1. Hierarchical Routing Scheme
Presented By: Raquel Whittlesey-Harris
5/1/03
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2. Contents Partial Routing Schemes – An Introduction
Hierarchical Scheme – An Overview
Regional Routing Schemes
The Hierarchical Scheme
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3. Partial Routing Schemes
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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4. 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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5. 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 it’s 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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6. 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 [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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7. 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 v’s 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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8. 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 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 C=O(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 , 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 , 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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9. Regional Routing Schemes
Corollary For every graph G and integers k,   1, there exists a regional (O(k2),)-routing scheme RSk,p using O(k·n1/k·log n) memory bits per vertex
Theorem [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 TC=TCk, for G with Depth(TC)(2k-1)2, Overlap(TC) 2k·n1/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(k·n1/k),
We get a regional (O(k2),)-routing scheme, RSk, using O(k·n2/k·log n) bits per vertex,
Substituting k=2k, we get the above bound with a dilation multiplied by 4
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10. Hierarchical Routing Scheme
Definition [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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11. Regional Routing Scheme
Routing Graph
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12. Regional Routing Scheme
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13. Regional Routing Scheme
Routing Tree
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14. 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 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 < d  2j)
u looks for the lowest level, i, on which it belongs to home(v)
By Lemma , u belongs to homej(v) and Rj is applicable if no previous level was.
By Lemma ,
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15. Hierarchical Routing Scheme
Theorem 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
Corollary 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(k·n1/k·log n·) memory bits per vertex
Using Corollary to construct the routing scheme containing tree covers having DR=O(k2) and memory requirements O(k·n1/k·log n) each.
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16. Hierarchical Routing Scheme
Routing Graph
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17. Hierarchical Routing Scheme
Routing Tree
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