Logical locality entails
frugal distributed computation
over graphs
Stéphane Grumbach, Zhilin Wu
INRIA-LIAMA
07/22/09
WG 2009
1
Introduction
Find a route to
node g
a
b
c
Build a spanning
tree with me as
the root
f
g
d
k
e
Use logics to describe globally
the expected results of network
functionalities
j
h
Logical formulas
l
i
m
Distributed computation
(complexity)
n
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Classical logics




First order logic (FO)
Fixpoint logics
Monadic second order logic (MSO)
…
The starting point: FO
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First Order Logic

Foundation of mathematics and theoretical
computer science


Used in databases: relational query languages
Expressive power

Locality


Local areas of the graphs sufficient to evaluate FO
formulas
Expressive power limited

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Connectivity, parity (whether a set is of even size) are
non-local properties, thus not expressible in FO
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Complexity of FO

Sequential complexity


Parallel complexity


Linear time over classes of locally tree
decomposable graphs (including bounded degree
graphs, planar graphs, etc)
Evaluated in AC0, constant time over Boolean
circuits with unbounded fan-in
Distributed complexity?
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Local distributed computation


Message passing model
Constant distributed time

Compute with information only from a bounded
neighborhood
Solve global problems
by local distributed computations around each node
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Does logical locality imply
local distributed computation?

The answer is “NO”
Simple properties cannot be checked by local
computations
“There are at least two distinct triangles”
Non-local communication is necessary
to check the distinctness of the two triangles
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Frugal distributed computations


A bounded number of messages
of size O(log n)
sent over each link
Frugal computations resemble
local computations
over bounded degree networks


Each node receives only
a bounded number of messages
Though the messages can come from
remote nodes
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Outline





Locality of first order logic (FO)
Frugal distributed computation
Frugality of FO over bounded degree
networks
Frugality of FO over planar networks
Frugality beyond FO properties
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Locality of FO: Intuition
Connectivity : non-local
Parity : non-local
2k
k
k
...
Versus
Versus
2k+1
2k
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...
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Neighborhood




First order logic over graphs
FO with vocabulary “E(x,y)”
k-neighborhood of v (Nk(v))
Vertices with distance from v no more than k
Let v be a list of vertices, then
N k (v ) :=  N k (vi )
i
“The distance between x and z is no more than k”
Expressed by an FO formula dist ( x , z ) ≤ k
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Local Formula

(k )
ϕ
(
y
)
(x)
Local formula
( x is a list of variables not occurring in ϕ ( y ) )
The quantifiers of ϕ ( y ) are bounded to the kneighborhood of x
 Example, if ϕ ( y ) := ∃ zψ ( y , z ) , then
(ϕ ( y ))
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(k )
(
( x ) := ∃ z dist ( x , z ) ≤ k ∧ (ψ ( y , z ) ) ( x )
(k )
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)
12
Gaifman theorem

Each FO formula ϕ (u ) ( u = u1 ⋯ u p )
can be written into a Boolean combination
of sentences of the form
∃ x1 ⋯ ∃ xs  ∧ d ( xi , x j ) > 2r ∧ ∧ ψ ( r ) ( xi ) 
i
 1≤ i < j ≤ s

ψ
(t )
( y ),
and formulae of the form
where y = y1 ⋯ yq and yi ∈ {u1,⋯, u p }
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Outline





Locality of first order logic (FO)
Frugal distributed computation
Frugality of FO over bounded degree
networks
Frugality of FO over planar networks
Frugality beyond FO properties
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Distributed computation model





Message passing model
Asynchronous system
Network topology modeled by a graph
G=(V,E)
Unique identifier for each node
A breath-first-search (BFS) tree
rooted at a distinguished node (requesting
node) distributively stored
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Frugal distributed algorithms
Only a constant number of messages
of size O(log n)
sent over each link
during its computation
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Frugal computation of FO sentences

Let C be a class of graphs, and ϕ be an FO
sentence. ϕ can be frugally computed over C
if
∃ a frugal distributed algorithm such that
∀ G ∈ C and any requesting node in G,
after the algorithm terminates, the requesting
node is in the accepting state iff G |= ϕ
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Outline





Locality of first order logic (FO)
Frugal distributed computation
Frugality of FO over bounded degree
networks
Frugality of FO over planar networks
Frugality beyond FO properties
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Locality of FO
over bounded degree networks

r-type of v
Isomorphic type of (〈Nr(v)〉G,v)

G1 and G2 are (r,m)-equivalent
For every r-type τ,
either G1and G2 have the same number of vertices
with r-type τ,
or both have at least m vertices with r-type τ.
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Locality of FO
over bounded degree networks
(continued)

Theorem. Fagin-Stockmeyer-Vardi
∀k,d, ∃r=f(k), m=g(k,d), s.t.
For all G1 and G2 with degree at most d,
if G1 and G2 are (r,m)-equivalent, then
G1 and G2 satisfy the same
FO sentences with quantifier rank at most k.
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Frugal algorithm for FO
over bounded degree networks



The requesting node ask each node to collect
the topology information of its rneighborhood.
The r-types of the nodes are
aggregated through the BFS tree
to the requesting node
up to the threshold m for each r-type.
Finally the requesting node
decides whether G |=ϕ
by using the information about the r-types.
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Complexity



The total number of distinct r-types of
graphs with degree bounded by d
depends only on r and d
Each r-type is counted up to the threshold m
Over each link,
only O(1) messages are sent during the
computation
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Outline





Locality of first order logic (FO)
Frugal distributed computation
Frugality of FO over bounded degree
networks
Frugality of FO over planar networks
Frugality beyond FO properties
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Frugality of FO
over planar networks

Idea
Transform
the linear time centralized evaluation algorithm
into
the frugal distributed evaluation algorithm
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Linear time centralized computation of
FO over planar graphs


Sufficient to consider FO sentences of the form
∃ x1 ⋯ ∃ xs  ∧ d ( xi , x j ) > 2r ∧ ∧ ψ
i
 1≤ i < j ≤ s
Centralized algorithm


( xi ) 

Select a distinguished vertex v0, decompose G=(V,E) into
subgraphs G[i,i+2r]:={v | i ≤ dist(v0,v) ≤ i+2r}.
Evaluate ψ(r)(x) over G[i,i+2r]



(r )
Each G[i,i+2r] is a graph with bounded tree width
Construct from ψ(r)(x) an automaton A and run it over a tree
decomposition obtained from G[i,i+2r]
Let P be the set of vertices v such that ∃i, Nr(v) ⊆G[i,i+2r] and
G[i,i+2r] |= ψ(r)(v).
Decide whether ψ ′ := ∃ x1 ⋯ ∃ xs  1≤ i∧< j ≤ s d ( xi , x j ) > 2r ∧ ∧i P ( xi )  holds over G or
not
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Distributed computation of FO over
planar networks


Transform the centralized algorithm
into a distributed one
while ensuring frugality
The most difficult part is
to evaluate distributively ψ(r)(x) over G[i,i+2r]
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Distributively Evaluate ψ(r)(x) over
G[i,i+2r]


ψ (r) (x) is a local formula, so
ψ (r) (x) can be evaluated over each connected
component of G[i,i+2r]
Still use automata-theoretic technique



Construct an automaton A from ψ (r) (x)
Construct and store distributively a tree
decomposition for each connected component of
G[i,i+2r]
Evaluate ψ (r) (x) by distributively running A over the
tree decomposition.
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Construct a tree decomposition for Ci

Let Ci be a connected component
of G[i,i+2r] and w1i ,⋯, wli be the
nodes of Ci with distance i from the
requesting node.


Let Ci′ be the graph obtained from Ci by
i
i
including all ancestors of w1 ,⋯, wl in
the BFS tree.
Let Ci* be the graph obtained from Ci by
i
i
contracting all the ancestors of w1 ,⋯ , wl
into one vertex v*
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Ci′ Ci
w1i
w1i
w1i
*
v
..
.
.
.. . . .
. . C. i
...
...
...
i
w
l
i
wl
wli
Ci* is a planar graph with
bounded diameter
28
Construct a tree decomposition for Ci
(continued)


A tree decomposition of
planar networks with bounded diameter
can be computed frugally
[Grumbach & Wu, manuscript]
Construct a tree decomposition for Ci
Construct distributively a tree decomposition for Ci*,
while doing special treatments for the virtual vertex,
and get a tree decomposition for Ci .
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Outline





Locality of first order logic (FO)
Frugal distributed computation
Frugality of FO over bounded degree
networks
Frugality of FO over planar networks
Frugality beyond FO properties
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Frugality beyond FO properties

FO unary queries: FO formulas ϕ(x) with one free
variable x



FO(#): Extension of FO with unary counting



Example: “Vertices of degree d”
ξd(x) := ∃y1…∃ yd (∧i≠j yi≠yj ∧ ∧i E(x, yi) ∧ ∀z (E(x,z) → ∨i z=yi))
Frugally computed over
bounded degree and planar networks
Example: “the same number of vertices of degree 2 and 3”
#x.ξ2(x) = #x.ξ3(x)
Frugally computed over
bounded degree and planar networks
Fixpoint, MSO, …
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Conclusion


Logical locality implies low distributed complexity
BFS tree assumption




Can be replaced by
a pre-computed tree 2-spanner,
independently of the choice of the requesting node
Best distributed message complexity:
O(∆) messages sent over each link
Open question:
Does a BFS tree can be frugally computed?
Open question: Does NP-complete problems like
Hamiltonicity, can be frugally computed?
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Thanks!
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