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