Routing Complexity of Faulty Networks

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Routing Complexity of Faulty Networks

• Omer Angel Itai Benjamini Eran Ofek Udi
  Wieder
• The Weizmann Institute of Science

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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).

G
Gp
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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.

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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.

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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 .

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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,BSH04.
• Diameter of giant component is .
  Short paths exist.
• When all components are of
  size O(n) w.h.p.

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The Faulty Hypercube
No giant component

• Graph is connected.
• Emulation (and routing) possible

Threshold for constant distortion embedding of
Hn in AB03
Question What probabilities in the range allow
local routing (inside the g.c.) with complexity
polynomial in n ?
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Local Routing Phase Transition

• Let 0 lt ? lt 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.

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The Faulty Hypercube
Local routing in poly(n) queries

• No efficient local routing

• Graph is connected.
• Emulation (and routing) possible

• No giant component

Threshold for constant distortion embedding of Hn
in AB03
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The Lower Bound Lemma

• 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
  .

e
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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 (0ltplt1)
• S the bottom tree, .
• Lemma implies for ,

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The Double Tree u,v are connected

• Double Binary Tree 2 depth n trees joined at
  their leaves.
• A path uv exists iff there is a leaf w and
  mirroring paths .
• The event uvis tantamount to a branching
  process with p2.
• Path exists with constant probability, when p is
  a constant gt .

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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.

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Lower Bound Lemma - Proof

• Assume
• For each probed edge ei entering S

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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.

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Applying the Lemma to the Hypercube

• Fix (almost surely
  ).
• Claim of paths s of length is
  at most .s

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Applying the Lemma to the Hypercube

• 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 changesn
  possible coordinates and possible
  indices in the path.

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Applying the Lemma to the Hypercube

• Fix (almost surely
  ).
• Claim of paths s of length is
  at most .s

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Applying the Lemma to the Hypercube

• Claim for any vertex of distance m
  from v
• Proof sketch paths inside S of
  length m2k is at most .

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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
  .

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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 xi1 is
  found.
• Requires at most probes.
• The algorithm does not know the embedding.

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The Mesh Md

• We will show An efficient local algorithm for
  the mesh.

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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.

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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 (jgti) is
  found.
• Assuming the algorithm will output a
  path.

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Proof Outline

• Claim Let xi be a vertex in the shortest path.
  Its potential contribution is expected to be
  O(1).
• Show

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Proof Bounding ?

• 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
  

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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.

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Double Binary Tree

• Find a mirror pathby 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.

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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) ).

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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.

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Summary
Gap Hyper-cube 1/n lt p lt n-1/2
Efficient oracle routing
Efficient local routing
connectivity
Gap double binary tree p2 gt ½ .
Gap in Gn,p for p c/n . Oracle router needs
O(n3/2) queries but diameter is poly(log n).