Hierarchical Routing Scheme

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Hierarchical Routing Scheme

• Presented By Raquel Whittlesey-Harris
• 5/1/03

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Contents

• Partial Routing Schemes An Introduction
• Hierarchical Scheme An Overview
• Regional Routing Schemes
• The Hierarchical Scheme

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Partial Routing Schemes

• Partial Routing Scheme
• Guaranteed to perform its task only for a
  restricted subset of sender-destination pairs and
  is allowed to fail for certain other pairs.
• Main Complication
• Handling of unknown destinations
• Needs to be accomplished efficiently
• Permit the employment of flexible and dynamic
  Trial and Error routing schemes

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Partial Routing Schemes

• Header functions are also utilized
• If the attempt fails, an alternative partial
  routing scheme with a different header and port
  is attempted
• ITR (Interval Tree Routing) scheme is used as the
  basic component
• Spans only a partial subnetwork, G
• Changes to protocol,
• Permit the case that none of the 4 possible
  choices for forwarding are applicable at a given
  instant
• Routing Failure
• Message is returned to sender with message
  complexity O(kRad(G))

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Hierarchical Scheme - Overview

• A Hierarchy is constructed of Tree Covers
• Each Level of the hierarchy, each tree of the
  cover has its own ITR routing mechanism
• Routing to and from the ROOT
• Memory is reduced
• Each cluster needs only know of its spanning
  tree
• Communication Cost is increased
• May not take shortest paths
• The two will be reduced by a proper choice of
  Tree Cover (i.e., balanced tree cover)

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Regional Routing Schemes

• ITR is used to route Messages in a subgraph, G.
• Tree Covers are used to form Regions
• Each level of the hierarchy constitutes a
  regional (C,?)-routing scheme
• Definition 26.2.1 Regional (C,?)-routing
  scheme A regional (C,?)-routing scheme is a
  scheme with the following properties,
• For every two processors u,v, if dist(u,v) ? ?,
  then the scheme succeeds in delivering messages
  from u to v. Otherwise, the routing might end in
  failure, in which case the message is returned to
  u.
• In either case, the communication cost of the
  entire process is at most C.

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Regional Routing Schemes

• Data Structures
• Given a ?-tree cover, TC,
• Each tree, T, is assigned a distinct label,
  Label(T)
• An ITR component, ITR(T) is set up on each tree,
  T
• Each vertex, v has home tree, Home(v) in TC
  containing its entire ?-neigborhood
• The routing label for v will be the pair,
  (Label(T),IntT(v)), where Label(T) is the label
  of the home tree, and IntT(v) is the vs routing
  label in ITR(T).
• v stores routing information for each ITR(T)
  component v participates in (overlap)
• Routing tables are stored by tree id so the next
  step can be determined in logarithmic time

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Regional Routing Schemes

• Forwarding Protocol
• For u to route a message to v
• u examines whether it belongs to the tree T
• If it does, it sends the message to v using the
  ITR(T) component
• If not, it immediately detects an unknown
  destination failure and terminates the routing
• Lemma 26.2.2 For every n-vertex weighted graph G
  (V,E,?) and ?-tree-cover TC for G, the scheme
  RSTC described above is a regional (C,?)-routing
  scheme with CO(Depth(TC)) and can be implemented
  using O(Overlap(TC)?TC(TC)log n) memory bits
  per vertex
• Suppose dist(u,v) ? ?, v ? ??(u), T Home(u) ?
  ??(u) ? V(T), so v ? T and by Lemma 26.1.1, the
  tree routing on T will succeed in passing the
  message from v to u, and
• The routing path is a length at most O(Depth(TC))
• By Lemma 26.1.1, each vertex v belongs to no more
  than Overlap(TC) trees in TC, and its degree in
  each tree ? ?TC(TC)

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Regional Routing Schemes

• Corollary 26.2.3 For every graph G and integers
  k, ? ? 1, there exists a regional
  (O(k2?),?)-routing scheme RSk,p using
  O(kn1/klog n) memory bits per vertex
• Theorem 15.5.3 Balanced Tree Cover Theorem
  For every weighted graph G(V,E,?), V n and
  integers k, ??1, it is possible to construct a
  (virtual)?-tree cover TCTCk,? for G with
  Depth(TC)?(2k-1)2?, Overlap(TC)? ?2kn1/k? and
  ?TC(TC) ?2n1/k
• is used to construct a tree cover TCk,?with
  Depth(TC) O(k2?), ?TC(TC)O(n1/k) and
  overlap(TC)O(kn1/k),
• We get a regional (O(k2?),?)-routing scheme,
  RSk,? using O(kn2/klog n) bits per vertex,
• Substituting k2k, we get the above bound with a
  dilation multiplied by 4

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Hierarchical Routing Scheme

• Definition 26.2.4 Tree Cover Hierarchy A
  hierarchical DR-family of tree covers is a family
  of ?i-tree covers TCi, where?i 2i for 1 ? i ?
  ?, with the property that there exists a bound DR
  such that Depth(TCi)O(DR?i)
• Data Structures
• For every i, 1 ? i ? ?,
• Construct a regional (O(DR?i),?i)-routing scheme
  Ri RSTC
• Each processor, v, participates in all ? regional
  routing schemes Ri,
• v has homei(v) in each Ri and Labeli(v) in each
  level i
• The complete routing label of v is
  (Label1(v),,Label?(v))

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Regional Routing Scheme

• Routing Graph

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Regional Routing Scheme
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Regional Routing Scheme

• Routing Tree

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Hierarchical Routing Scheme

• Forwarding Protocol
• u wishing to send a message to v
• u identifies the lowest-level regional scheme R,
  that can be used for this routing operation
  (homei(v))
• u sends the message to v on the ITR(homei(v))
  component of the regional scheme Ri
• Lemma 26.2.5 The hierarchical routing scheme RS
  has Dilation(RS)O(DR)
• Suppose u needs to send a message to v and d
  dist(u,v) and j ?log d? (i.e, 2j-1 lt d ? 2j)
• u looks for the lowest level, i, on which it
  belongs to home(v)
• By Lemma 26.2.2, u belongs to homej(v) and Rj is
  applicable if no previous level was.
• By Lemma 26.2.2,

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Hierarchical Routing Scheme

• Theorem 26.2.6 For every n-vertex weighted graph
  G(V,E,?) with a hierarchical DR-family of tree
  covers TCi,it is possible to construct (in
  polynomial time) a hierarchical routing scheme RS
  with Dilation(RS)O(DR) using Mem(RS)
  
  memory bits
• By constructing the ? regional schemes Ri as
  above, the memory requirements of the
  hierarchical scheme are composed of ? terms
  bounded as in Lemma 26.2.2
• Corollary 26.2.7 For every n-vertex weighted
  graph G(V,E,?) and fixed integer k?1 it is
  possible to construct (in polynomial time) a
  hierarchical routing scheme RSk with
  Dilation(RSk)O(k2) using Mem(RSk) O(kn1/klog
  n?) memory bits per vertex
• Using Corollary 26.2.3 to construct the routing
  scheme containing ?tree covers having DRO(k2)
  and memory requirements O(kn1/klog n) each.

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Hierarchical Routing Scheme

• Routing Graph

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Hierarchical Routing Scheme

• Routing Tree