Jointly Optimal Routing & Caching for Arbitrary Network - - PowerPoint PPT Presentation

Jointly Optimal Routing & Caching for Arbitrary Network Topologies

Stratis Ioannidis and Edmund Yeh ICN 2017

A Caching Network

Jointly Optimal Routing and Caching 1

User nodes generate requests for content items with certain arrival rates

?

λ

User Webserver

A Caching Network

Jointly Optimal Routing and Caching 2

Requests are routed towards a designated server ? User Webserver

?

λ

Jointly Optimal Routing and Caching 3

?

A Caching Network

Nodes have caches with finite capacities User

?

λ

Webserver

Jointly Optimal Routing and Caching 4

?

A Caching Network

User Routers have caches with finite capacities

?

λ

Webserver

Jointly Optimal Routing and Caching 5

Requests terminate early upon a cache hit ?

A Caching Network

User

?

λ

Webserver

Jointly Optimal Routing and Caching 6

Design network so that expected routing costs are minimized.

Goal

User ? Webserver 5 4 6 3 4 3 3 10 10 1 1 2 2 10 6 7

Jointly Optimal Routing and Caching 7

Joint Optimization

User ?

q Both caching and routing decisions are part of

• ptimization

Caching decisions Routing decisions Webserver

?

λ

Jointly Optimal Routing and Caching 8

Additional Challenge

Challenge: Caching and routing algorithm should be q adaptive, and q distributed.

User ? Caching decisions Routing decisions Webserver

Jointly Optimal Routing and Caching 9

Fixed Routing Setting

User ?

q Routes are given, e.g.: q determined by BGP, OSPF, etc. q Route to Nearest Server (RNS) q Only caching decisions are part of optimization

Caching decisions Webserver [Ioannidis, Yeh, SIGMETRICS 2016]

qShould we jointly optimize caching and routing? Is RNS sufficient? qCan we devise algorithms that: 1) have performance guarantees? 2) are distributed and adaptive? Challenges/Questions

Jointly Optimal Routing and Caching 10

q Mathematical model for joint optimization of routing & caching

q Source-routing q Hop-by-hop

q RNS is arbitrarily suboptimal q A distributed, adaptive algorithm within 1-1/e factor from

• ptimal

q Evaluation over 14 network topologies:

q Routing costs reduced by factor 103 compared to RNS.

Our Contributions

Jointly Optimal Routing and Caching 11

qModel qMain Results

qRNS is suboptimal qOffline Algorithm qDistributed, Adaptive Algorithm

qEvaluation Overview

Jointly Optimal Routing and Caching 12

qModel qMain Results

qRNS is suboptimal qOffline Algorithm qDistributed, Adaptive Algorithm

qEvaluation Overview

Jointly Optimal Routing and Caching 13

Model: Network

Jointly Optimal Routing and Caching 14

Network represented as a directed, bi-directional graph G(V, E)

G(V, E)

Model: Edge Costs

Jointly Optimal Routing and Caching 15

Each edge has a cost/weight

G(V, E)

5

Edge costs:

wuv

wuv, (u, v) ∈ E

(u, v) ∈ E

Model: Node Caches

Jointly Optimal Routing and Caching 16

Node has a cache with capacity

G(V, E)

v ∈ V

cv ∈ N

Node capacities: cv, v ∈ V Edge costs:

wuv, (u, v) ∈ E

Model: Cache Contents

Jointly Optimal Routing and Caching 17

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

Items stored and requested form the item catalog C

wuv, (u, v) ∈ E

Model: Cache Contents

Jointly Optimal Routing and Caching 18

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

For and , let v ∈ V

i ∈ C

xvi = ( 1, 0,

wuv, (u, v) ∈ E

if stores

• .w.

v

i

Then, for all , v ∈ V X

i∈C

xvi ≤ cv Caching Strategy of :

xv = [xvi]i∈C

v ∈ V

Caching strategy: xv = [xvi]i∈C

Model: Designated Sources

Jointly Optimal Routing and Caching 19

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

For each and , there exists a set of nodes (the designated servers of ) that permanently store .

i ∈ C

wuv, (u, v) ∈ E

Si ⊂ V i i

I.e., if then

v ∈ Si xvi = 1

Caching strategy: xv = [xvi]i∈C

Model: Requests

Jointly Optimal Routing and Caching 20

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

A request is a pair such that:

wuv, (u, v) ∈ E

q is an item in q is the source node of the request.

i

C

?

(i, s)

s ∈ V

s

Caching strategy: xv = [xvi]i∈C

Model: Demand

Jointly Optimal Routing and Caching 21

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

wuv, (u, v) ∈ E

? Demand : set of all requests

R

Request arrival process is Poisson with rate

(i, s)

λ(i,s)

s

?

λ

Caching strategy: xv = [xvi]i∈C Demand: λ(i,s), (i, s) ∈ R

Jointly Optimal Routing and Caching 22

Routing Variant I : Source Routing

Source node generating request selects entire path that request should follow.

User ? ? ?

Jointly Optimal Routing and Caching 23

Routing Variant II: Hop-By-Hop Routing

Each node selects only next-hop in the path.

User ? ? ? ?

Source Routing: Routing Strategy

Jointly Optimal Routing and Caching 24

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

For each request let be a set of possible paths the request can follow do a node in

wuv, (u, v) ∈ E

?

(i, s)

P(i,s) Si

s

Caching strategy: xv = [xvi]i∈C Demand: λ(i,s), (i, s) ∈ R

P(i,s)

Set of possible paths:

Source Routing: Routing Strategy

Jointly Optimal Routing and Caching 25

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

wuv, (u, v) ∈ E

?

P(i,s)

Routing strategy : probability distribution over

r(i,s)

X

p∈P(i,s)

r(i,s),p = 1

r(i,s),p ≥ 0, p ∈ P(i,s)

s

Caching strategy: xv = [xvi]i∈C Demand: λ(i,s), (i, s) ∈ R Routing strategy:

r(i,s) = [r(i,s),p]p∈P(i,s)

P(i,s)

Set of possible paths:

Source Routing: Routing Costs

Jointly Optimal Routing and Caching 26

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

wuv, (u, v) ∈ E

?

P

p∈P(i,s) r(i,s),p

P|p|−1

k=1 wpk+1pk

Qk

k0=1(1 − xpk0i)

Expected routing cost: Upper bound:

P

p∈P(i,s)

P|p|−1

k=1 wpk+1pk

Caching Gain: P

p∈P(i,s)

P|p|−1

k=1 wpk+1pk

h 1 − r(i,s),p Qk

k0=1(1 − xpk0i)

i

s

Caching strategy: xv = [xvi]i∈C Demand: λ(i,s), (i, s) ∈ R Routing strategy:

r(i,s) = [r(i,s),p]p∈P(i,s)

P(i,s)

Set of possible paths:

Caching Gain Maximization

Jointly Optimal Routing and Caching 27

G(V, E)

C={

}

Node capacities: cv, v ∈ V Edge costs:

wuv, (u, v) ∈ E

R: demand

5 3 4 6 ?

Maximize:

MaxCG

Subject to:

X

i∈C

xvi = cv for all v ∈ V xvi ∈ {0, 1} for all and v ∈ V

i ∈ C

xvi = 1

for all and

v ∈ Si

i ∈ C

F(r, X) = X

(i,s)∈R

λ(i,s) X

p∈P(i,s) |p|−1

X

k=1

wpk+1pk " 1 − r(i,s),p

k

Y

k0=1

(1 − xpk0i) #

r(i,s),p ∈ [0, 1]

p ∈ P(i,s) (i, s) ∈ R

,

for all

X

p∈P(i,s)

r(i,s),p = 1 for all

(i, s) ∈ R

s

Caching strategy: xv = [xvi]i∈C Demand: λ(i,s), (i, s) ∈ R Routing strategy:

r(i,s) = [r(i,s),p]p∈P(i,s)

P(i,s)

Set of possible paths:

Jointly Optimal Routing and Caching 28

Fixed Routing Setting (e.g., RNS)

User ? Caching decisions Webserver

|P(i,s)| = 1, ∀(i, s) ∈ R

qNP-hard qOffline algorithm achieves 1-1/e approximation qDistributed, adaptive algorithm achieving 1-1/e approximation

[Shanmugam et al. 2013] [Shanmugam et al. 2013] [Ioannidis & Yeh 2016]

qModel qMain Results

qRNS is suboptimal qOffline Algorithm qDistributed, Adaptive Algorithm

qEvaluation Overview

Jointly Optimal Routing and Caching 29

Jointly Optimal Routing and Caching 30

Is Joint Optimization Really Necessary?

User ? Why not just use shortest path routing towards the nearest designated server?

Is Joint Optimization Really Necessary?

Jointly Optimal Routing and Caching 31

Shortest path routing to nearest server is arbitrarily suboptimal.

RNS is Suboptimal

Jointly Optimal Routing and Caching 32

M

1

M

s t 2 c = 1 c = 1

= = 0.5

λ

?

λ

?

requests per sec Shortest path for both

Routing to Nearest Server is Suboptimal

Jointly Optimal Routing and Caching 33

M

1

M

s t 2 c = 1 c = 1

= = 0.5

λ

?

λ

?

requests per sec

Irrespective of caching decision, cost under shortest path routing is Θ(M)

? ?

M

1

M

s t 2 c = 1 c = 1

Routing to Nearest Server is Suboptimal

Jointly Optimal Routing and Caching 34

= = 0.5

λ

?

λ

?

requests per sec

Irrespective of caching decision, cost under shortest path routing is

? ?

Cost under split routing strategy is .

O(1)

Shortest path routing to nearest server is arbitrarily suboptimal. Θ(M)

Key Intuition

Jointly Optimal Routing and Caching 35

Increasing path diversity creates more caching opportunities.

?

Offline Algorithm

Jointly Optimal Routing and Caching 36

Maximize:

MaxCG

Subject to:

X

i∈C

xvi = cv for all v ∈ V

for all and

v ∈ V

i ∈ C

xvi = 1

for all and

v ∈ Si

i ∈ C

F(r, X) = X

(i,s)∈R

λ(i,s) X

p∈P(i,s) |p|−1

X

k=1

wpk+1pk " 1 − r(i,s),p

k

Y

k0=1

(1 − xpk0i) #

r(i,s),p ∈ [0, 1]

p ∈ P(i,s) (i, s) ∈ R

,

for all

X

p∈P(i,s)

r(i,s),p = 1 for all

(i, s) ∈ R

xvi ∈ {0, 1}

Offline Algorithm

Jointly Optimal Routing and Caching 37

Maximize: Subject to:

X

i∈C

xvi = cv for all v ∈ V

for all and

v ∈ V

i ∈ C

xvi = 1

for all and

v ∈ Si

i ∈ C

F(r, X) = X

(i,s)∈R

λ(i,s) X

p∈P(i,s) |p|−1

X

k=1

wpk+1pk " 1 − r(i,s),p

k

Y

k0=1

(1 − xpk0i) #

r(i,s),p ∈ [0, 1]

p ∈ P(i,s) (i, s) ∈ R

,

for all

X

p∈P(i,s)

r(i,s),p = 1 for all

(i, s) ∈ R

xvi ∈ [0, 1]

Offline Algorithm

Jointly Optimal Routing and Caching 38

Maximize: Subject to:

X

i∈C

xvi = cv for all v ∈ V

for all and

v ∈ V

i ∈ C

xvi = 1

for all and

v ∈ Si

i ∈ C

F(r, X) = X

(i,s)∈R

λ(i,s) X

p∈P(i,s) |p|−1

X

k=1

wpk+1pk " 1 − r(i,s),p

k

Y

k0=1

(1 − xpk0i) #

r(i,s),p ∈ [0, 1]

p ∈ P(i,s) (i, s) ∈ R

,

for all

X

p∈P(i,s)

r(i,s),p = 1 for all

(i, s) ∈ R

xvi ∈ [0, 1]

q Key idea: There exists a concave function such that

(1 − 1 e)L(r, X) ≤ F(r, X) ≤ L(r, X)

L(r, X)

Offline Algorithm

Jointly Optimal Routing and Caching 39

Maximize: Subject to:

X

i∈C

xvi = cv for all v ∈ V

for all and

v ∈ V

i ∈ C

xvi = 1

for all and

v ∈ Si

i ∈ C r(i,s),p ∈ [0, 1]

p ∈ P(i,s) (i, s) ∈ R

,

for all

X

p∈P(i,s)

r(i,s),p = 1 for all

(i, s) ∈ R

xvi ∈ [0, 1]

q Key idea: There exists a concave function such that

(1 − 1 e)L(r, X) ≤ F(r, X) ≤ L(r, X)

L(r, X)

L(r, X) = X

(i,s)∈R

λ(i,s) X

p∈P(i,s) |p|−1

X

k=1

wpk+1pk min{1, 1 − r(i,s),p +

k

X

k0=1

xpk0i}

Follows from Goemans-Williamson inequality:

• 1 − 1

e

• max
• 1,

X

i

yi ≤ 1 − Y

i

(1 − yi) ≤ max

• 1,

X

i

yi , for all yi ∈ [0, 1]

Offline Algorithm

Jointly Optimal Routing and Caching 40

Maximize: Subject to:

X

i∈C

xvi = cv for all v ∈ V

for all and

v ∈ V

i ∈ C

xvi = 1

for all and

v ∈ Si

i ∈ C r(i,s),p ∈ [0, 1]

p ∈ P(i,s) (i, s) ∈ R

,

for all

X

p∈P(i,s)

r(i,s),p = 1 for all

(i, s) ∈ R

xvi ∈ [0, 1]

q Offline Algorithm: q Step 1: Solve relaxed convex optimization problem q Step 2: Round caching solution via pipage rounding [Ageev and Sviridenko 2004]

L(r, X) = X

(i,s)∈R

λ(i,s) X

p∈P(i,s) |p|−1

X

k=1

wpk+1pk min{1, 1 − r(i,s),p +

k

X

k0=1

xpk0i}

Theorem: Resulting solution is within 1-1/e from the optimal solution to MaxCG

Distributed, Adaptive Algorithm

Jointly Optimal Routing and Caching 41

Each node maintains the following state information:

C={

}

v

0.5 0.9 0.6

s

r(i,s)

0.1 0.5 0.4

r = [r(i,s)](i,s)∈R X = [xv]v∈V

s

r(i,s)

q Each source maintains routing probabilities

xv

v xv = [xvi]i∈C

q Each cache maintains caching probabilities routing probabilities: caching probabilities:

Distributed, Adaptive Algorithm

Jointly Optimal Routing and Caching 42

Time is slotted.

C={

}

v

0.5 0.9 0.6

s

r(i,s)

0.1 0.5 0.4

xv

v

q Each cache estimates

s

q Each source estimates rr(i,s)L(r, X)

rxvL(r, X)

During each timeslot: via control packets exchanged between nodes during timeslot

r = [r(i,s)](i,s)∈R X = [xv]v∈V

routing probabilities: caching probabilities:

Distributed, Adaptive Algorithm

Jointly Optimal Routing and Caching 43

At the end of a timeslot, all nodes adapt their state:

C={

}

v

0.5 0.9 0.6

s

r(i,s)

0.1 0.5 0.4

xv

v

q Each cache :

s

q Each source :

xv ΠDv(xv + γk\ rxvL(r, X))

Stochastic Projected Gradient Ascent (PGA)

r(i,s) ΠDr(i,s) (r(i,s) + γk \ rr(i,s)L(r, X))

r = [r(i,s)](i,s)∈R X = [xv]v∈V

routing probabilities: caching probabilities:

Distributed, Adaptive Algorithm

Jointly Optimal Routing and Caching 44

At the end of a timeslot:

C={

}

v

0.5 0.9 0.6

s

r(i,s)

0.1 0.5 0.4

xv v

q Each cache reshuffles its contents independently according to

s

q Each source routes packets independently according to

r(i,s)

xv r = [r(i,s)](i,s)∈R X = [xv]v∈V

routing probabilities: caching probabilities:

Theorem: For , under Projected Gradient Ascent, the randomized allocation and the randomized routing strategy at timeslot are such that: where is an optimal solution to the offline problem.

Convergence

Jointly Optimal Routing and Caching 45

γk = 1/ √ k

Xk rk k

lim

k→∞ E[F(r, X)] ≥ (1 − 1

e)F(r∗, X∗)

(r∗, X∗)

qModel qMain Results

qRNS is suboptimal qOffline Algorithm qDistributed, Adaptive Algorithm

qEvaluation Overview

Jointly Optimal Routing and Caching 46

Experiments

Jointly Optimal Routing and Caching 47

Graph |V | |E| |C| |R| cycle 30 60 10 100 grid-2d 100 360 300 1K hypercube 128 896 300 1K expander 100 716 300 1K erdos-renyi 100 1042 300 1K regular 100 300 300 1K watts-strogatz 100 400 300 1K small-world 100 491 300 1K barabasi-albert 100 768 300 1K geant 22 66 10 100 abilene 9 26 10 90 dtelekom 68 546 300 1K | cv |P(i,s)| 2 2 3 30 3 30 3 30 3 30 3 30 3 2 3 30 3 30 2 10 2 10 3 30

Graph Topologies Routing Algorithms q Shortest Path (RNS) q Uniform q Dynamic routing: PGA on L for routes alone q LRU q LFU q FIFO q RR q PGA on L Caching Algorithms

Performance Comparison

Jointly Optimal Routing and Caching 48

Ratio of expected routing cost to routing cost under our algorithm

small-world

100 101 102 103

dtelekom

cost(·) cost(PGA)

Performance Comparison

Jointly Optimal Routing and Caching 49 LRU-S LFU-S FIFO-S RR-S PGA-S LRU-U LFU-U FIFO-U RR-U PGA-U PGA-U LRU-D LFU-D FIFO-D RR-D PGA

small-world

100 101 102 103

dtelekom

shortest path uniform dynamic routing Ratio of expected routing cost to routing cost under our algorithm

cost(·) cost(PGA)

Performance Comparison

Jointly Optimal Routing and Caching 50 LRU-S LFU-S FIFO-S RR-S PGA-S LRU-U LFU-U FIFO-U RR-U PGA-U PGA-U LRU-D LFU-D FIFO-D RR-D PGA

small-world

100 101 102 103

dtelekom

Path diversity Ratio of expected routing cost to routing cost under our algorithm

cost(·) cost(PGA)

Performance Comparison

Jointly Optimal Routing and Caching 51 LRU-S LFU-S FIFO-S RR-S PGA-S LRU-U LFU-U FIFO-U RR-U PGA-U PGA-U LRU-D LFU-D FIFO-D RR-D PGA

small-world

100 101 102 103

dtelekom

Optimizing Routing Ratio of expected routing cost to routing cost under our algorithm

cost(·) cost(PGA)

Performance Comparison

Jointly Optimal Routing and Caching 52 LRU-S LFU-S FIFO-S RR-S PGA-S LRU-U LFU-U FIFO-U RR-U PGA-U PGA-U LRU-D LFU-D FIFO-D RR-D PGA

small-world

100 101 102 103

dtelekom

Jointly Optimizing Caching & Routing

cost(·) cost(PGA)

Performance Comparison

Jointly Optimal Routing and Caching 53

cycle grid-2d hypercube expander erdos-renyi regular

100 101 102 103 LRU-S LFU-S FIFO-S RR-S PGA-S LRU-U LFU-U FIFO-U RR-U PGA-U LRU-D LFU-D FIFO-D RR-D PGA

watts-strogatz small-world barabasi-albert geant abilene dtelekom

100 101 102 103

qRNS is suboptimal qPath diversity caching opportunities qJoint optimization with guarantees over arbitrary topologies Conclusions & Open Problems

Jointly Optimal Routing and Caching 54

Open Problems qOptimize cost rather than gain qSpeed up convergence & reduce state qIntroduce queueing delays

Thank you!

Gradient Estimation

Jointly Optimal Routing and Caching 56

How can nodes estimate gradient in a distributed fashion?

C={

}

yv v

Y = [yv]v∈V

0.5 0.9 0.6

s

r(i,s)

0.1 0.5 0.4

r = [r(i,s)](i,s)∈R

Gradient Estimation

Jointly Optimal Routing and Caching 57

When request is generated, create a new control message

C={

}

?

(i, s)

Y = [yv]v∈V r = [r(i,s)](i,s)∈R

s

Gradient Estimation

Jointly Optimal Routing and Caching 58

Forward it to a path selected u.a.r. from

C={

}

?

Y = [yv]v∈V r = [r(i,s)](i,s)∈R P(i,s)

s

Gradient Estimation

Jointly Optimal Routing and Caching 59

C={

}

? 5 3 4 6

Y = [yv]v∈V r = [r(i,s)](i,s)∈R

r(i,s)

0.1 0.6 0.3 If is chosen, initialize counter in message to

p ∈ P(i,s)

1 − r(i,s),p

0.4

s

Gradient Estimation

Jointly Optimal Routing and Caching 60

Forward control message over path until:

C={

}

yv v

? 5 3 4 6 0.6 0.2 0.7 0.5 0.9 0.6 0.5 0.9 0.6

p Y = [yv]v∈V r = [r(i,s)](i,s)∈R

0.4

1 − r(i,s),p +

k

X

k0=1

ypk0i > 1

s

s

Gradient Estimation

Jointly Optimal Routing and Caching 61

Send control message over reverse path, collecting sum of edge costs.

C={

}

yv v

? 5 3 4 6 0.6 0.2 0.7 0.5 0.9 0.6 0.5 0.9 0.6

+3 +8

Each node on reverse path, sniffs upstream costs, and maintains average per item . Forward until: Average at end of slot is estimate of i ∈ C

∂L(r, Y ) ∂yvi

Upon receipt, source node averages same values to estimate ∂L(r, Y )

∂r(i,s),p

Y = [yv]v∈V r = [r(i,s)](i,s)∈R

1 − r(i,s),p +

k

X

k0=1

ypk0i > 1

qPath to closest cache learned adaptively qOptimization of caches and paths happens jointly

qCaching decisions affect routing and vice-versa qSame control plane for both

Key Insights

Jointly Optimal Routing and Caching 62

Jointly Optimal Routing and Caching 63

Routing Variant I : Source Routing

Source node generating request selects entire path that request should follow.

User ? ? ?

Jointly Optimal Routing and Caching 64

Routing Variant I : Source Routing

Source node generating request selects entire path that request should follow.

User ?

Jointly Optimal Routing and Caching 65

Routing Variant II: Hop-By-Hop Routing

Each node selects only next-hop in the path.

User ? ? ? ?

Jointly Optimal Routing and Caching 66

Routing Variant II: Hop-By-Hop Routing

Each node selects only next-hop in the path.

User ?

Jointly Optimal Routing and Caching 67

Routing Variant II: Hop-By-Hop Routing

Each node selects only next-hop in the path.

User ? ? ? ?

Jointly Optimal Routing and Caching 68

Routing Variant II: Hop-By-Hop Routing

Each node selects only next-hop in the path.

User ?

Jointly Optimal Routing and Caching 69

Routing Variant II: Hop-By-Hop Routing

Each node selects only next-hop in the path.

User ? ? ? ? ?

Same Principles Apply to Hop-By-Hop Routing

Jointly Optimal Routing and Caching 70

0.4 0.1 0.6 0.3

yvi + 1 − r(i,v),e v v

When receives a control message, it forwards it to random edge , adding to it

e

Randomized Placement

Jointly Optimal Routing and Caching 71

How can place exactly items in its cache, so that marginals are satisfied?

C={

}

0.5 0.9 0.6

yv

L(Y ) = X

(i,p)∈R

λ(i,p) P|p|−1

k=1 wpk+1pk min{1, Pk k0=1 yp0

ki}

v

5 3 4 6

v

cv

cv = 2

Y = [yv]v∈V

Randomized Placement

Jointly Optimal Routing and Caching 72

Suppose that I give you a such that .

X

i∈C

yvi = cv

yv ∈ [0, 1]|C|

Is there a way to select exactly items at random, so that the probability that item is selected is ?

cv

i

yvi yv1

yv2 yv3

yv4

= 0.82 = 0.77 = 0.77 = 0.64

X

i∈C

yvi = 3

cv = 3

C = {1, 2, 3, 4}

Randomized Placement: Sketch of Algorithm

Jointly Optimal Routing and Caching 73

µ1 µ2 µ3 µ4

1

cv = 3

yv1

yv2 yv3

yv4

Triplets: