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 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.
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.
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