The document discusses landmark-based routing techniques for networks. It proposes a new routing scheme that improves upon an existing scheme with O(n2/3 log4/3(n)) table size and stretch of 3, by achieving a table size of O(n1/2 log(n)) with the same stretch of 3. This is done by using a new labeling technique that assigns each node a (1+o(1))log2n-bit label to enable constant time routing decisions based on the labels.

2.  In Routing schemes there is trade-off between the Routing table
size and stretch.
Stretch of path p(u; v) from node u to node v is defined as
|p(u; v) |/|d(u;v)| , where |d(u; v)| is the length of the
shortest u-v path.
 Naive Scheme : Each node holds the next hop to all nodes and
employs optimal routing
 Routing Table size is O(n log(n)) at each node
where n : number of nodes
 Stretch =1 ( optimum path)

3. By this Compact Routing Method
 Routing Table size bound - O(n2/3 log4/3 (n))
 Maximum stretch ≤ 3

4.  Nodes are connected with arbitrary weighted
undirected edges
 Reassign the node names in lexicographic
order, with bound O(log(n))
 Edges identified by port names, locally
relevant
 Concept of Landmark based routing

5.  Re-Labeling of nodes
 Storage in Routing Table
 Routing Procedure

6.  Every node name is represented as a
 Triplet –
( orig. node name, name of its landmark, edge from landmark to the
node)
eg . V <- (v, lv, elv(v)) where
elv(v) : is the edge from landmark to the node on shortest path,
lv : Landmark of v

7.  For each v ∈ V
lv argminl∈L d(l; v) //closest landmark to node v
 For each l ∈ L perform truncated-Dijkstra(nα )
 For each v ∈ VL
V <- (v; lv ; elv(v)) // the first link on the
shortest path from lv to v

8.  1) Extended Dominating Set – spans the
neighborhood of all nodes in the network,
obtained by Greedy Algorithm
 2) Nodes in neighborhood of maximum
nodes, selected as Landmark

9. 
V1
V2 V3
V4
V5
V6
V9
V7
V11
V10
V12
V8

10.  Find set D, the Extended Dominating Set, using Greedy Algorithm
(here α is a parameter s.t. 0<α <1)
Ǝ D ⊂ V such that
• |D| = O(n1-α log n)
• ∀ v ∈V , D ∩ Bv ≠ φ
Where Bv is the neighborhood of vertex ‘v’ of size nα
 Find set C, set of nodes which lie in neighbourhood of maximum nodes
C ⊂ V s.t. ∀ c ∈ C , |Rc| ≥ n(1+α)/2
 Set of Landmarks, L =D ∪ C

11.  For non-landmark nodes – information of
neighbour and all landmarks is stored .
 For landmarks - information of landmarks
only is stored , otherwise routing table is
huge at landmark

12. //For neighbours of node v
For each v ∈ V, perform truncated-Dijkstra(nα)
For each u reached from v:
If no landmark is on the path from v to u:
store(v, eu(v)) at u
// shortest paths from landmarks to every node
For each l ∈ L, perform full-Dijkstra(nα)
For each v ∈ V
Store (l, eu(l)) at u

13. V1
V2 V3
V4
V5
V6
V9
V7
V11
V10
V12
V8
NODE PORT
V2 1
V4 1
V3 2
V10 2
1
1
2
3
1
2
2
1
V3 4
….
4
1 2
3
neighbours
landmarks

14. At node u, a packet with destination (v; lv;elv(v))
is routed as –
 If u=lv (landmark)-> route along elv(v).
 If not, but (v; eu(v)) is in u's local routing table
-> route along eu(v).
 Else route along (lv ; eu(lv)).

15.  A set of landmarks (L) is |L|= O(n1-α log n + n(1+α)/2)
 If d(u; v) < d(lv ; v) then u is not a landmark and ∄
a landmark on the shortest path from u to v.
 For each v, lv is among v's n closest neighbors.
 For each x ≠ lv on the shortest path from lv to v,
(x; ex(v)) is stored at x.

16.  Let d(u; v) denote the length of the shortest
path from u to v. Then the routing algorithm
returns a path of length at most 3d(u; v).
 The local storage space used at each
node is O((n 1- α log n + n(1+α)/2) log n).
 Using α = 1/3 + (2 log log n)/(3 log n), we get the
bound for size as O(n2/3 log4/3 (n))

17. By Mikkel Thorup & Uri Zwick

18.  Suggests new routing for trees
 Improvement of Routing Table size
(in landmark based routing technique)
from O(n2/3 log4/3 (n)) to O(n1/2 log(n)) for stretch 3
 General Routing technique for Graphs

19. Stretch Table Size Handshaking?
3 O(n1/2) no
5 O(n1/3) yes
7 O(n1/3) no
2k-1 O(n1/k) yes
4k-5 O(kn1/k) no
New Routing Schemes
Authors Stretch Table Size
Cowen 3 O(n2/3)
Eilam,Gavoille 5 O(n1/2)
Awerbuch, Peleg O(k2) O(kn1/k)
Awerbuch O(k29k) O(kn1/k)
Previous Available Schemes

20. Each vertex is assigned a (1+o(1))log2n–bit
label.
Given label(u) and label(v), it is possible to
find, in constant time, the right edge to take
from u.
Similar result by Fraigniaud and Gavoille
[ICALP’01]
u
v

21. 4 5 6
1
2
3 8
10
9
11
7
12
DFS Enumeration of Nodes

22. 1
2
3
4
5 6
8
7
9
10
11
1-2
3-11
Stretch=1
RT=O(d log n)
Header= O(logn)
d= degree(node)
12

23. Single source shortest path routing:

24.  Root the tree arbitrarily
 Perform depth first enumeration of the vertices
 let fw be the largest descendent of w
 vertex v is descendent of w iff v ∈(w , fw)
 else is sent to parent of w using parent pointer of
w

25.  O(log2n)-bit labels.
Arbitrary port numbers.
DFS numbering:
For every vertex u, let fu
be the largest descendant of u. Then v is a
descendant of u iff 4
],[ ufuv
1
2 10
113
65
12
1413
7
98
10
7
A trivial solution with O(deg(v)) memory.

26.  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.
14
8 2
17
1 41
3
11
3
11

27. 0
1
2
2
3
3
4
The light-level lv of a vertex v is the number of light vertices
on the path to it from the root.
Claim: lv<log2n
label(v)=(v,port(e1),port(e2),…)
At v we store:
(v, fv, hv, lv, port(v,pv) and port(v,hv))
e1
e2
e3
r
v
e4

28.  Each vertex ‘v’ assigned (1+O(1))log2n-bit label
 Label is the only information stored at the vertex
 Label serves as header attached to messages sent
to the vertex
 Routing decision takes constant time

29.  Weight sv of a vertex v is number of descendents in the tree
 A child v’ is said to be heavy if sv’>sv/b Else light
 Light level lv is def as the number of light vertices on the path
from r to v
 Enumerate tree in depth first order – where light vertices
visited before heavy children
 Routing information stored at v =(v,fv,hv,Hv,Pv) = O(b) words
 hv is the first heavy child of v
 Hv -> array of heavy children of v
 Pv ->array of port no to parent & heavy nodes
 < v0,v1,v2,….,vk> where v0=r and vi is the light nodes from r
to node v , vk=v
 LV=(port(vi1-1), port(vi2-1),……, port(vilv-1))= O(logbn)

30.  Label(v)=(v , Lv)
 At node w for header (v,Lv)
 If w=v – done
 Else if v ∈(w , fw) – if not not a descendent
forward to parent of w using Pv[0]
 Else if descendent check v ∈(hw , fw) – search
Hw and get corresponding Pw
 Else light descendent – search Lv[lw]
 Eg. b=2
 ((v>=w && v <h) ? L[1] : P[v>=h && v<=f])

32. centA(v) = a center closest to v

33. clusterA(v) = vertices that are closer to v than to all centers.
cluster

35. u vw
For any w on the shortest
path we have v clusterA(w).

36. u
v
centA(v)
),(3)),(())(,(
),(2))(,(
),()),((
vuvvcentvcentu
vuvcentu
vuvvcent
AA
A
A
Label(v)=
(v,centA(v),port(centA(v),v))

37.  We want A such that
 |A|=O(n1/2)
 clusterA(v)=O(n1/2), for every v
 [Cowen does this with O(n2/3)]

38.  Weight sv of a vertex v is the number of
descendants in the tree
 v’ is the heavy child & v0,v1,v2,….,vd-1 be its
light children in decreasing order of weight
 sv’>sv0>sv1>sv2>sv3>……>svd
 All the strings are concatenated and stored,
masking bits are to identify the lengths of
each string
 label(v)=(v,Lv,Mv)
 Header size =3.4logn

39.  Code(s)=s.bin(||s||,|s|).bin(||s||,||s||)
 Label(v)=ID(v) + RT(v)
 ID(v) consists of –
 Binary representation of i, the index of heavy
path containing v
 String s corresponding to v¯ =v
 ID(T; v) =ID(Tv ; v):code(i):code(sj ):code(port( v; v)) if v ≠ v,
code(i):code(sj ) otherwise
 Label(v) = code(ID(v)):code(RT(v)):code(pnt(v))

40. Algorithm center(G)
A ; W V;
While W
{
A A choose(W,n1/2);
W {w V | clusterA(w)>4n1/2 };
}
Return A;
The expected size of A is O(n1/2log n).
Improvement over Cowen’s landmark based
routing scheme

41.  Use a hierarchy of centers.
 Construct a tree cover
of the graph.
 Identify an appropriate tree from the cover and route on it.
Generalized routing scheme

42. 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.

43.  Table size = O(n1/k)
 Label size = O(log n)
 No handshaking
???