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Routing Complexity of Faulty Networks
Omer Angel Itai Benjamini Eran Ofek Udi Wieder
The Weizmann Institute of Science
Routing in a Faulty Network
Node u knows the topology of the graph. Can choose a path to node v.
Each link survives independently with probability p . u has partial knowledge on the topology of the graph. How many links (edges) should u probe before a path
to v is found (if a path exists).
u v
G:Gp:
Routing in a Faulty Network
Local Router – an algorithm which: Starts at node u. Probes edges which it has reached. Outputs a path to v.
Local Routing Complexity of A (with respect to u,v): The random variable counting the number of probed edges until a path is found (given that a path exists). Interesting when is bounded away from 0.
Efficiency: a local algorithm is efficient if its complexity is polynomial in the diameter of the largest component of Gp.
u v
Pr[u » v]
Routing in a Faulty Network
The existence of short paths does not guarantee the ability of finding them. A cycle with a random matching has diameter O(log n)
[BC88]. Finding a path requires time [Kleinberg00]. On the other hand: The Small World Phenomenon…
Our perspective: fault tolerance of networks. Study the effect of random failures on routing. Related to percolation theory – studies the effect of
random failures on connectivity.
(n1=2)
O(logn)
Outline
The Hypercube Lower bound: if local routing is not efficient. Tight upper bound: if . For short paths exist but are hard to
find.
The Mesh Tight upper and lower bounds. Whenever short paths
exist (as a function of p), they can be found.
The importance of the locality assumption Local and non local routers may have exponential gap. Another example: tight analysis of Gn,p .
p<< 1pn
p>> 1pn
1n
< p< 1pn
The Faulty Hypercube – Some History
– The n-dimensional hypercube in which each edge fails independently with probability 1-p . If then w.h.p. is connected [Burtin77].
Disconnected w.h.p. when .
If then w.h.p. Hn can emulate Hn with constant slowdown [HLN85] (considered node failures). Implicit: local routing in is possible.
If then w.h.p contains a giant component [AKS82]. Sharpened by [BKL92],[BSH+04]. Diameter of giant component is . Short paths
exist. When all components are of size O(n) w.h.p.
H pn
1¡ p
H pn
p< 12
p> 12
p> 12 H p
n
H pn
p> (1+²)n¡ 1 H pn
poly(n)p< (1¡ ²)n¡ 1
The Faulty Hypercube
Question: What probabilities in the range allow local routing (inside the g.c.) with complexity polynomial in n ?
• Graph is connected.
• Emulation (and routing) possible
No giant component
Threshold for constant distortion embedding of Hn in [AB03]
1n
< p< 12
1n
1pn p= 1p = 0
12
H pn
Local Routing Phase Transition
Let 0 < < ½. Lower bound (for ):
Any local routing algorithm makes at least queries w.h.p. . Short paths exist but cannot be efficiently found!
Upper bound (for ):There exists a local routing algorithm that finds a path between u,v in poly(n) time with high probability.
2 (n¯ )p= n¡ 1=2¡ ¯
p= n¡ 1=2+¯
The Faulty Hypercube
• Graph is connected.
• Emulation (and routing) possible
• No giant component
Threshold for constant distortion embedding of Hn in [AB03]
Local routing in poly(n) queries
• No efficient local routing
H pn
p= 0 p= 11n
1pn
12
Lemma: Assume V . Denote:
-– v is connected to u inside S. Q – the number of queries of a local router from u
to v. For each e crossing the cut .
The Lower Bound Lemma
V = S [ ¹S, v 2 S
S ¹S
vu
e
f (u S» v)g
(S; ¹S) , Pr[(v S» e)] < ³
8t Pr[Q < t] · t³ + Pr[(u S» v)]
The Lower Bound Lemma – Simple Example
Lemma: Assume V . Denote:
-– v is connected to u inside S. Q – the number of queries of a local router from u
to v. For each e crossing the cut .
Double Tree (0<p<1): S = the bottom tree, . Lemma implies: for ,
V = S [ ¹S, v 2 S
v
u
S
³ = pn
f (u S» v)g
(S; ¹S) , Pr[(v S» e)] < ³
8t Pr[Q < t] · t³ + Pr[(u S» v)]
t = ² 1pn
Pr[Q < t] · tpn = ²
The Double Tree – u,v are connected
Double Binary Tree – 2 depth n trees joined at their leaves. A path u~v exists iff there is a
leaf w and mirroring paths .
The event {u~v}is tantamount to a branching process with p2. Path exists with constant
probability, when p is a constant > .
u
v
u » v
u » w;v » wf u » vg
1p2
Lower Bound Lemma proof – Relaxed Model
If , the algorithm stops successfully (complexity = 0).
When a cut edge is probed, its entire component in S is given to the algorithm for free. If this component contains v the algorithm stops successfully. S ¹S
v
(u S» v)
Assume:
For each probed edge ei entering S:
Lower Bound Lemma - Proof
u
v
C0
C1 C2
Ci
u 2 S
Pr[(ei » v) 2 SjC0; : : : ;Ci ¡ 1] · ³
· t³ S
¹S
Pr[Q < t] · Pr[Q < t j (u S¿ v)]+ Pr[(u S
» v)]
Hyper Cube - Lower Bound
Fix: (almost surely ).
For any two vertices u,v , any local routing algorithm (almost surely) makes at least queries to find a path between u,v.
p = n¡ 1=2¡ ¯ (u » v)
n¡ (n¯ =2)
Fix: (almost surely ).
Claim: #of paths s of length is at most .s
Applying the Lemma to the Hypercube
(v » x) 2 S `+ 2knk 2̀k !̀
p = n¡ 1=2¡ ¯ (u » v)
v x ye
¹SS
` = n¯ =28e2 S £ ¹S : Pr[(v S
» e)] < ³
) Pr[Q < t] · t³ + Pr[(u S» v)]
· 2n¡ (n¯ =2)³ = Pr[(v S» x)] ·
1X
k=0
pl+2knk l2k l!
Lemma: #of paths s inside S of length is at most .s
Proof: Let Ak be the set of such paths of length . A0 = l!
There exists a mapping between Ak and Ak-1 that maps at most Ak-paths into one Ak-1-path.
A path is a list of coordinate changes:
n possible coordinates and possible indices in the path.
Applying the Lemma to the Hypercube`+ 2k
nk 2̀k !̀
jAkj · n 2̀jAk¡ 1j =) jAkj · nk 2̀k !̀
`+ 2k
1 8 31 7 20 8 3 .... 12
`+1
A0 = !̀
n ¢̀ 2
¡`+1
2
¢
(v » x)
Fix: (almost surely ).
Claim: #of paths s of length is at most .s
Applying the Lemma to the Hypercube
(v » x) 2 S `+ 2knk 2̀k !̀
p = n¡ 1=2¡ ¯ (u » v)
v x ye
¹SS
` = n¯ =2
³ = Pr[(v » x) 2 S] ·1X
k=0
pl+2knkl2kl!
8e2 S £ ¹S : Pr[(v S» e)] < ³
· 2n¡ (n¯ =2)
= o(1)) Pr[Q < t] · t³ + Pr[(u S
» v)]
Applying the Lemma to the Hypercube
Claim: for any vertex of distance m from v:
Proof sketch: #paths inside S of length m+2k is at most .
nk 2̀km!
vu
S
` = n¯ =2
[= n¡ m¯ =2]
u 2 S
Pr[(u S» v)] = o(1)
(v » u)
Pr[(u S» v)] ·
1X
k=0
pm+2knk l2km!= o(1)
Hyper Cube
So far we have shown: if , then #queries made by any local algorithm is exponential.
We will now show: a local algorithm which (almost surely) makes only poly(n) queries for .
p<< 1pn
p>> 1pn
The Hypercube – Efficient Algorithm for
We observe that the embedding of [AB03]: For any adjacent u,v in Hn: with probability
u,v are mapped to themselves and their distance in is at most .
The Algorithm: Fix a shortest path in Hn: .
With high probability all nodes are mapped to themselves. Any two adjacent vertices in the above path are at distance from each other in .
Exhaustively search balls around xi until xi+1 is found. Requires at most probes. The algorithm does not know the embedding.
p = n¡ 1=2+¯
` = (̀¯ )
u = x0;x1; : : : ;xk = v
`
n`
1¡ e¡ (p
n)
H pn
H pn
The Mesh Md
We will show: An efficient local algorithm for the mesh.
The Infinite Mesh Md
M - Each edge fails with probability . For each dimension d there exists such that:
If then contains one infinite component with prob .
If then with prob. all components of are finite.
The value of is not always known: . and decreasing.
For finite meshes: translates to high probability bounds on the existence of giant components.
M dp 1¡ p
11
pc
pdc
M dp
p< pdc
p> pdc
M dp
p2c = 1
2pd
c = (1+ o(1)) 12d
Routing in the Faulty Mesh
Theorem: let u,v be two vertices at distance k in Md. Assuming , there exists a routing algorithm which finds a path using O(k) probes in expectation.
The Algorithm – similar to the hypercube algorithm: Fix a shortest path – .
Once in xi – exhaustively search inside increasing balls
around xi until xj (j>i) is found.
Assuming the algorithm will output a path.
u » v
u » v
u = x0;x1; : : : ;xk = v
Proof Outline
x1 xi+1
¿xk = v
Pr[¿ > a] < e¡ ca
xi
1X
a=1
(2a)de¡ ca = O(1)
u = x0
Claim: Let xi be a vertex in the shortest path. Its potential contribution is expected to be O(1).
Show:
Let xi be a vertex in the shortest path and in the giant component:
[AP96] The next vertex in g.c. is not likely to be far:
Let d,D be the metrics before and after the percolation.
[AP96]: There is such that for any :
Proof – Bounding :
xi
xi+1
a > ½¢d(x;y)
u = x0
xk = v
l
xi+l+1
Pr[l > a] < e¡ (a)
Pr[(D(x;y) > a) ^(x » y)] < e¡ (a)
Local Routing vs. Oracle Rounting
Oracle Routing: The algorithm may probe any edge of the graph (even edges it did not reach).
Oracle Routing adds power: there are graphs in which there is a noticeable gap between Oracle and Local Routing complexities. Double binary tree – exponential gap. - polynomial gap.Gn;c=n
Double Binary Tree
Find a “mirror path”by querying simultaneouslyfrom both sides (using DFS).
Equivalent to finding a path from a root to a leaf in a super-critical tree. Bad branches are expected
to have constant size.
u
v
p2 > 12
Gn,c/n Lower Bound for Local Routing
Lower bound: Any local algorithm almost surely needs (n2/c2) queries.
Proof Sketch: After k queries the algorithm reveals roughly kp
vertices. Any new revealed vertex has probability p to be
connected to v.
total probability of connection to v after k queries is kp2
(=o(1) for k = o(n2/c2) ).
Gn,c/n Oracle Routing using O(n3/2) queries
Grow a (n1/2) size component around each of u,v.
Roughly n3/2/c queries are needed.
Almost surely there is an edge between Cu,Cv (and only O(n) queries are needed to find it).
Remark: the above algorithm is optimal up to constant factors.
Cu Cv
u v
Summary
connectivity Efficient oracle routing
Efficient local routing
Gap: double binary tree p2 > ½.
Gap: in Gn,p for p= c/n . Oracle router needs O(n3/2) queries but diameter is poly(log n).
Gap: Hyper-cube1/n < p < n-1/2