We prove that the directed graph reachability problem (transitive closure) can be solved by monotone fan-in 2 boolean circuits of depth (1/2+o(1))(log n)^2, where n is the number of nodes. This improves the previous known upper bound (1+o(1))(log n)^2. The proof is non-constructive, but we give a constructive proof of the upper bound (7/8+o(1))(log n)^2.
Deep Dive into Improved Monotone Circuit Depth Upper Bound for Directed Graph Reachability.
We prove that the directed graph reachability problem (transitive closure) can be solved by monotone fan-in 2 boolean circuits of depth (1/2+o(1))(log n)^2, where n is the number of nodes. This improves the previous known upper bound (1+o(1))(log n)^2. The proof is non-constructive, but we give a constructive proof of the upper bound (7/8+o(1))(log n)^2.
arXiv:0809.3614v1 [cs.CC] 22 Sep 2008
Impro
v
ed
Monotone
Cir uit
Depth
Upp
er
Bound
for
Dire ted
Graph
Rea
habilit
y
S.
A.
V
olk
o
v
(Mos o
w,
2008)
Abstra t
W
e
pro
v
e
that
the
dire ted
graph
rea
habilit
y
problem
an
b
e
solv
ed
b
y
monotone
fan-in
2
b
o
olean
ir uits
of
depth (1/2 + o(1))(log2 n)2
,
where n
is
the
n
um
b
er
of
no
des.
This
impro
v
es
the
previous
kno
wn
upp
er
b
ound (1+o(1))(log2 n)2
.
The
pro
of
is
non- onstru tiv
e,
but
w
e
giv
e
a
onstru tiv
e
pro
of
of
the
upp
er
b
ound (7/8 + o(1))(log2 n)2
.
De nitions
By [x]
denote
the
o
or
of x
.
F
or
real-v
alue
fun tions f, g
w
e
will
write f ≍g
,
if f = O(g)
,
and g = O(f)
;
w
e
will
write f ∼g
,
if lim(f/g) = 1
.
By G(g11, g12, . . . , g1n, . . . , gn1, gn2, . . . , gnn)
denote
the
dire ted
graph
su
h
that
the
set
of
v
erti es
of
this
graph
is {1, . . . , n}
,
and
the
adja en y
matrix
of
this
graph
is {gij}.
By V (G)
denote
the
set
of
v
erti es
of
the
graph G
.
Similarly
,
b
y E(G)
denote
the
set
of
edges
of
the
graph G
.
W
e
sa
y
that
the
path p
has
length l
,
if p
has l
edges.
The
graph G′
is
alled l
- losur
e
of
the
graph G
,
if V (G′) = V (G)
,
and
E(G′) = {(i, j) : G
has
a
path
from i
to j
with
length ≤l }.
Supp
ose G
is
a
graph, S ⊆V (G)
;
b
y GS
denote
the
graph
su
h
that V (GS) = S
,
E(GS) = {(i, j) ∈E(G) : i, j ∈S}
.
In
the
follo
wing
de nitions G
is
an
abbreviated
notation
for G(g11, . . . , gnn)
.
Put
Reachn(g11, . . . , gnn) =
(
1,
if
the
graph G
has
a
path
from 1
to n,
0
in
the
other
ase,
Reachn,l(g11, . . . , gnn) =
1,
if
the
graph G
has
a
path
from 1
to n
with
length l
,
0
in
the
other
ase,
Reach
p
n,l(g11, . . . , gnn) =
1,
if
the
graph G
has
a
path
from 1
to n
with
length ≤l
,
0,
if
there
is
no
path
from 1
to n
in
the
graph G ,
unde ned
in
the
other
ases.
1
In
this
pap
er
w
e
onsider
b
o
olean
monotone
ir uits
with
fan-in 2
,
i.e.,
ir uits
o
v
er
the
basis {x&y, x ∨y}.
By d(Σ)
denote
the
depth
of
the
ir uit Σ.
The
ordered
family
of
sets (S1, . . . , Sm)
is
alled
an (n, m, s, l, d)
-family
,
where n
, m
,
s
, l
, d
are
natural
n
um
b
ers,
if
the
follo
wing
onditions
hold:
1. | Sm
i=1 Si| ≤n
;
2.
for
an
y i
w
e
ha
v
e |Si| ≤s
;
3.
F
or
an
y A ⊆{1, . . . , m}
su
h
that |A| ≥md
l
,
w
e
ha
v
e | S
i∈A Si| ≥n −d + 1
.
Order
is
needed
for
al ulation
on
v
enien e
only
(in
the
pro
of
of
statemen
t
4).
Supp
ose M = (S1, . . . , Sm)
is
a
family
of
sets;
b
y S M
denote
the
set | Sm
i=1 Si|
.
Pro
of
of
the
upp
er
b
ound
Statemen
t
1.
Ther
e
exists
a
family
of
monotone
ir
uits Σn,l
su h
that
for
any n, l
, Σn,l
r
e
alizes Reachn,l,
and
d(Σn,l) = log2 n · log2 l + O(log n).
Äîêàçàòå
ëüñòâî.
By
rep
eated
matrix
squaring.
Statemen
t
2.
L
et M
b
e
an (n, m, s, l, d)
-family, A0, . . . , Al′
( l′ ≤l
)
b
e
dier
ent
elements
of S M
;
then
ther
e
exists
a
set S ∈M
and
numb
ers k ≤l
d
, 0 = i0 < i1 < . . . < ik = l′
su h
that Ai1, . . . , Aik−1 ∈S
,
and
for
any j
( 0 ≤j < k
)
we
have ij+1 −ij ≤2d
.
Pr
o
of.
Without
loss
of
generalit
y
w
e
an
assume
that l′ ≥2d + 1
.
Supp
ose k =
l′
d
,
Bi = {Aid, Aid+1, . . . , Aid+d−1}
, 1 ≤i ≤k −1
.
W
e
no
w
pro
v
e
that
there
exists S ∈M
su
h
that
for
an
y i
( 1 ≤i ≤k −1
)
w
e
ha
v
e S ∩Bi ̸= ∅
.
Assume
the
on
v
erse.
Then
for
an
y
set Bi
there
exists
an S ∈M
su
h
that S ∩Bi = ∅
.
Then
there
exists
a
set Bi
su
h
that
there
exists
at
least
m
k−1 ≥md
l
sets S ∈M
su
h
that S ∩Bi = ∅
.
Sin e |Bi| = d
,
w
e
see
that
the
ardinalit
y
of
the
union
of
these
sets
S
do
es
not
ex eed n −d
.
This
on
tradi ts
the
de nition
of
an (n, m, s, l, d)
-family
.
T
ak
e
the
set S ∈M
su
h
that
for
an
y Bj
w
e
ha
v
e S ∩Bj ̸= ∅
.
Also,
tak
e i1, . . . , ik−1
su
h
that Aij ∈Bj ∩S
( 1 ≤j ≤k −1
).
Put i0 = 0
, ik = l′
.
Ob
viously
,
the
n
um
b
ers
i0, . . . , ik
satis y
the
onditions
of
the
statemen
t.
Statemen
t
3.
L
et M
b
e
an (n, m, s, l, d)
-family;
supp
ose
a
monotone
ir
uit Σ′
r
e
alizes
Reach
p
s+2,[ l
d]
;
then
ther
e
exists
a
monotone
ir
uit Σ
su h
that Σ
r
e
alizes Reach
p
n,l
,
and
d(Σ) = log2 m + log2 n · log2 d + d(Σ′) + f(n)
for
some
xe
d
fun tion f(n) = O(log n)
.
2
Pr
o
of.
Without
loss
of
generalit
y
,
w
e
an
assume
that
all
sets
of
the
family M
are
subsets
of {1, . . . , n}
.
A
dding 1
and n
to
ea
h
set
of M
,
w
e
obtain
the (n, m, s + 2, l, d)
-family
M′
.
No
w
w
e
des rib
e
a
onstru tion
of
the
ir uit Σ
.
Let G
b
e
an
input
graph.
Supp
ose G′
is
the 2d
- losure
of
the
graph G
.
F
rom
statemen
t
1
it
follo
ws
that
the
adja en y
matrix
of
G′
an
b
e
obtained
b
y
a
monotone
ir uit
of
depth log2 n · log2 d + O(log n)
.
F
or
ea
h
set S ∈M′
w
e
onstru t
a
blo
k ΣS
whi
h
is
a
lone
of
the
ir uit Σ′
;
it
tak
es
the
adja en y
matrix
of G′
S
to
it’s
input
( ΣS
determines
the
existen e
of
a
needed
path
in
G′
S
from
the
v
ertex 1
to
the
v
ertex n
);
if |V (G′
S)| < s
,
then
the
input
matrix
of ΣS
is
expanded
b
y
zeros.
The
disjun tion
of
all
outputs
of
the
ir uits ΣS
, S ∈M′
is
de lared
to
b
e
the
output
of Σ
.
It’s
lear
that d(Σ)
satis es
the
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