Clustering lines: classification of incomplete data

Report
Index Coding
Part II of tutorial
NetCod 2013
Michael Langberg
Open University of Israel
Caltech (sabbatical)
1
Outline
This part of tutorial:
Will show an equivalence between the network
coding and index coding problems.
Outline:
•
•
• Preliminary: Network Coding model.
• Preliminary: Index Coding model.
• Equivalence for linear encoding/decoding
.
• Equivalence for general encoding/decoding
.
• Multicast vs. Unicast Index Coding
• Open Questions.
[ElRouayhebSprintsonGeorghiades]
[EffrosElRouayhebLangberg]
[MalekiCadambeJafar].
2
• Network communication challenging: combines topology
with
information.theme
General
•Reduction separates information from topology.
•Significantly
simplifies
the study
of Network
Comm.
Will show an
equivalence
between
the network
•Index
Coding
is aindex
simplecoding
but representative
coding
and
problems. instance of
• An efficient reduction that allows to solve NC
general network communication.
using any scheme to solve IC.
s1
NC
s2
s1
s2
s3
s4
s5
s6
IC
t3
t1
t2
Obtain solution to NC
t1
t2
t3
t4
t5
t6
Solve IC
3
General Network Coding
• Directed acyclic network N.
• Edge e has capacity c .
• Source vertices S.
• Terminal vertices T.
• Requirement matrix:
s1
s3
s2
e
• Transfer information from S to T.
t3
t1
t2
• Objective:
• Information flow using Network Coding that satisfies terminals.
4
S1
Assumptions
• Sources S hold independent information.
• Zero error in communication.
• We consider the multiple unicast communication
t1
i
S2
t2
requirement (w.l.o.g. [DoughertyZeger]):
• k source/terminal pairs (S ,T ) that wish to communicate over N.
i
i
S1
T1
S2
T2
S3
S4
N
T3
T4
5
NC preliminaries
s1
t1
s3
t3
s2
t2
s4
t4
Communication at rate R = (R1,…,Rk) is achievable over
instance NC with block length n if  random variables
{Si},{Xe}:
• Rate: Source S = R.V. independent and uniform over [2 ].
• Edge capacity: For each edge e: X = R.V. with support [2 ].
• Functionality: for each edge e we have f = function from incoming
R.V.’s X ,…,X
to X (i.e., X =f (X ,…,X
)).
X
f
• Decoding: for each terminal t we define
X
X
Rin
i
cen
e
e
e1
e,in(e)
e
e
i
e
e1
e,in(e)
a decoding function yielding sources Si reqired.
1
2
e
e
X3
• R=(R ,…R ) is ”n-feasible” if  code with block length n.
• Alternatively we say that NC is (R,n)-feasible.
1
k
6
Index Coding
[Birk,Bar-Yossef et al.]
• IC is a special case of NC
• A set S of sources.
• A set T of terminals.
• Each terminal has some subset of sources (as side
info.) and wants some subset of sources.
• Broadcast link has capacity c .
•Other links have unlimited cap. s s s
• Objective: To satisfy all terminals.
B
1
2
3
s4
cB
using broadcast rate cB.
t1
t2
t3
t4
Index Coding
Communication at rate R = (R1,…,Rk) is achievable with
block length n if  random variables {Si},XB:
• Rate: Source S = R.V. independent and uniform over [2 ].
• Encoding: X = f (S ,…,S ) is R.V. with support [2 ].
• Decoding: for each terminal t we define a decoding function g
Rin
i
B
B
1
cBn
k
i
taking as input the broadcasted message XB and the side
information of ti; and returning the sources Si wanted by ti.
s1
s2
s3
• R=(R ,…R ) is ”n-feasible” if
1
i
s4
k
 code with block length n.
cB
•ICWill
use notation: IC is (R,n)-feasible.
is a simple instance of the NC problem:
only a single encoding node.
t1
t2
t3
t4
Connecting NC to IC
s1
s2
s1
s2
s3
s4
s5
s6
NC
IC
t1
t2
Obtain solution to NC
t1
t2
t3
t4
t5
t6
Solve IC
• Step 1: Need to define reduction from NC to IC.
• Step 2: Need to prove
NC is (R,n)-feasible iff IC is (R’,n)-feasible.
• Would like: Reduction/code const. to be very efficient.
9
Outline
s1
s2
s1
s2
s3
s4
s5
s6
NC
IC
t1
t2
t1
t2
t3
t4
t5
t6
Theorem: For any NC, R one can construct IC, R’ such that
for any n: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
• Step 1: Present reduction from NC to IC.
• Step 2: Equivalence for linear and general
encoding/decoding.
[EffrosElRouayhebLangberg].
[ElRouayhebSprintsonGeorghiades],
10
The reduction
NC sources
NC
NC edges
NC sources
X1
Network:
Xe
edges
IC
X2
X3
NC terminals
•Index Coding instance:
NC term.
NC edges
•Sources corresponding to NC sources, and NC edges.
•Terminals corresponding to NC term., NC edges, special terminal.
•For edge e: terminal t in IC wants IC source X and has as
e
side information all IC sources incoming to e in NC.
e
IC encodes topology of NC in its terminals!
The reduction in more detail
NC sources
X1
Network:
ti
edges
NC
NC edges
NC sources
IC
X2
X3
NC terminals
NC term.
NC edges
•Sources: |S|+|E| sources, one for each source of NC and
one for each edge of NC: {S ’} and {S ’}.
•Terminals: |T|+|E|+1 terminals:
i
e
•One terminal t ’ for each t : wants S ’ and has {S ’} for e in In(t ).
•t ’ for each edge e: wants S ’ and has {S ’} for edge a in In(e).
•One special terminal t : wants {S ’} and has {S ’}.
i
i
e
i
e
all
e
a
e
i
i
The reduction in more detail
ReductionNC sources
NC
Sources
IC
S1,…,Sk
{Si’}, {Se’}
Network:
t1,…,tk
edges
Terminals
Capacities
{ti’},{te’},tall
ce
cB=ce
NC terminals
R1,…,Rk
Rate
NC edges
NC sources
NC edges
NC term.
{Ri’}, {Re’}
Ri’=Ri
Sources: |S|+|E| sources,
Re’=ce one for each source of NC and one for each
•
edge of NC: {S ’} and {S ’}.
Wants
•Terminals: Has
|T|+|E|+1 terminals:
foreeach
terminal
t ’ •One
{S ’} for
in In(t
). S ’ t : wants S ’ and has {S ’} for e in In(t ).
foraeach
edgeSe:’ wants S ’ and has {S ’} for edges a in In(e).
t ’ •One
{S ’} for
in In(e).
t •One
{S ’} special terminal
{S t’} : wants {S ’} and has {S ’}.
X
•Theorem:
Bottle neck edge of capacity c =c
For any NC, R one can construct IC, R’ such that
•for
Given rate vector R=(R ,…,R ) we construct rate vector R’=({R
X ’};{R ’}): t
n: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
•Rany
’=R and R ’=c .
X
i
i
e
e
a
all
i
e
i
i
i
i
e
e
1
i
i
e
e
k
i
a
e all
e
B
e
e.
i
1
2 i
3
e
i
Theorem
Reduction
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
t1,…,tk
{ti’},{te’},tall
Capacities ce
cB=ce
Rate
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
R1,…,Rk
Has
Wants
ti’
{Se’} for e in In(ti).
S i’
t e’
{Sa’} for a in In(e).
Se ’
tall {Si’}
•Theorem:
{Se’}
NC sources
NC edges
NC sources
NC edges
NC terminals
s1
t1
s2
t2
.
NC term
NC edges
s1
s2
s3
s4
s5
s6
t1
t2
t3
t4
t5
t6
NC is (R,n)-feasible iff IC is (R’,n)-feasible.
Geometric view
Reduction
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
t1,…,tk
{ti’},{te’},tall
Capacities ce
cB=ce
Rate
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
R1,…,Rk
•Theorem:
NC sources
NC edges
NC sources
NC edges
NC terminals
s1
t1
s2
t2
.
NC term
NC edges
s1
s2
s3
s4
s5
s6
t1
t2
t3
t4
t5
t6
NC is (R,n)-feasible iff IC is (R’,n)-feasible.
e: Re’=ce
What now?
Outline:
NC feasible implies IC feasible (works for
both linear and non-linear).
IC feasible implies NC feasible (will show new
proof for linear that modifies to non linear).
•
•
Theorem: For any NC, R one can construct IC, R’ such that
for any n: NC is (R,n)-feasible iff IC is (R’,n)-feasible.
16
NC
Reduction
IC
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
t1,…,tk
{ti’},{te’},tall
Capacities
ce
cB=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
•
Use global NC encoding functions.
••Rate:
Source S = R.V. independent and uniform over [2 ].
Seen that X = f (X (e)).
••Edge
capacity: For each edge e: X = R.V. with support [2 ].
Edge e also has a function F :
•Functionality:
for each
edge e we have f = function from
•
X = F (S ,…,S
).
incoming R.V.’s X ,…,X
to X (i.e., X =f (X ,…,X
)).
•
IC: We need to define X of rate c .
•Decoding:
for each terminal t we define a decoding function
•
Recall that c =Σc .
yielding sources S required.
• Recall that X of rate c .
•Theorem:
For all eFor
let X
(Scan
’,…,S
’).
any(e)=S
NC, R’+F
one
construct
IC, R’ such that
There
is a separation
between iff
{S ’}IC
andis{S(R’,n)-feasible.
’}.
for•any
n: NC
is (R,n)-feasible
• Lets see that this works (decoding …).
e
Has
i
e
In
e
e
e
e
1
k
e1
i
B
B
RWants
in
ti’
{Se’} for e in In(ti).
t e’
{Sa’} for a in In(e).
tall e{Si’}
e,in(e)
e
B
e
B
e
Si’
cen
Se’
{Se’}
e1
e,in(e)
i
e
e
e
e
e
1
k
i
e
17
NC
•
•
•
•
•
Reduction
IC
NC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
t1,…,tk
{ti’},{te’},tall
Capacities
ce
cB=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
Use global encoding functions of NC.
Each edge e has a function Fe such that Fe(S1,…,Sk)=Xe.
Recall Xe of support [2cen].
Recall cB=Σce.
We need to define XB of total support [2c n].
Basic idea: simulate the NC solution!
• Let X (e)=S ’+F (S ’,…,S ’).
• Decoding:
B
B
e
e
1
IC
Xa1
fe(Xa1,Xa2,Xa3)=Xe Xa2
k
Xe
Xa3
• Consider terminal t ’: wants S ’ and has {S ’} for edges a in In(e).
• t ’ also receives the broadcast X .
• For each a compute X (a)-S ’ = S ’+F (S ’,…,S ’)-S ’ = F (S ’,…,S ’).
• Usetlocal
encoding function f to compute:
t’
’ will
solution
on
edge
e. ’)
f (F simulate
(S ’,…,S the
’),…, NC
F (S
’,…,S
’))
=
F
(S
’,…,S
t’
• Compute X (e)-F (S ’,…,S ’) = S ’+F (S t’,…,S ’)-F (S ’,…,S ’) = S ’.
• Same process for other terminals.
e
e
a
e
B
B
e
a
a
a
e
e
a1
B
1
k
e
1
a3
k
e
1
e
1
k
Has
a
a
1
Wants
k
i
{Se’} for e in In(ti).
Si’
ek
{Sa’} for
e a1 in In(e).
k
Se’
k i’} e
{S
{See’}
1 all
1
k
18
What now?
Outline:
NC feasible implies IC feasible (works for
both linear and non-linear).
IC feasible implies NC feasible (will show new
proof for linear that modifies to non linear).
•
•
19
Reduction
Linear: IC
NC
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
T1,…,Tk
{ti’},{te’},tall
Capacities
ce
cB=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
• Given a linear code for IC, how do we build one for NC?
• Encoding for IC includes a linear encoding function f
t only has {S ’} and
A
f ({S ’}, {S
= ’},{S
•Rate: Sources
S’ =’})({S
’})+of Asupport wants
[2 ],all[2{S ’}].
••Bottleneck:
= fA ({S
’}) of
[2 ].
Can prove X
that
is ’},{S
square
andsupport
full rank.
• Crucial property:
• Fix any value s ’ for S ’=S ’,…,S ’
• There exists unique value s ’ for S ’=S ’,…,S ’ such
that f (s ’, s ’) =0.
• This will allow the construction of a NC!
B
B
i
e
B
i
B
E
I
i
S
e
e
I
I
E
Re’n i
e
cBn
1
k
E
B
all Ri’n
E
ti’
t e’
tall
Has
{Se’} for e in In(ti).
E
e1
em
{Sa’} for a in In(e).
{Si’}
Wants
Si’
Se’
{Se’}
20
Reduction
Linear: IC
NC
fB({Si’}, {Se’}) = AS
+
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
T1,…,Tk
{ti’},{te’},tall
Capacities
ce
cB=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
AE
• A is square and full rank.
• Crucial property: For all s ’ exists s ’ s.t. f (s ’, s ’) =0.
E
I
E
B
I
E
sE’
sI’
Will define NC by
‘projecting’ fB onto
the white curve!
Value = 0
fB(sI’, sE’)
21
Reduction
Linear: IC
NC
• Consider edge e in NC.
• We will define local encoding
function f (X ,X ,X )=X .
• Will define f based on decoding
e
a1
a2
e
a3
e
tall
{Si’}
{Se’}
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
T1,…,Tk
{ti’},{te’},tall
Capacities
ce
cB=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
sE’
sI’
ge’ function of IC for terminal te’.
ge’(Sa1’,Sa2’,Sa2’, fB({Si’}, {Se’}))=Se’.
fe(Xa1,Xa2
,Xa3)=ge‘(X
a1,Xa2,Xa3, 0).
Has
Wants
valid
function.
{Sea’} for
e in local
In(ti). encoding
Si’
ti’fe is
decoding
defined
similarly.
{S
Se’
teNC
’
a’} for a in In(e).
•
•
•
NC
Value = 0
Xa1
Xa2
Xe
Xa3
22
Reduction
Linear: IC
NC
• Consider terminal i in NC.
• We need to define local decoding
function g (X ,X ,X )=S .
• Will define g based on decoding
i
a1
i
a2
a3
all
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
T1,…,Tk
{ti’},{te’},tall
Capacities
ce
cb=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
sE’
i
s’
I
gi’ function of IC for terminal ti’.
gi‘(Sa1’,Sa2’,Sa2’, fB({Si’}, {Se’}))=Si’.
gi(Xa1,Xa2,Xa3)=gi‘(Xa1,Xa2,Xa3, 0).
Recall: feHas
(Xa1,Xa2,Xa3Wants
)=ge‘(Xa1,Xa2,Xa3, 0).
and
valid
{Sef
’} efor
e ingIn(t
Si’
ti’Both
i are
i).
encoding/decoding
functions.
{Sa’} for a in In(e). S
t e’
e’
{Si’}to prove correct
{Se’} decoding!
t Need
•
•
•
•
NC
Value = 0
Xa1
Xa2
Si
Xa3
23
Reduction
xe=fe(xa1,xa2,xa3) =
gDecoding:
e’(xa1,xa2,xa3, 0) =
ge’(xa1,xa2,xa3, fB(sI’,sE’)) =
ge’(sa1’,sa2’,sa3’, fB(sI’,sE’)) = se’
IC
NC
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
T1,…,Tk
{Ti’},{Te’},Tall
Capacities
ce
cE=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
,x ’(X
,x ,X) =,X , 0).
•Decoder
f (X ,X =g,X(x )=g
We get a valid NC!
0) =’(X ,X ,X , 0).
•g ’(x
g (X ,x,X ,x,X , )=g
s ’
g ’(s ’,s ’,s ’, f (s ’,s ’)) = s ’
s’
s
•= Consider
source info s =s ’=s ,…,s
Value = 0
• Let s ’ be corresponding value on curve.
• Will show by induction that running NC
e
a1
ia2 i a3a1
ea2 a1a3 a2
i i a1a1 a2a2 a3a3
i
a1
a2
i
a3
i
B
a1
I
a2
a3
a3
E
E
i
I
I
1
k
I
E
on input sI corresponds to running IC on
input (sI’, sE’).
Inductive claim: information xe in NC is
exactly se’.
Now idea:
for decoding
si at terminal
usesolution!
gi
Basic
NC is simulating
thei IC
•
•
Xa1
Xa2
Si/Sa
Xa3
24
What now?
Outline:
NC feasible implies IC feasible (works for
both linear and non-linear).
IC feasible implies NC feasible (will show new
proof for linear that modifies to non linear).
•
•
25
Reduction
Has
Wants
{S ’} for e in In(t
S’
General
: ).IC
ti’
e
i
NC
i
t e’
{Sa’} for a in In(e).
Se’
tall
{Si’}
{Se’}
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
T1,…,Tk
{ti’},{te’},tall
Capacities
ce
cB=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
• We use exact same proof!
Number of different X values is exactly
equal to number of different S ’ values.
• Where did we use linearity?
Crucial property: For all s ’ exists s ’ s.t. f (s ’, s ’) =0.
• Need to prove property for general encoding functions.
• Property follows from terminal t .
B
E
I
E
B
I
E
• Given S ’ and X =f (S ’, S ’) we must be able to decode S ’ s ’
• Thus fixing s ’, f is 1-1 as a function of S ’
• Support of X equals support of S ’.
s’
Value = 0
• Each row is a permutation.
• Thus property holds!
all
I
B
I
B
B
I
E
E
B
E
E
E
I
differ
26
What now?
Outline:
NC feasible implies IC feasible (works for
both linear and non-linear).
IC feasible implies NC feasible (will show new
proof for linear that modifies to non linear).
Multicast IC can be represented by Unicast
IC (linear only) [MalekiCadambeJafar].
•
•
•
27
Multicast vs Unicast
• In previous reduction we use an IC instance which
“multicasts” information to different terminals.
• Same information is wanted by more than one terminal.
Has
Wants
ti’
{Se’} for e in In(ti).
Si’
t e’
{Sa’} for a in In(e).
Se’
tall
{Si’}
{Se’}
Reduction
NC
IC
Sources
S1,…,Sk
{Si’}, {Se’}
Terminals
T1,…,Tk
{ti’},{te’},tall
Capacities
ce
cB=ce
Rate
R1,…,Rk
{Ri’}, {Re’}
Ri’=Ri
Re’=ce
• In NC, any (multiple) multicast can be reduced to
(multiple) unicast
.
• Does the same phenomena hold for IC?
[DoughertyZeger]
28
Multicast vs Unicast
•
•
•
In previous reduction we use an IC instance which “multicasts” information
to different terminals.
Same information is wanted by more than one terminal.
In NC, any (multiple) multicast can be reduced to (multiple) unicast.
Does the same phenomena hold for IC?
•
• Recent work by
•
show that unicast
suffices in case (if restricted to linear
encoding/decoding).
Implies that for linear encoding: NC reduces to
(multiple) unicast IC!
[MalekiCadambeJafar]
• Each terminal wants different message.
• Same number of sources and terminals.
• IC can be characterized by side information graph, rather than
side information hypergraph.
29
Some open problems
• Multicast vs. Unicast for general encoding.
• Would be surprising:  problems known to be more difficult
in the multiple-multicast setting (e.g., IC via cycle packing
[ChaudhryAsadSprintsonLangberg]).
• Capacity: can one determine if rate R is in capacity
region of NC via knowledge of capacity region of IC?
• Reduction is not robust enough to withhold the closure
operation in the definition of capacity.
• Answer is yes for linear case. Also for co-located NC sources
[WongLangbergEffros].
e: Re’=ce
35
Some open problems
•  vs zero error in communication:
• Does allowing some error increase rate in NC/IC?
• IC there is no advantage
to allowing small
error in communication … can this extend to NC?
• NC – not known! In NC “no advantage” known for co-located
and other cases.
• Can we use equivalence between NC and IC?
• Intriguing connections to other problems such as the edge
[LangbergEffros]
[ChanGrant] [LangbergEffros]
removal problem [HoEffrosJalali].
• Algorithms!:
• Wide open … both in NC setting and IC setting …
36
Conclusions
NC sources
NC sources
NC edges
NC edges
NC terminals
.
NC term
NC edges
• Network communication challenging: combines topology
with information.
• Discussed equivalence between the network coding and
index coding problems.
• Reduction separates information from topology.
• Significantly simplifies the study of Network Comm.
• Index Coding is a simple but representative instance of
general network communication.
Thanks!
37

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