Post on 04-May-2022
Function computation in networks: feasibility and rates
Bikash Kumar Dey
IIT Bombay30th October, 2017
Some computation problems
S
S
S
SS
S
R
RT
S
X
Y
f(X,Y)
X
max(X,Y)
Y
max(X,Y)
Outline of the talk
• Computing the sum function [with Brijesh Rai and Sagar Shenvi]
• Network flow for functions [with Virag Shah and D. Manjunath]
Computing the modulo-sum function
Multicasting over a network
1T T2
S
X, Y
X, YX, Y
For unicast:Capacity = min−cut{S-T}
For multicast:Capacity ≤ mini min−cut{S − Ti}
Can we multicast at rate 2?
Multicasting over a network
1T T2
S
X, Y
X Y
X
X Y
Y
X
X X
X, YX
Can we multicast at rate 2?
NO!
Multicasting over a network
1T T2
S
X, Y
X Y
X
X Y
Y
X
X X
X, YX
Can we multicast at rate 2?
NO!
Network coding
1T T2+X Y+X Y
+
+X Y
S
X, Y
X Y
X
X Y
Y
X, YX, Y
Can we multicast at rate 2?
YES, with network coding
Multicast capacity = mini min−cut{S − Ti}
Network coding for multicasting:
• Easy to compute capacity (unlike under routing)
• Easy to design codes
• Linear coding sufficient
• Random linear codes work
Network coding
1T T2+X Y+X Y
+
+X Y
S
X, Y
X Y
X
X Y
Y
X, YX, Y
Can we multicast at rate 2?
YES, with network coding
Multicast capacity = mini min−cut{S − Ti}
Network coding for multicasting:
• Easy to compute capacity (unlike under routing)
• Easy to design codes
• Linear coding sufficient
• Random linear codes work
Network coding
1T T2+X Y+X Y
+
+X Y
S
X, Y
X Y
X
X Y
Y
X, YX, Y
Can we multicast at rate 2?
YES, with network coding
Multicast capacity = mini min−cut{S − Ti}
Network coding for multicasting:
• Easy to compute capacity (unlike under routing)
• Easy to design codes
• Linear coding sufficient
• Random linear codes work
General communication networks
S1 S2 S3
1T 2T
X11 X12 X2 X31XX32 X33
X11 X2 XX32 X11 X12X31X33
Capacity= sup{ kn}
• Coding capacity is difficult to compute
• Linear coding is not sufficient
Multiple unicast networks (MUN)
T2 1T
S S1 2
X Y
• For every network, there is anequivalent multiple unicast network.
• MUNs are as difficult as generalnetworks.
MUNs ⇔ Polynomial equations
The most general function computation problem
S1 S2 S3
1T 2T
X11 X12 X2 X31 XX32 X33
f ( )1 X f ( )2 X
Detour: Computing function without network coding
X1 X2X3
Θ
*
+
X1X2
Computation tree of Θ(X1,X2,X3) = X1X2 + X3
Embedding
s1 s2 s3
t
X1 X2X3
Θ
*
+
X1X2
s1 s2 s3
t
X1
X3
X3
Θ
X2
X1X2
s1 s2 s3
t
X1 X2
X1*X2
X3
X3X1*X2
Flow for functions
• Packing of the embeddings (LP not efficient)
• Flow conservation based LP (efficient)
• More efficient algorithm using min-cost embedding
X1 X2X3
Θ
*
+
X1X2
[Shah, Dey, Manjunath 2013]
The most general function computation problem
S1 S2 S3
1T 2T
X11 X12 X2 X31 XX32 X33
f ( )1 X f ( )2 X
A special case: sum-networks
S1 S2 S3
1T 2T
X2 X3X1
X +X +X1 2 3X +X +X1 2 3
• m sources, n terminals
• Source alphabet is a field (F )or an abelian group (G)
• Linear coding over the samefield
Min{m, n} ≤ 2 [Ramamoorthy 2008]
The sum can be communicated if and only if every source-terminal pair isconnected.
S1 S2 S3
1T 2T
1 2 3 1 2 3f(Y ,Y ,Y ) = Y +Y +Y
X1 X2 X3
A ’connected’ Network with no solution
s1 s3 s21x 3x 2x
t1
u
v
1
1
t3 t2
u
v
2
2
The network S3
Capacity of S3 = 2/3
s1 s3 s21x 3x 2x
t1
u
v
1
1
t3 t2
u
v
2
2
• t3 can recover x1, x2, x3.
• But mincut(s1, s2, s3; t3) = 2• So, q2n ≥ q3k ⇒ k/n ≤ 2/3.• Achievability: ...
Capacity of S3 = 2/3
s1 s3 s21x 3x 2x
t1
u
v
1
1
t3 t2
u
v
2
2
• t3 can recover x1, x2, x3.• But mincut(s1, s2, s3; t3) = 2
• So, q2n ≥ q3k ⇒ k/n ≤ 2/3.• Achievability: ...
Capacity of S3 = 2/3
s1 s3 s21x 3x 2x
t1
u
v
1
1
t3 t2
u
v
2
2
• t3 can recover x1, x2, x3.• But mincut(s1, s2, s3; t3) = 2• So, q2n ≥ q3k ⇒ k/n ≤ 2/3.
• Achievability: ...
Capacity of S3 = 2/3
s1 s3 s21x 3x 2x
t1
u
v
1
1
t3 t2
u
v
2
2
• t3 can recover x1, x2, x3.• But mincut(s1, s2, s3; t3) = 2• So, q2n ≥ q3k ⇒ k/n ≤ 2/3.• Achievability: ...
Capacity of S3 = 2/3
Achievability:
• Two sums in three slots
• Two realizations: (X11,X21,X31), (X12,X22,X32)
• Two sums: S1 = X11 + X21 + X31, S2 = X12 + X22 + X32
• Slot 1: S1 → t1, t2
• Slot 2: S2 → t2, t3
• Slot 3: S1 + S2 → t1, t3
Capacity of S3 = 2/3
Achievability:
• Two sums in three slots
• Two realizations: (X11,X21,X31), (X12,X22,X32)
• Two sums: S1 = X11 + X21 + X31, S2 = X12 + X22 + X32
• Slot 1: S1 → t1, t2
• Slot 2: S2 → t2, t3
• Slot 3: S1 + S2 → t1, t3
Capacity of S3 = 2/3
Achievability:
• Two sums in three slots
• Two realizations: (X11,X21,X31), (X12,X22,X32)
• Two sums: S1 = X11 + X21 + X31, S2 = X12 + X22 + X32
• Slot 1: S1 → t1, t2
• Slot 2: S2 → t2, t3
• Slot 3: S1 + S2 → t1, t3
Capacity of S3 = 2/3
Achievability:
• Two sums in three slots
• Two realizations: (X11,X21,X31), (X12,X22,X32)
• Two sums: S1 = X11 + X21 + X31, S2 = X12 + X22 + X32
• Slot 1: S1 → t1, t2
• Slot 2: S2 → t2, t3
• Slot 3: S1 + S2 → t1, t3
Capacity of S3 = 2/3
Achievability:
• Two sums in three slots
• Two realizations: (X11,X21,X31), (X12,X22,X32)
• Two sums: S1 = X11 + X21 + X31, S2 = X12 + X22 + X32
• Slot 1: S1 → t1, t2
• Slot 2: S2 → t2, t3
• Slot 3: S1 + S2 → t1, t3
Capacity of S3 = 2/3
Achievability:
• Two sums in three slots
• Two realizations: (X11,X21,X31), (X12,X22,X32)
• Two sums: S1 = X11 + X21 + X31, S2 = X12 + X22 + X32
• Slot 1: S1 → t1, t2
• Slot 2: S2 → t2, t3
• Slot 3: S1 + S2 → t1, t3
Polynomial equations
s1 s2 sm−1 sm1x 2x m−1x mx
t t t t1 2 m−1 m
u u u
v v v
1
1
2
2
m−1
m−1
• Linearly solvable iff
m − 2 = 0
[Rai-Dey 2012]
Polynomial equations
s1 s2 sm−1 sm1x 2x m−1x mx
t t t t1 2 m−1 m
u u u
v v v
1
1
2
2
m−1
m−1
• Linearly solvable iff
m − 2 = 0
[Rai-Dey 2012]
Polynomial equations
t t t t
s ss1 2
1 2
u u u
v v
1
1
2
2v
m−1
m−1
m−1
m−1 m
• Linearly solvable iff
(m − 2)X − 1 = 0 has a solution
[Rai-Dey 2012]
How general/rich is the class of problems?
• Difficulty of computing capacity, solvability, designing codes
• Equivalence with other classes of problems
• Sufficiency of linear codes
Special classes:
• Multicast networks are not general enough - no network forf (X ) = 2X − 1.
• Multiple-unicast networks are very general ∼ systems of polynomials.
Equivalence with MUN [Rai-Dey 2012]
v2z 2
w2
u2
1 w
1z z
L
m
L1 R1 mR
v1 vm
um1
1 m m+1
m
m
w
s s
u
s s2
tR2ttL2t t t
network
Multiple−unicast
N1
• Equivalent under linear coding
• Equivalent under non-linear coding
• Same holds for the reverse networks
Equivalence with MUN [Rai-Dey 2012]
v2z 2
w2
u2
1 w
1z z
L
m
L1 R1 mR
v1 vm
um1
1 m m+1
m
m
w
s s
u
s s2
tR2ttL2t t t
network
Multiple−unicast
N1
• Equivalent under linear coding
• Equivalent under non-linear coding
• Same holds for the reverse networks
Equivalence with MUN [Rai-Dey 2012]
s1 s2 s3
u1u2 um
v1v2
rnmr31r21r1211r
s2n
t2nt2 v2n
u2n
s23
t23r23
u23
v23
u22
v
s22
t22r22 22
u
v
s12
tr 12
u13
v13
u1n
12
1212
s
r13 t
13
13
s1n
v1nt1 t1n
u
v t
v
m2
m2m2
sm2
rm2
sm3um3
rm3 m3 tm3
smnumn
tm vmn tmn
1mr r22 r2m r32 r3m rn2rn1
N
1 2 3 n
1 2 3 n
z z z z
z z z z
sm
vm
w w1 w2 w3 m
• Equivalent under linear coding
• Equivalent under non-linear coding
Equivalence with MUN [Rai-Dey 2012]
s1 s2 s3
u1u2 um
v1v2
rnmr31r21r1211r
s2n
t2nt2 v2n
u2n
s23
t23r23
u23
v23
u22
v
s22
t22r22 22
u
v
s12
tr 12
u13
v13
u1n
12
1212
s
r13 t
13
13
s1n
v1nt1 t1n
u
v t
v
m2
m2m2
sm2
rm2
sm3um3
rm3 m3 tm3
smnumn
tm vmn tmn
1mr r22 r2m r32 r3m rn2rn1
N
1 2 3 n
1 2 3 n
z z z z
z z z z
sm
vm
w w1 w2 w3 m
• Equivalent under linear coding
• Equivalent under non-linear coding
Sum-networks as a class
• Sum-networks are as (but not more) rich a class as general communicationnetworks
• Linear codes not sufficient
• Equivalence with polynomial equations
• Many more theoretical insights in future works [Rai and Das 2013, ...]
• Capacity can be arbitrarily less than the min − cut [Tripathy andRamamoorthy 2016] - also follows from [Rai and Dey 2012].
Thank You