New Protocols for Asymmetric Communication Channels

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New Protocols for Asymmetric Communication
Channels

• Presented by Nick Farnan
• 11-11-08

2
Asymmetric Communications

• Setup
• Client needs to send server an n-length bit
  string x
• The communication channel to send from server to
  client has much greater bandwidth than the one
  from client to server
• The server knows D, the distribution of
  probabilities of receiving any give n-length bit
  string from the client (Note that the client has
  no knowledge of D)?

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Asymmetric Communications

• In his A Mathematical Theory of Communication,
  Shannon provided a limit to the amount of
  lossless compression of any communication with
  Shannon entropy
• Entropy is defined to be a measure of uncertainty
  in a random variable (in this case a bit string)?
• A repeating string (i.e. 00000...) has minimal
  entropy as it is completely predictable, while a
  random bit string (could be any bit string of
  length n with equal probability) has maximum
  entropy

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Asymmetric Communications

• From this, we can draw the conclusion that the
  absolute minimum expected number of bits that the
  client will have to send to the server to
  describe x to the server is the entropy of x,
  H(x)?

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Asymmetric Communications

• Consider a list of all n length bit string
  ordered from least to greatest
• Let F(x) be the sum of D(y) for all y lt x
• Let F'(x) F(x) 1/2(D(x))?
• Given a real number v in the range (0, 1, a bit
  string y is considered to be a v-split string if
  it is the largest n bit string such that F'(y) lt
  v

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Asymmetric Communications
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Split-String Algorithm
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Split-String Algorithm
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Bit-Efficient-Split

• By using split strings, the server can do a sort
  of binary search through the space of n bit
  strings to determine the string x that the client
  wishes to send.

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Bit-Efficient-Split
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Bit-Efficient-Split

• Fact F'(x) is the midpoint of the interval
  (F(x), F(x) D(x))?
• After m rounds, given that D(x) gt 2(1-m), and
  that F'(x) is contained within (um, vm), at the
  end of the current round, y x z, and hence
  the string has been discovered!
• Hence, if it takes at most m1 rounds to discover
  the string being sent, the client m1 bits
  indicating which side of the split string the x
  falls on

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Bit-Efficient-Split

• Now, asm1 ceil(lg(1/D(x)))1 lt
  lg(1/D(x))2
• The expected number of rounds can be bounded by
  the sum over all x ofD(x) (lg(1/D(x))
  2sum(D(x) lg(1/D(x))) 2H(D) 2
• A constant factor from optimal!

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Asymmetric Communication Protocols via Hotlink
Assignments

• Presented by Nick Farnan
• 11-11-08

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The Hotlinking Problem

• Proposed for the Web to make particular pages on
  a site more accessible
• Given a rooted tree where each leaf has a
  probability attached to it (let D be the
  distribution of these probabilities), the
  Hotlinking problem moves subtrees up the tree
  closer to to the root in order to minimize the
  cost of the tree as a whole.

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The Hotlinking Problem

• Definitions
• c(u) is the sum of all of the probabilities of
  the leaves of the subtree rooted at node u, this
  is the cost of u
• c(u, v) where u is the parent of v, is c(v), this
  is the cost of edge u-v
• d(u, T) is the length of the path from the root
  of T to u
• c(T) is the sum of the costs of all the edges in
  tree T
• L(T) is the set of leaves of T

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The Hotlinking Problem

• Adoption
• A node u can adopt one of its decedents (w) by
  removing w from it's parents list of children and
  adding it instead to u's list of children

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The Hotlinking Problem

• A k-hotlink-assignment of T is obtained by
  performing up to k adoptions (where k is some
  integer) at the root of T, creating T1, and the
  further find a k-hotlink-assignment for each
  subtree of T1 that is a child of the root.
• This problem has been show to be generalized as
  at each node u, adopt a its descendants until all
  descendants of u are either leaves or have a cost
  lt (2/(k2)) c(u)?

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The Hotlinking Problem

• The resulting tree will have a cost of at most
  (H/(lg(k 2) 1)) 1, where H is the
  entropy of D, the distribution of the leaves

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Hotlinking in Asymmetric Communications

• Let T be a complete binary tree with 2n leaves
  (where n is the length of the string being sent
  from client to server) and a label on each edge
  such that all edges leading to left children have
  the label 0 and all the right are labeled 1
• Each vertex is then labeled with its path from
  the root (i.e. going left, right then left from
  the root would lead to a node labelled 010)?

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Hotlinking in Asymmetric Communications

• Both client and server maintain r1 and r2 where
  r1r2 is the string r that the client wishes to
  send.
• Initially, r1 is set to empty and the server does
  not know r2
• Find T', the k-hotlink-assignment of T
• For each round of the protocol, the server
  considers all of the children (w) of node u in T
  whose label is r1, and send to the client all l
  where l is label(w) - r1

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Hotlinking in Asymmetric Communications

• The client then replies with the index in the
  last server message of the longest l that is a
  prefix of r2
• The client then removes l from the beginning of
  r2, and both client and server add l to r1
• If r1 n, the server outputs r1 and the
  communication terminates

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Hotlinking in Asymmetric Communications

• The client can then be expected to send
  c(T') ceil(lg(k 2))?
• Given that c(T') (H/(lg(k 2) 1)) 1, the
  client can then be expected to send
  H(ceil(lg(k2))/(lg(k2)-1))
  ceil(lg(k2)) bits
• setting e (ceil(lg(k2))/(lg(k2)-1)) 1, and
  ce ceil(lg(k2)), then, a protocol where
  client is expected to send (1e)H ce bits is
  achieved