Compact%20Routing%20Schemes

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Compact Routing Schemes

• Mikkel Thorup Uri Zwick
• ATT Labs Research
• Tel Aviv University

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Routing
3
v
2
u
1
3
Handshaking
header(u,v)
Packet
information
header(u,v)
The same header is used for all messages sent
from u to v
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Routing in Trees
u
v
5
Routing in General Graphs
Handshaking? Table Size Stretch
no n1/2 3
yes n1/3 5
no n1/3 7
yes n1/k 2k-1
no n1/k 4k-5
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Previous Results
Authors Table Size Stretch
Cowen 99 n2/3 3
Eilam, Gavoille Peleg 98 n1/2 5
Awerbuch Peleg 92 n1/k O(k2)
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Our Results Are Essentially Optimal!

• Labels must be at least log2n bit long.
• In graphs, for stretchlt3, the total size of the
  routing tables must be ?(n2). For stretchlt5, the
  total size must be ?(n3/2).
• Conjecture For stretchlt2k1, the total size of
  the tables must be ?(n11/k). (Equivalent to a
  well known girth conjecture of Erdös.)

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Tree Routing A Practical Scheme

• O(log2n)-bit labels. Arbitrary port numbers.

1
10
7
2
10
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DFS numbering For every vertex u, let fube the
largest descendantof u. Then v is a
descendantof u iff
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3
14
13
4
6
5
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A trivial solution with O(deg(v)) memory.
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8
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Tree Routing A Practical Scheme (Cont.)

• Let s(v) be the number of descendants of v.

Let pv be the parent of v. Then, vertex v is
heavy if s(v)?s(pv)/2, and light otherwise.
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Tree Routing A Practical Scheme (End)
r
0
The light-level lv of a vertex v is the number
of light vertices on the path to it from the
root. Claim lvltlog2n label(v)(v,port(e1),port(
e2),) At v we storev, fv, hv, lv, port(v,pv)
and port(v,hv).
e1
1
e2
2
2
e3
3
3
e4
4
v
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Routing in Graphs
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Choose a Set of Centers
centA(v) a center closest to v
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Construct Clusters
cluster
clusterA(v) vertices that are closer to v than
to all centers.
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Keep Routing Info from v to A?clusterA(v)
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If v?clusterA(u), Route Directly
u
v
w
For any w on the shortestpath we have
v?clusterA(w).
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If v?clusterA(u), Route through centA(v)
centA(v)
Label(v) (v,centA(v),port(centA(v),v))
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How do we choose centers?

• We want A such that
• AO(n1/2)
• clusterA(v)O(n1/2), for every v
• Cowen does this with O(n2/3)

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Algorithm center(G) A?? W?V While W??
A?A ? choose(W,n1/2) W?w?V
clusterA(w)gt4n1/2 Return A
The expected size of A is O(n1/2log n).
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Smaller Tables, Larger stretch

• Use a hierarchy of centers.
• Construct a tree cover of the graph.
• Identify an appropriate tree from the cover and
  route on it.

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Tree Cover
Each vertex contained in at most n1/k trees. For
every u,v, there is a tree with a path of
stretch at most 2k-1 between them.
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Is there a routing scheme with

• Table size O(n1/k)
• Label size O(log n)
• No handshaking

???