Plaxton%20Routing

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Plaxton Routing
2
History

• Greg Plaxton, Rajmohan Rajaraman, Andrea Richa.
  Accessing nearby copies of replicated objects,
  SPAA 1997
• Used in several important P2P networks like
  Microsofts Pastry and UC Berkeleys Tapestry
  (Zhao, Kubiatowicz, Joseph et al.) that is also
  the backbone of Oceanstore, blueprint of a
  persistent global-scale storage system

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Goal

• A set A of m objects reside in a network G.
  Plaxton et al. proposed an algorithm of accessing
  such a shared object using a nearby copy of it.
  Supported operations are insert, delete, read
  (but not write) on shared objects.
• What is the challenge here? Think about this
  if every node maintains a copy, then read is
  fast, but insert or delete are very expensive.
  Also, storage overhead grows. The algorithm must
  be efficient w.r.t both time and space

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Main idea

• Embed an n-node virtual height-balanced tree T
  into the network. Each node u maintains
  information about copies of the object in the
  subtree of T rooted at u.
• To read, u checks the local memory of the
  subtree under it. If the copy or a pointer to the
  object is available, then u uses that
  information, otherwise, it passes the request to
  its parent in T.

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Plaxton tree

• Plaxton tree (some call it Plaxton mesh) is a
  data structure that allows peers to efficiently
  locate objects, and route to them across an
  arbitrarily-sized network, using a small routing
  map at each hop. Each node serves as a client,
  server and a router.

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Object and Node names

• Objects and nodes acquire names independent of
  their location and semantic properties, in the
  form of random fixed-length bit-sequences
  represented by a common base (e.g., 40 Hex digits
  representing 160 bits). using the output of
  hashing algorithms, such as SHA-1 (leads to
  roughly even distribution in the name space)
• Assume that n is a power of 2b, where b is a
  fixed positive integer. Thus, name of object B
  1032 (base 4 22, so b2), name of object C
  3011 etc. n 256

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Neighbor map
x

• Level i matches i suffix entries
• Number of entries per level
• ID base (here it is 16)
• Each entry is the suffix matching
• node with least cost. If no such
• entry exists, then pick the one that
• with highest id largest suffix match
• These are all primary entries

0123
Destination 4307
Destination3623
A027
3187
9623
y1
y1
L0
L2
c(x,y1) lt c(x.y1)
2A53
1553
L1
c(x,y2) lt c(x.y2)
y2
y2
Destination 7353
Size of the table b log b (n)
8
Neighbor map of 5642
L2
L0
L1
L3

• Level i matches i suffix entries.
• Number of entries per level
• ID base (here it is 8)
• Each entry is the suffix matching
• node with least cost. If no such
• entry exists, then pick the one
• that with highest id largest
• suffix match
• These are all primary entries

y2
y2
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Neighbor map

• In addition to primary entries, each level
  contains secondary entries.
• A node u is a secondary entry of node x at a
  level i, if
• it is not the primary neighbor y, and
• c(x,u) is at most d.c(x,w) for all w with a
  matching suffix of size i
• Here d is a constant
• Finally, each node stores reverse neighbors for
  each level.
• Node y is a reverse neighbor of x iff x is a
  primary neighbor of y.
• All entries are statically chosen, and this needs
  global knowledge (-)

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Routing Algorithm

• Assume that the destination node is a tree root.
  To route to node (xyz)
• Let shared suffix n so far
• Look at level n1
• Match the next digit of the destination id
• Send the message to that node
• Eventually the message gets relayed to the
  destination.

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Example of suffix routing

• Consider 218 namespace, 005712 ? 627510

How many hops?
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Choosing the root

• For each object, the root node simply stores a
  pointer to the server that stores the object.
  The root does not hold a copy of the object
  itself.
• An objects root or surrogate node is chosen as
  a node whose id matches the objects id in the
  largest number of trailing bit positions. In case
  there are multiple such nodes, choose the node
  with the largest such id.

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Pointer list
Root of O

• Each node x has a pointer list P(x)
• a list of triples (O,S, k), where O is
• the object, S is the node that holds
• a copy of O, and k is the upper bound
• of c(x,y). The pointer list is updated
• By insert and delete operations.

(O,S,3)
(O,S,1)
(O,S,2)
Server S
(O,S,2)
(O,S,1)
(O,S)
Server S
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Inserting an Object

• To insert an object, the publishing process
  computes the root, and sends the message (O,S)
  towards the root node. At each hop along the way,
  the publishing process stores location
  information in the form of a mapping
  ltObject-ID(O), Server-ID(S)gt

Root of O
(O,S,5)
(O,S,3)
(O,S,4)
(O,S,2)
(O,S,1)
(O,S)
Server S
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Inserting another copy
Root of O
(O,S,3)

• Intermediate nodes maintain the cost of access.
  While inserting a duplicate copy, the pointers
  are modified so that they direct to the closest
  copy

(O,S,1)
(O,S,2)
Server S
(O,S,2)
(O,S,1)
(O,S)
Server S
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Read
Root of O

• Client routes to Root of O, route to S when
  (O,S) found.
• If any intermediate node stores (O,S), then that
  leads to a faster discovery. Also, nodes handling
  requests en route communicate with their primary
  and secondary neighbors to check the existence of
  another close-by copy.

(O,S,3)
(O,S,1)
(O,S,2)
Server S
(O,S,2)
(O,S,1)
(O,S)
Server S
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Deleting an object
Root of O
(O,S,5)

• Removes all pointers to that copy of the object
  in the pointer list on the nodes leading to the
  root
• Otherwise, it uses the reverse pointers to to
  update the entry.

(O,S,3)
(O,S,3)
(O,S,4)
(O,S,1)
(O,S,2)
Server S
(O,S,2)
(O,S,1)
(O,S)
Server S
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Results
Let C (maxc(u,v) u,v in V Cost of read
O(f(L(A))c(x,y)) where L is the length of the
message. If there is no shared copy, then it is
O(C) Cost of insert O(C) Cost of delete O(C
log n) Auxiliary memory to store q objects O(q
log2n) w.h.p
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Benefits and Limitations

• Scalable solution
• Existing objects are guaranteed to be found
• - Needs global knowledge to form the neighbor
  tables
• - The root node for each object may be a possible
  bottleneck.

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Benefits and limitations

• Simple fault handling
• 1234 1238 1278 1678 5678
• 3128 --gt 3178 --gt 3678
• Scalable
• All routing done using locally available data
• Optimal routing distance