The document presents a compact routing scheme that routes on power law graphs with additive stretch. It breaks the graph into a dense core and fringe. It uses known ultra-compact routing algorithms like Thorup-Zwick to route within trees covering the core and fringe. Node names are topology dependent and routing tables use O(e log2 n) bits. The scheme finds the highest degree node, grows a shortest path tree to define the core, and additional trees to cover the remaining fringe edges.

1. Compact Routing on Power Law
Graphs with Additive Stretch
-Arthur Brady & Lenore Cowen
Presented by – Meenakshi Tripathi

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BC Scheme
• Compact Routing with Additive Stretch on power-law graphs.
• Breaks graph into a dense core, of diameter d; the rest of the graph is
called the fringe.
• Let e be the number of edges that need to be removed from the fringe to
make it into a forest.
• Theorem. For any unweighted, undirected network, there is a routing
scheme that uses O(e log2
n)-bit headers and routing tables, and has
additive stretch d.
• Node names are topology dependent.
• Guarantees
– Additive stretch d
– Routing table sizes O(e log2
n)

3. BC Scheme
– find the highest-degree node in the graph (h)
– grow the shortest path tree rooted at it
– find the core, which is all nodes located within maximum distance d from
the highest degree node (I)
– grow small number (e) of trees to cover all the edges that do not belong
to the main tree and lie outside of the core (F)
•the larger d, the smaller e
– use known ultra-compact routing algorithms to route on these trees
– Constructed using a shortest path tree to route in the core, and e
spanning trees in the fringe, and a distance labeling to choose between
them.
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4. Compact Routing on Trees
 Given a tree T = (V,E), let TZT table[u] be the local routing
table assigned to vertex u & by TZT header[w] be the header
assigned to vertex w, by the Thorup-Zwick tree routing scheme on T.
 Let dT (u, v) represent the length of the unique (u, v)-path in T.
 By Peleg in tree T, each vertex v of T is assigned an O(log2
n)-bit
label lT(v), s.t. given the labels of any two vertices v,w in T, the
distance d(v,w) b/w them can be computed exactly
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5. Routing Scheme in Tree
Let T = (V,E) be any tree with root r.
1.Initially, store a routing table TZT table[u] at each node u & let TZT
header[w] be the header of node w by TZ scheme.
2. For each vertex u, add the label lT (u) to TZT table[u] to form
TZ’Ttable[u].
3. For each vertex v, add the label lT(v) to TZT header[v] to form
TZ’Theader[v].
4. Routing decisions are same as the TZ tree-routing scheme
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6. Routing Scheme in Tree
• Lemma: TZ’ uses O(log2
n)-bit headers, O(log2
n)- bit routing
tables, and allows any vertex u, given the header of any
destination v, to compute dT (u, v).
• dT(u, v) can be computed from lT(u) and lT(v) by Peleg scheme
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7. Routing Scheme in Tree
Define a set T of trees Ti to be the union of the following two sets:
1. a set of spanning trees {T' = T, T1, . . . , T|E'|} on G, constructed as follows:
• i <− 1.
• For each edge uv ∈ E’,
– Grow a single-source shortest path tree Ti rooted at u which includes uv.
– Increment i.
2. the connected components of TF , with each assigned an arbitrary root
vertex.
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8. Routing Scheme in Tree
• cfroute consists of four parts:
1)a preprocessing step, in which we construct temporary data structures using
TZ’,
2) a labeling step, in which nodes are assigned Headers
3) a storage step, in which a local routing table is constructed at each node
4) the routing procedure
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9. Routing Scheme in Tree
1) Preprocessing : Process each
tree Ti ∈ T using TZ’. Let
TZ’Titable[u]) be the routing table
for node u in tree Ti, and let
TZ’Tiheader[v]) be the header
assigned to node u in tree Ti.
2) Labeling : Assign to each vertex v
a list of pairs (i, TZ’Tiheader[v])
(ordered by increasing i), one for
every tree Ti which contains v.
3) Storage : The routing table stored
at each vertex u consists of a list
of pairs (i, TZ’Titable[u]) (ordered
by increasing i), one for every tree
Ti which contains u
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10. Routing Procedure
Given Source vertex u to a destination v, routing using cfroute
1.For each tree Ti containing both u and v, extract lTi (u) from
TZ'Titable[u], extract lTi(v) from TZ’Tiheader[v] & compute dTi (u, v)
according to the Peleg scheme.
2. Choose some tree Tj such that dTj (u, v) is minimized.
3. Route from u to v in Tj according to TZ’.
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11. Analysis of the routing
procedure
If dTF(u, v) > d(u, v) i.e. ∃ edge u‘v’ ∈ P, ∉ TF . Two cases exist –
1. If u', v' ∈ F, then because u'v' ∉ TF , u'v' ∈ E', so the preprocessing routine of
cfroute constructed a single-source
shortest path tree Ti spanning G with source u’ (or v’)
dTi (u, u')+dTi (u', v) = d(u, u')+d(u', v).
2. Now assume that u' ∈ I or v' ∈ I (or both) ∀ edges u'v' ∈ P which are not in TF .
P contains at least one vertex in I, so d(u, I) + d(I, v)<=|P| = d(u, v).
Hence dT (u, v) <= dT (u, h) +dT (h, v) <= [d(u, I)+ d/2 ]+[ d/2 +d(I, v)] <= d(u, v)+d.
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12. Hybrid Scheme
• Superimpose BC scheme and TZ scheme – hence a hybrid scheme
concatenate headers for both schemes, and store tables for both
schemes at every node.
• Packet is routed using which results in a shorter path to its
destination
• Better stretch than either the universal TZ scheme or BC scheme
Stretch is min{(1, d), (3, 0)} & RT size O(n1/2
log2
n) bits.
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13. Results
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14. Results
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