Lexicographic products in metarouting Alexander Gurney & - - PowerPoint PPT Presentation
Lexicographic products in metarouting Alexander Gurney & Timothy Griffin University of Cambridge ICNP 2007 Beijing Lexicographic choice is important (BGP , OSPF , ...) But it doesnt always work: not all metrics allow us to find
ICNP 2007 – Beijing
Lexicographic choice is important (BGP , OSPF , ...) But it doesn’t always work: not all metrics allow us to find best paths. When precisely can we find globally or locally
lexicographic choice of exterior metric followed by interior metric
delay 20 30 + = 50 300 min 400 = 300 bandwidth smaller is better bigger is better (N, +, ≤) (N, min, ≥)
compute path weights from link weights a ⊗ b ⊗ c ⊗ d = ...
and compose them along a path compare path weights p ≤ q ?
(like min, max) to choose the best path that makes four possible families of structures p ~ q = same preference p # q = incomparable
(10, 4) (10, 2) (20, 2) (20, 7) choose (20, 9)
have (10, 11) and not (10, 6)
x y z
bandwidth delay – not monotonic delay bandwidth – is monotonic hop-count delay reliability bandwidth – ???
lexicographic product
“one-to-one” or “cancellative”
“constant”
Proved for total orders: Tôru Saitô 1970
integers with addition paths with concatenation sets with disjoint union probabilities with multiplication BUT NOT integers with min
choose left operand “everything’s equivalent”
delay reliability bandwidth constant × × × OK! delay reliability bandwidth × × NOT OK (need at least one N or at least one C)
distinct areas delay timestamp sequence × × 08:22:01 ⊗ [...] = [08:22:01, ...] everything is weighted the same reliability × 0.98 × 0.96 = 0.9408 higher probabilities are better
18 + 30 = 48 lower delays are better
{15} ∪ {12, 3, 100} = {15, 12, 3, 100} use the subset order {15} ∪ {3, 15} = {3, 15}
exterior link reset interior metric interior link preserve exterior metric
a ⊗ b = a a ⊗ b = b
C(LEFT(T)) always N(RIGHT(S)) always
z ⊗ x = z ⊗ y = z z ⊗ x = z ⊗ y ⇒ x = y
bandwidth delay bandwidth delay delay
The scoped product is more permissive than the
We can find global optima this way
x < z ⊗ x
INCREASING
x ≤ z ⊗ x
NONDECREASING
ND( S T ) ⇔ I(S) ∨ (ND(S) ∧ ND(T)) I( S T ) ⇔ I(S) ∨ (ND(S) ∧ I(T))
OK: (10, 4) (10, 2) (20, 2) (20, 7)
bandwidth-delay yields local optima just fine for path-vector
(need at least one I) based on results by João Sobrinho
Lexicographic choice is ubiquitous We have an easier way of telling whether or not a particular product will work This extends to more elaborate situations, like the scoped product Aids exploration of the routing metric design space
What about other ways of combining metrics? EIGRP: use the formula “d + k/b” to compare How do we give structure to this technique? What properties do we need to track?