Distributed computation and reconfiguration in actively dynamic networks

Distributed computation and recon
f
iguration
in actively dynamic networks
Peter Kos | CSCI 761 | 4.25.22

Overview
• Recon
f
igurable networks
• Cost measurement
• 3 Transformation algorithms
• Overview, Analysis, Runtime
• Impact
• Summary

Asynchronous LOCAL, but with two 💖 fun 💖 features:

Asynchronous LOCAL, but with two 💖 fun 💖 features:
1. Nodes can activate connections to new neighbors locally
2. Nodes can eliminate some of their existing connections

What is our goal?
Solve the depth-d tree problem!

What is our goal?
What is are our goals?
Gs → Gf
Solve the depth-d tree problem!
where is a rooted tree of depth
Gf d
is of (poly)log diameter
Gf
& via (poly)log time algorithms

Gs → Gf
Elect a unique leader
1)
2)
What is our goal?
Gf d

Gs → Gf
Gf d

Gs → Gf
R
Gf d
=“wreath” = ring + spanning binary tree
Gs

Gs → Gf
R
Gs
R
more than one total !
Gf d

Gs → Gf
R
Gf d
R
R
R

Gs → Gf
R
Gf d
Gs
R
more than one total !

(not necessarily runtime & message complexity…)
separate analysis for later

(polylog runtime & polylog diameter…)

(polylog runtime & polylog diameter…)
(deterministic & bounded-degree runtime…)

We have problems?

We have problems?
This is expensive…

We have problems?
1. Nodes can activate connections
2. Nodes can eliminate some of the
This is expensive…
Assume a cost of 1
for each connection
Θ(n2
) cost for clique formation

3 cost measures
Total number of
edge activations
Max number of
activated edges
in any round
Max activated
degree of a node
in any around
1
1
1 3
0

Total number of
edge activations
Max number of
activated edges
in any round
Max activated
degree of a node
in any around
1
1
1 3
0
T
∑
i=1
|Eac(i)|
T is runtime
max
i∈[T]
|E(i) E(1)| max
i∈[T]
Δ(D(i) D(1))
Graph di
ff
erence is de
f
ined as the
di
ff
erence of their edge sets
is max degree of
Δ(G) G

Total number of
edge activations
Max number of
activated edges
in any round
Max activated
degree of a node
in any around
1
1
1 3
0
Trying to optimize the worst-case
number of activations per round

De
f
initions
gadget network - something close to and that is helpful in reaching
Gf Gf
Gs → Gf

De
f
initions
star graph (gadget) spanning star ( )
Gf
gadget network - something close to and that is helpful in reaching
Gf Gf
Gs → Gf

De
f
initions
committee - subgraph that makes up Gs
Gs → Gf
means activated with in the previous round
C(u) ↔ C(v) u v
is a committee led by node
C(u) u
(this is my notation)

Algo 1
Gadget: star graph
: spanning star
Runtime:
Total edge act:
Final diameter: 2
Gf
O(logn)
O(n logn)
GraphToStar
Algo 2
Gadget: wreath
: binary spanning tree
Runtime:
Total edge act:
Final diameter:
Gf
O(log2
n)
O(n log2
n)
O(logn)
GraphToWreath
Algo 3
GraphToThinWreath
Gadget: wreath
Runtime:
Total edge act:
Final diameter:
Gf
o(log2
n)
O(n log2
n)
O(logn)
Runtime: ………………………
o(log2
n)

graph
g star
ct:
er: 2
logn)
O(n logn)
ar
Algo 2
Gadget: wreath
Runtime:
Total edge act:
Final diameter:
Gf
O(log2
n)
O(n log2
n)
O(logn)
GraphToWreath
Algo 3
GraphToThinWreath
Gadget: wreath
Runtime:
Total edge act:
Final diameter:
Gf
o(log2
n)
O(n log2
n)
O(logn)
Runtime: ………………………….
o(log2
n) O
(
log2
n
loglogkn)

Algo 1
Gadget: star graph
: spanning star
Runtime:
Total edge act:
Final diameter: 2
Gf
O(logn)
O(n logn)
GraphToStar
Algo 2
Gadget: wreath
Runtime:
Total edge act:
Final diameter:
Gf
O(log2
n)
O(n log2
n)
O(logn)
GraphToWreath
Algo 3
GraphToThinWreath
Gadget: wreath
Runtime:
Total edge act:
Final diameter:
Gf
o(log2
n)
O(n log2
n)
O(logn)
Runtime: ………………………
o(log2
n)
maintains bounded max degree
throughout runtime

Algo 1
Gadget: star graph
: spanning star
Runtime:
Total edge act:
Final diameter: 2
Gf
O(logn)
O(n logn)
GraphToStar
Algo 2
Gadget: wreath
Runtime:
Total edge act:
Final diameter:
Gf
O(log2
n)
O(n log2
n)
O(logn)
GraphToWreath
Algo 3
GraphToThinWreath
Gadget: wreath
Runtime:
Total edge act:
Final diameter:
Gf
o(log2
n)
O(n log2
n)
O(logn)

Algo 1
GraphToStar
Each committee is a star graph with leader
C(u) u
Leader of each committee is node with greatest UID
in committee
uC
UID of is the UID of
C(u) u
Winning committee is C(umax)

Algo 1
GraphToStar
Selection Merging Pulling Waiting Termination
Every node starts as leader, forms committee
n = 1
In every phase, executes in one of the following
modes, starting with selection mode
C(u)

Algo 1
GraphToStar

Algo 1
GraphToStar
Selection
If and are neighboring s.t. ,
and is not in pulling mode,
C(u) C(z) UIDz > UIDu
C(z)
Merging Pulling Waiting Termination
selects by doing the following:
C(u) C(z)
Activate with potential neighbor in
E(u, e1) C(v)
Activate , and deactivate
E(u, v) E(u, e1)
If did not select, then and form a pair and
enters merging mode.
C(v) C(u) C(v) C(u)
no diagram

Algo 1
GraphToStar
Now, the leader of has an edge
with the leader of
C(u)
C(v)
Each follower x in activates edge
and deactivates
C(u) xv
xu
and are now merged into ,
where is a star rooted at spanning all nodes
C(u) C(v) C(v)
C(v) v
V(C(u)) ∪ V(C(v))

Algo 1
GraphToStar
Now, the leader of has an edge
with the leader of
C(u)
C(v)
Each follower x in activates edge
and deactivates
C(u) xv
xu
and are now merged into ,
where is a star rooted at spanning all nodes
C(u) C(v) C(v)
C(v) v
V(C(u)) ∪ V(C(v))
u v

Algo 1
GraphToStar
Given and
C(u) ↔ C(v) C(v) ↔ C(w)
If leader of did not activate in previous phase {
enters merging mode
} else if and is now empty {
activates
enters merging mode
} else {
activates , deactivates
remains in pulling mode
}
C(v)
C(u)
C(u) ↔ C(v) C(v)
u uw
C(u)
u uw uv
C(u)
u
w
v

Algo 1
GraphToStar
Given and
C(u) ↔ C(v) C(v) ↔ C(w)
u
w
v
umerge
enters merging mode
activates
enters merging mode
} else {
}
C(v)
C(u)
C(u) ↔ C(v) C(v)
u uw
C(u)
u uw uv
C(u)

Algo 1
GraphToStar
Given and
C(u) ↔ C(v) C(v) ↔ C(w)
u
w
v
BREAK
umerge
enters merging mode
activates
enters merging mode
} else {
}
C(v)
C(u)
C(u) ↔ C(v) C(v)
u uw
C(u)
u uw uv
C(u)

Algo 1
GraphToStar
Given and
C(u) ↔ C(v) C(v) ↔ C(w)
u
w
v
enters merging mode
activates
enters merging mode
} else {
}
C(v)
C(u)
C(u) ↔ C(v) C(v)
u uw
C(u)
u uw uv
C(u)

Algo 1
GraphToStar
Given and
C(u) ↔ C(v) C(v) ↔ C(w)
u
w
v
umerge
BREAK
enters merging mode
activates
enters merging mode
} else {
}
C(v)
C(u)
C(u) ↔ C(v) C(v)
u uw
C(u)
u uw uv
C(u)

Algo 1
GraphToStar
if has no neighboring committees {
enters termination mode
} else if in prev phase, no committee activated edge with {
enters selection mode
} else {
remains in waiting mode
}
C(u)
C(u)
u
C(u)
C(u)
no diagram

Algo 1
GraphToStar
deactivates every edge in
C(u) E(Gs E(C(u))
Each follower in deactivates all incident active edges
except .
x C(u)
xu
no diagram

Algo 1
GraphToStar
Proof of
Correctness
Does it solve the tree problem?
Depth − 1
One committee eventually enters termination,
and this can only* be . (proved on next slide)
C(umax)
Then by de
f
inition, must be a unique leader
of the spanning star rooted at
umax
umax
This satis
f
ies all requirements of the tree problem.
Depth − 1

Algo 1
GraphToStar
Proof of
Correctness
Further conjecture:
How do we ensure is the only committee left alive?
(i.e., all other committees die or grow)
C(umax)
We need to prove that this occurs for any of the following modes:
Pulling Selection Waiting
? ? ?

Algo 1
GraphToStar
Proof of
Correctness
? ? ?

Algo 1
GraphToStar
Proof of
Correctness
Set of committees in pulling mode are Cpull
These form a forest such that belongs to a pulling
tree of
F C(u) ∈ Cpull
T F
? ? ?
Any pulling tree mimics TreeToStar algorithm on the leaders of
committees C(u)

tree of
F C(u) ∈ Cpull
T F
? ? ?
😠
committees C(u)

Algo 1
GraphToStar
Proof of
Correctness
tree of
F C(u) ∈ Cpull
T F
? ? ?
committees C(u)
All non-root committees in will eventually merge with
T Cr
Invariant: ’s root committee, is always in waiting mode,
and ’s children are always in merging mode
C(u) Cr
Cr
These eventually “telescope in” as ’s children merge with
Cr Cr

Algo 1
GraphToStar
Proof of
Correctness
tree of
F C(u) ∈ Cpull
T F
✔ ? ?
committees C(u)
All non-root committees in will eventually merge with
T Cr
Invariant: ’s root committee, is always in waiting mode,
and ’s children are always in merging mode
C(u) Cr
Cr
These eventually “telescope in” as ’s children merge with
Cr Cr

Algo 1
GraphToStar
Proof of
Correctness
Committee can enter either mode above
C(u)
✔ ? ?
If it enters merging or pulling, will eventually die
C(u)
If it enters waiting, will eventually grow
What if we stay in section mode inde
f
initely?
Can only happen if all current & neighboring committees
have a smaller than
UID UIDu
Contradicts inde
f
inite local maximality of in
neighbor selection process for some neighbor such that
UIDu
C(w)
UIDw > UIDu

Algo 1
GraphToStar
Proof of
Correctness
Committee can enter either mode above
C(u)
✔ ?
If it enters merging or pulling, will eventually die
C(u)
If it enters waiting, will eventually grow
What if we stay in section mode inde
f
initely?
Can only happen if all current & neighboring committees
have a smaller than
UID UIDu
Contradicts inde
f
inite local maximality of in
neighbor selection process for some neighbor such that
UIDu
C(w)
UIDw > UIDu
✔

Algo 1
GraphToStar
Proof of
Correctness
Any committee is either a root of a pulling tree in forest ,
or a root of star of committees in which all leaf-committees
are merging with
C(u) F
C(u)
✔ ?
✔
Either way, eventually enters selection mode
as soon as all other committees in its pulling tree or star
have merged to it.
C(u)

Algo 1
GraphToStar
Proof of
Correctness
Any committee is either a root of a pulling tree in forest ,
or a root of star of committees in which all leaf-committees
are merging with
C(u) F
C(u)
✔ ✔
Either way, eventually enters selection mode
as soon as all other committees in its pulling tree or star
have merged to it.
C(u)
✔

Algo 1
GraphToStar
Proof of
Correctness
✔ ✔ ✔
QED.

Algo 1
GraphToStar
Proof of
Time Complexity

Algo 1
GraphToStar
Proof of
Time Complexity
For now, focus on number of phases before
a single committee is left.
is the size of committee in round
|C(u)s | C(u) s

Algo 1
GraphToStar
Proof of
Time Complexity
Lemma 2
Committee between phases and
If , then in selection mode
(i.e., in two following phases, the size is at least )
C(u)waiting s s + j
|C(u)s | > 2k
|C(u)s+j+1 | > 2k+j−2
2k+j−2
Any is a root of either:
(i) a pulling tree T in the forest F
(ii) a star of committees which all leaf-committees
are merging with
C(u)waiting
C(u)

Algo 1
GraphToStar
Proof of
Time Complexity
Lemma 2
C(u)waiting s s + j
|C(u)s | > 2k
|C(u)s+j+1 | > 2k+j−2
2k+j−2
C(u)waiting
, so any other committee in is either in
pulling or merging mode
C(u)waiting T
All non-root committees in will eventually merge in
in some phase
T C(u)
s + j

Algo 1
GraphToStar
Proof of
Time Complexity
Lemma 2
C(u)waiting s s + j
|C(u)s | > 2k
|C(u)s+j+1 | > 2k+j−2
2k+j−2
C(u)waiting
W.l.o.g. assume every committee
(i) entered pulling or merging mode in phase
(ii) will have merged with by phase
C(v) ∈ T
s
C(v) s + j

Algo 1
GraphToStar
Proof of
Time Complexity
Lemma 2
C(u)waiting s s + j
|C(u)s | > 2k
|C(u)s+j+1 | > 2k+j−2
2k+j−2
C(u)waiting
W.l.o.g. assume every committee
(i) entered pulling or merging mode in phase
(ii) will have merged with by phase
C(v) ∈ T
s
C(v) s + j
(Left here)

Algo 2
GraphToWreath
Each committee is a wreath graph with leader
C(u) u
Proceeds in phases, always start in selection mode.
Selection Ring merging Tree Merging Termination

Algo 2
GraphToWreath
Selection Ring merging Tree Merging Termination
deactivates every edge in
C(u) E(Gs E(C(u))
Each follower in deactivates all incident active edges
except .
x C(u)
xu

Distributed computation and recon
f
iguration
in actively dynamic networks
Peter Kos | CSCI 761 | 6.25.22

Distributed computation and reconfiguration in actively dynamic networks

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