristic This heuristic approximates k-1 metrics w

ith k-1 bounded finite values. For 2≦i≦k, the range (0,ci] is mapped int

o Xi elements, ri1,ri

2,…,riXi

, where 0<ri1<ri

2<…<ri

Xi=ci.

The wi(e) (0,є ci] is approximated by rij if an

d only if rij-1<wi(e)≦ri

j. awi(p) ： denote the approximated wi(p)

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 recor

ds the path that has the smallest w1 weight among all paths p from the source to node u that satisfy wl(p)≦rl

il for 2

≦l≦k.

Limited Granularity Heuristic

Time complexity ： O(X2X3 … Xk)

Time complexity ： O(X|N||E|)

X=X2X3 … Xk

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.

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 {ri

l=(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.

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.

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.

Limited Path Heuristic For a set S of |S| paths of the same leng

th, we derive the probability probi that set S contains i optimal QoS paths.

When X=O(|N|2lg(|N|)), ΣXi=1probi is very l

arge (or Σ|S|i=X+1probi is very small), whic

h indicates the heuristic have very high probability to record all optimal QoS paths in each node.

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 Pk

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)

Limited Path Heuristic

Amk(|S|,0) ： The probability that the set S

contains exactly m optimal QoS paths.

Limited Path Heuristic To determine the value X such that

Σ|s|m=X Am

k(|S|,0) is very small. Theorem ： Given a N-node graph with k in

dependent 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)

Simulation

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

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.

Simulation

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

Degradation

Simulation

2-constrained problems on (a) 8*8 meshes (b) 16*16 meshes by limited path heuristic.

Almost the same

Simulation

3-constrained problems on 8*8 meshes.(a) limited granularity heuristic (b) limited path heuristic.

Increase dramatically

Increase

slightly

Simulation

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

Simulation

Conclusion The limited granularity heuristics must mai

ntain 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.