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