Degree. Spanning. Tree to within. One from the. Optimal. Degree. Martin. Furer*. Abstract. We consider the problem of constructing a spanning tree for a graph G ...
Chapter
Approximating
the Minimum
Martin
38
Degree
Spanning
Optimal
Degree
Furer*
Balaji
Abstract the problem
of constructing
tree for a graph G = (V, E) with degree
of G. We
is the
This
problem
describe
tion
for
a spanning
O(A*
+ log n),
tree.
The
some
our tree the
A*
can
of degree best
be
at most
bound
of some
case where
refined A*
(Steiner
It to
this is
the input
time.
minimum
Introduction
1 The
problem
graph
of computing
that
studied
satisfies
before
constraints,
classes.
a minimum
with
edge
the
sequential
weights
constraints hensive tree
and on the
frequently
make list
problems Consider
(MDST)
Though
the problem
can be found
problem,
tree
minimum which
both
cases, even of the
resulting
AT-hard. in [12], degree is that
in
eral
minimum
An
tree tions
namic
degree
[15]. tree
quickly
of constructing
net,
t computer sity, University
Science Department, Pennsylvania State tJniverPark, PA 16802, f urer~cs .psu. edu Science Department,
Pennsylvania
Park, PA 16802, rbkQcs.
State Univer-
317
general
case where
Agrawal,
Klein
more
problem
a log n factor.
algorithm
gen-
can be They
flow
for
non-critical
use prob-
[11].
basis.
different
may
sites
by their
have
Solu-
mainly
there
as
are inst antes
not
be executed
of the
parameters
to reduce
is the
site.
con-
and news on the Interneed
One want
in dy-
the broadcasting
But
of mail
applica-
trees
broadcast.
problem
the broadcast
on a priority
finds
of spanning
broadcast
as possible
done
of
The min-
is more
special
tree
on how to complete
in which
of work
psu. edu
is a tree
even the
Steiner
computation
like the distribution * ~omputer sity, University
a distinguished
work,
that
with
solution.
to the
centrated
graphs. along
to the multicommodity
net works
tions
earlier
approximation
in the
auxiliary
problem
shown
prob-
a sequence
a solution
is the
to within
lem in their
simple
spanning
spanning
[5] have
approximations
A compre-
constrained
In another
Ravi
approximated
in a graph
tree
tree
the
spans the set D.
problem
and
approxi-
problem,
and
which
Steiner
MDST
and
on some
~ V)
guarantee Furer
paper,
we are also given (D
degree
D = V.
of com-
solvable
the parallel
been
of computing
degree
and the
on the
problem
is efficiently
of AY’-complete the
the
imum
of a
by different
spanning
structure
has
Depending
characterized
cost
tree
constraints
[6], [7], [8], [19].
it has been
complexity puting
given
a spanning
this
be
degree
a spanning
to that
graph,
set of vertices
T*
approxima-
a parallel
produces
of this
prob-
Let
which
In
version
small-
maximal
log n).
matchings
Steiner
the
quality.
given
with
This
finding
problem
which
O(A*
is reduced
In the
whose in
have
is
IVP-hard,
tree,
for this [10]
of maximal
a spanning
P = NP,
in polynomial
lem
that
the
(V, E)
of G.
approximation
algorithm
of degree
and
shown
produce
+ 1. Unless
achievable
case)
is then
mation
only
be
are interested
algorithms
=
degree trees
to
spanning
We
G
maximal
shown
Raghavachari
optimal
a graph
spanning
some nontrivial
is at most
to the
graphs.
tion
com-
degree
is easily
is A*.
approxima-
degree
all
an optimal
NP-hard.
algorithm
maximal
is the
be
time
to be connected
case of directed
algorithm
to
for
whose
among
lem
trees
tree
vertices
est
whose max-
spanning
This
is generalized
need
all
shown
problem. whose
where
vertices
to the
among
polynomial
this tree
result
n vertices
is easily
an iterative
algorithm
putes
smallest
n
a spanning
One from
Raghavacharit
a spanning
We consider
imal
Tree to within
Broadcasting
that amount
informa-
FURER
318
tion
on a minimum
such
solution.
lem
is in the
the
cost
grow
area
the
with
right.
of small
this
problem
Recently
there
on approximating
sult
shows that
problem
The
to have
similar
edge coloring of planar In this at most bound and
in
then
the
[18],
a spanning observe
tree
[21].
Our
re-
spanning
an additive
We
with
the property
term
We
now
are the
edges
3-colorability
an approximation
show
of the
algorithm
that
a similar
NP,
the
graphs
problem.
at most
=
are
of degree
which
(LOT)
+ 1.
is the
results
any
We best
of
this
paper.
One the
approximation spanning
There
algorithm
tree problem
tree of degree O(A*
a polynomial
is
time
the minimum
jor
produces
which
THEOREM
ask the
could
such
difficulty.
Steiner
tree problem
of degree O(A*
question,
1.3.
approximation the
degree
produces
O(A*
+ logn). THEOREM
approximation spanning
a directed
in
utilize
a tree
is
at most
a polynomial the
one.
edges
maximal
degree
is
This
in our first
a LOT,
and
an optimum
idea
poses
of how
could
a little
such
a local
be implemented
However
time.
some crucial
approximate
far from
aware
algorithm
to
properties
we
of LOTS
to
wish
to
which
we
algorithm.
k be the maximal
degree
of vertices
of a LOT
of degree
i in T.
T.
Let
We define
S~ as follows:
time tree
time
directed
version
tree
problem
spanning
is for
which A*
maximum at least
degree
a Steiner
spanning
There
algorithm
called
of the non-tree
ideas
be?
polynomial
Let
Si=(jxj
(2.1)
tree of degree
In
other
which
a polynomial the minimum produces
+ 1.
a spanning
Proof. rate
[Si_ll/[S;l) times
contains at least For
2.1.
k of a locally
+ Pogb nl.
the
an
S;
degree
THEOREM
time degree
words,
have
degree bA*
1.4.
we
j=;
for
degree
tree problem
tree of degree
a polynomial the minimum
produces
There
algorithm
minimum
which
which
by the
tree
construct
how
Xi be the set of vertices
is for
T of
optimal
Its
is to
+ log n).
THEOREM
of
There
algorithm
none
We are not
observe
a spanning
+ log n).
1.2.
locally
attractive
problem
then
groups approximation
in one
by
C +2.
by k.
of the
MDST
run 1.1.
A
improvements.
improvement
THEOREM
w in p(v))
the
decreased
2.1.
denoted
(u, v) is
that
because
has
is a tree in which
always
Note
Let
We
time.
main
p(w)}
when
deleting
w.
improvement
p(v),
produce
produces
A*
this
in polynomial
following
an
to
Con-
in T.
z max(p(u),
and
de-
scheme. is not
“improvement”
(u, v)
DEFINITION
tree
an
edge
We try
“large”
is a vertex
p(w)
C’ incident
step
with
generated there
that
introduce the
in
this
cycle
We
T of G’. We
u in T.
the following
Suppose
tree
properties
of vertices
using
T.
the funda-
tree
of vertex
(u, v) of G which
unique
to
adding
case of directed
version
P
C be the
we know
with
degrees
an edge
added
spanning
the degree
the
provides
we use in all our algorithms.
an arbitrary
by p(u)
of {p(u),
in the
unless
sider
which
[22] and
of degree
achievable
The
[20],
a spanning
a refined
that
bound
e problems
degree
[14],
+ log n).
Steiner
present
of activity
NP-complet
we start
is achievable
with
grees iteratively
a flurry
problems
finds
start denote
own
its
[1], [13].
which O(A*
We need
Algorithm
we present
that
in
approximation
paper,
algorithm
may
algorithm ideas
is interesting
within
problem
graphs
of a station
The first mental
to reduce
[17],
other
Here
RAGHAVACHARI
Approximation
Ad-
the minimum
only
grids.
A Simple
2
degree.
is approximable
of one.
prob-
of the split.
has been
[16],
is one
this
maximal
various
[3], [4], [2], [10],
tree for
power
output
the degree
networks
ditionally,
spanning
application
of designing
of splitting
rapidly
reliable
degree
Another
AND
in
i with
First of
Hence
any
a row.
that of
be larger To
of T
the
maximal
is less
than
i = k, k – 1,...,
sets
S’i (the
b at most
more
than
be
vertices
tree
k = O(A*
we note
k– [log~ n]
b
optimal
expansion can
those i.
precise, i
of S; from
F with
has degree incident
~
because
S; itself
the
vertices
(POT).
spanning
degree
of T.
reduce
the
between
k and
;Isil – 2(1s;]
t ~
stops By
the
local
optimality
trees
in
between vertex [Sil
s&~.
such
the
of G, there these
of
are at least
degree
to
at
least
have
identified
each
edge
S;_l
edge
show
t +
mial
are
called to
each
one of these
of vertices
in S;_l.
in Si_l
critical
tree
Proof
edges
Hence
is the
of
The
is bounded
to
show
by
a polynomial
the
degree
of these
vertices.
least
Hence
algorithm improve-
argument in
function
total
the
We
to
a polyno-
that
O(A* n.
show
above
that
the
satisfies
followed
degree
of
this
Theorem
the
+ log n).
number in
is defined
The
1,1.
the
by
that
resulting
We just
of phases Consider
need
is bounded
an exponential
+ on the vertex
of a vertex
be the
is at
The
function
sketched
that
set of T.
If the
u is d in the tree T, the potential to be cd, for O(T)
potential
any
of the
sum of the potentials
k is the maximal A*
can be standard
Si has a local
observation
establishes
degree
degree
is
improve-
using
converges
Theorem
theorem. 2.1
potential
in any spanning
t+ls; l–1 1s,-1[ maximal
to
algorithm
graphs.
algorithm
algorithm
~(u)
The
local
tries degree
of steps.
iterative
tree
of G is at least
(2.4)
the
whose
time
in
prob-
an arbi-
algorithm
of the
searching
no vertex
number
of
a lower
In any spanning
one vertex
phase
tree
the
with
using
polynomial
+ 1.
optimal
k be the maximal
the
We use a potential that
start
for
between
one vertex
is a witness
for
when
ment.
one
t + ] Si ] – 1 edges connecting
and
to at least
every
at least
n) for A*.
components
average
that
set S;-l
of Q(Ic –log
incident
We to
vertices
and
bound
is incident
is incident
The
vertices
condition,
i – 1.
components
components
tree
F
of degree
in
techniques
– 1)
We
hog nl,
k –
[logb nl
solves
of some vertex
Each
implemented (2.3)
k –
algorithm
In each phase,
steps.
holds
condition
a pseudo
time.
de-
optimality
tree 2’ of G. Let
degree
on “high”
same result
i =
a tree
following
in polynomial
trary
local
Si with
such
only
the
the in
call The
ment
we have
Therefore
satisfying
We might lem
condition
degree
constant tree
c >2.
is defined
to
of all the vertices.
If
of T,
average
equation
(2.3)
(2.6)
@(T)
s nck
implies Any
improvement
some (2.5)
vertex
step on T reduces
in
the reduction
Si for
the degree
i = k – Pog nl.
in potential
of
Therefore
due to any improvement
is at least (2.7)A@
>— Combining the range This time follows.
equation
with
of i, we get k < bA* can now
algorithm Note
that
the
the in the
desired above
condition
+ (log~ nl.
be converted
with
(ci + 2-
=
(c-1)
.(c-2).
c~-’
>
Ck c.—
where
c=
>
~
b (2.5)
into
a
on
u
theorem
Ci-2)
-
v
;(T) .— ~2
polynomial
performance
(3 . c;-’)
>
as
we used
In least
other
words,
a polynomial
the factor.
potential
reduces
Therefore
in
by 0(n2)
at
320
FWRER
steps,
the
Hence
the number
potential
Alternatively, the
reduces
we could
same
for
value
of
again
implying
more
decrease
an
O (n3)
be implemented
in
above
runs
The
3
that
n2 /c
As mentioned
algorithm
is bounded
argue
than
k cannot
of phases.
by a constant
of phases
k cannot
more
than
bound
on the each
Problem
a graph
G = (V, E)
vertices.
are no useless
In
at least
other
edges because
tree are distinguished
Hence time.
2. None
can
all
two
words,
there
leaves
of the
vertices.
of the non-tree
provement
the
for
Pog nl,
0
tree.
paths
any
where
of the then
Steiner
tree separates
number
phase
time.
in polynomial
edge in the
distinguished
the
n times,
before,
polynomial
1. Every
stay
Also
RAGHAVACHARI
properties:
factor.
by O(n3).
phases.
AND
vertex
k denotes
If
this
produce
any
im-
S~ for
i =
k –
in
the maximal
property
the tree is a locally
is true
optimal
degree for
all
Steiner
i,
tree
(LOST). Consider set
of vertices
Steiner
D
Tree
minimum
~
V.
problem
degree,
which
Klein
[5] have
Ravi
can be approximated tree.
of O(log Their
problem They the by
the
toughness
tree
algorithms on the
of
and
toughness
D
this
graph,
on reducing flow
which
degree
which
+ log n).
by
T.
those Let
For
improvement
through
vertices
points.
We
Adding
any
cycle.
remove
from
in
such
this on
the notion
Steiner
case.
DEFINITION
which
path
T
extend
Theorem This
entirely the
one edge
of high
of a locally
again,
time
end path.
A
is a Steiner
pseudo tree
we can
trees
algorithm runs
in
tree
into
produces
needs
to
With
extension
of
Cl a polynomial
a Steiner Theorem
tree 1.2.
of An
to the one for spanning
polynomial degree
has
we identified.
is a simple
satisfying
similar
whose
component
to be connected
to the reader.
which + log n),
formed vertices.
connection
can be converted
O(A*
Steiner
every
that
2.1 and is left
to
in a crucial
certain
needs
possible
the proof
we need
3.1) of
and
vertices
ideas,
idea
that
on
in turn
time
and
is O(A*
produces
a
+ log n).
of T which
degree. optimal
We tree
optimal with
critical
iterative a
Every
in
bound
which
of components
that
vertex
to
of T is a distinguished
removal
ensures
algorithm
degree
also
to the
4
Directed
We
now
handle 3.1.
the
condition
the above
Note
leaf
the number
by
a distinguished
any path
introduces
any
tree
a lower
vertices,
on A*.
we can need
set of vertices
1 of Definition
We count
the
gives
i
For such
set and
small
of these
every
(condition
use the
for
a tree
that
This
as a non-tree to
bound
terms
goes
it into
cycle
is a lower
in
This
tree. degree
in We
except
a path
a vertex
Steiner
some between
of components
this
to the
we find the ratio
a critical
number
through
edge.
Consider
in structure
As before,
forms
a large
to the others.
problem. W
of i, Si_l
that
problem,
defined
2.1.
is at
+ log n).
by a constant.
This
of vertices
tree
another
V – W
To make
extend
(POST)
this
non-tree
unique
is incident
of
call
for
spans
separate
set
spanning
as follows.
vertices
which
the
was just
in
idea
two
be
the
edge
is insufficient
the exchanging
edges
W
step one
tree T which
k = O(A*
1S;-1 I is bounded
way.
an arbitrary
Hence
is similar
tree T
lS~l and
vertex
bounds
1, the maximal
Steiner
where
use the fact
of a graph.
T.
between
a
Our
b >
k and k – Pog nl
the average
finds
proof
of Theorem
any
defined
is O(A*
The
be connected
approximation
problem
+ llog~ nl.
any
optimal
between
show
computing was
an
bA*
a value
this
problem.
for
provide
most
proof
to within
For
k of a locally
Proof.
problem
of an optimal
degree
is based
only
in
of exchanging
tree
that
degree
Agrawal,
of the
Steiner
with
retain
set
idea
is
can also be used to give better
spanned the
This
time
a
the
of
in polynomial
We
whose
We start D
shown
a tree
set D.
approximations
[9]. for
Steiner
the
multicommodity
also provide
Chvii.tal
of finding
3.1.
THEOREM
Degree
MD ST problem.
algorithm
to
algorithm
the
n)
a distinguished
Minimum
spans
of the
a factor
The
is that
a generalization and
and
Steiner
the following
directed which
Spanning
show
directed graph is the root
how
to
graphs.
Trees extend In this
G together of the tree.
with
our
algorithm
to
case the input
is a
a special
The root
vertex
is reachable
r
APPROXIMATING
from
THE
all vertices
tree
MINIMUM
DEGREE
of the graph.
T of G is a subgraph
A rooted
of G with
SPANNING
TREE
spanning
321
Proof.
the following
of
properties:
The
S; and
Suppose
1. T does not
contain
from
any cycles.
out degree
exactly
degree
maximal
of a rooted
graphs
Consider
in that
cycle from
for directed
i.
subtrees
We
The
improvement move
from
subtree.
T’ the
connection.
to
and
every
by
attaching
vertex
one
the
size
leaf.
starting
base
Each
set Sz can remove
consists
of two
subtree
of
the
addition
at most
trees,
but
in that vertex
convenient
graph
induced edges
new
root
one
adds
by the
are
of the
root
vertices
from
of T’
vertices
of
at
locally the
k – Pog nl using following
and pseudo
as before. degrees
optimal
The of vertices
the improvement
lemma
is useful
in
tries
Si for
step above.
in the proof
trees. the
directed
Let T be a directed
4.1.
Let S; consist
indegree
at
is
least
of Si from there
are
vertices in T.
at
i. T,
least
of those Suppose
breaking
spanning vertices we
The
T into
do not have descendants
trees
that
were
the
of
G.
F.
adding
F
blocks
of degree i
in
then
rest
of
which
to
of the
form
a
proof
is
to the reader.
Cl
should
be
of vertices
on
that
edge.
of
Suppose
a spanning
optimality
tree
condition
an
algorithm
to reduce In
y
the number
doing
can increase
to
so, the
arbitrarily
k.
Let (u, v) @ T be an edge in
5.1.
a local
no edge
less than
w is a vertex
can
tree
optimalit
degree.
vertices
(u, v) to T.
a spanning local
For
it is sufficient
of maximal
w from (u, v)
by degree
remain
can be refined
degree
necessary.
of other
idea
is that
the local
that
as they
Suppose
through
vertices
order
The
maximal
maximal
DEFINITION
the
i or larger.
the
produces + 1.
iadumd
than
of vertices
A*
the
cycle
Observe
as long
Si.
whose
of components The
we used,
to reduce basic
whose
a jorest
with
of S;-l
which
at most
property
T,
in
trees
lSi[ . (i – 1)+1
1.1 and is left
an algorithm
show progress,
tree
along
right
of some
A* + 1 algorithm
of degree able
by the
work
of degree
We now show how the previous into
to we
vertices
are at least
tree.
leads case,
possibility
on those
number
spanning
The
remove
IS; I . (i – 1) + 1
large
to Theorem
degrees LEMMA
only
trees
use vertices
to
is stricter
1.3.
of degree k.
These
not
k
“expand”
generated
descendent
4.1, there
range
This
does
have no descendants
i =
of Theorem
This
having
Si form
k
directed
algorithm
T.
of the
the
undirected
due to the
have
the
i – 1 or greater.
the
in the forest
graphs trees
proof
an i in
factor.
In
we concentrate
similar
a
a constant S~_l.
of Sz from
Lemma
5
has
vertices
part
removed
We
vertex to
connected
parts.
another
of
set
degree.
As in the for
the set Si does not
at all the trees
such i
is being
the
to
v of
T’ that
which
trees
set
vertices
one of the
tree
non-tree
than
By
1.3.
we look
where
more
of these
of the
of smaller
case,
removal
work
a vertex
attached
in the
or greater
scratch,
produces
of Theorem
in directed
can poten-
indegree
v to a “convenient”
strongly
decrease
whose
from
the
observation.
El
a critical look
an improvement
is then
We define
Then
on
simple
set of candidate
to k – Pog nl by
(u, v) in
does not
Consider
of the
in the
vertices
the
Proof
is no
such
v)
This
see if the
step
The
spanning to
u and
we define
way.
try
the root
removed
degree
at least
i more.
Hence
of v to another
all
with
undirected
di-
adding
a cycle
edge.
graphs
first
of T’
edge
graphs,
(except this
that
There
a non-tree
can be reduced
with
is the
in the case of di-
produces
step in the following indegree
those
tree
This
to the
from
tree
root.
of a vertex least
graphs.
For directed
outside
is
to the
problem.
case of undirected
benefit
vertex
r
vertex
Note
improvements
to T always
an edge
vertex.
pose a severe
graphs.
In the
vertex
every
spanning
of any
easy way to define
tially
in T from
indegree
rected
except
the
induction
this
r.
The rected
vertex
tree
is a path
root
every
one.
3. There
G.
of
uses
from
we build
the
induction 2. The
proof
follows
If p(u)
(u, v).
in the cycle
by
~ k – 1, we say that
If neither
u nor
be used to reduce improvement
generated
step.
the
v blocks degree
In such
u w,
of w
a case,
322
FfiRER
we say w benefits Let
S be the
algorithm to
in phases.
the
we move
on
S reduces
size to
by
there
maximal
of
the
show
degree
next
is k.
that
in
T
conditions degree
a minimum
from F.
are
Then
ksA*+l,
ProoJ the
the
k–l–([Bl
vertices
tree
time
Theorem of degree that
A* The
gorithm.
be the
Let S U B be into that,
trees
in
F.
in
Sk.
case.
any
average
to
these
with
tree
and hence A*
F
A*
every
of degree
the
and
our procedure of F.. case,
ments with
each other
Observation:
Let
above.
within
a blocking
following
is a top-down
directly The
a tree
T
set B is a proof
as follows.
sk U S,&l
from
the
T
and
are marked
as bad. If there
components,
case,
we will
remaining A*
components
to T and observe of degree
Making
to
vertices
F.
this
al-
the
degree
set S~, the
set
find
a way
and
In
this
vertices
that
k
on this
we
s
the
to
reduces
cycle.
as good
is a
identified
propogate
on this
(u, v)
If there have
the
is at least
cycle
(k – 1) vertices.
We add
generated. cycle,
(k – 1) on the
to reduce
fact
F..
which
of all components
of our
set the
there
of degree
all bad
stops.
changes
vertex
in Sk US&l
are no edges between
these
one.
connected
All vertices
Otherwise
Sk by
the
in
(u, v) be an edge between
of improvements
set
a bottom vertices
of bad
the cycle k in
in all
all
can be
Sk.
the
that let
is possible
from
remove
for
by the
of l?.
algorithm
witnesses
Otherwise
good
vertex
the
show
are
+ 1.
good.
become
to look
that
mark
as being
blocks
to
induced
a vertex
components
good
which
u tries
outside
We
interfere
non-blocking.
is implemented
up fashion
union view
to benefit
and
observation
do not
if it tries
going
by the
of improve-
following
improvement
edges
by one by
induced
the subgraph
Any
algorithm
u.
a vertex
it is sufficient
used
contains
non-blocking
When
improvements of F..
of w.
to become
u be
by
blocks
vertices
non-blocking,
vertex.
that
The
in trying
u by using
k vertices
a sequence
why
w as described
problems
Note
to w. shows
a
which
in the graph u is made
by
improve-
w be a vertex
of F which
we say that
and
blocked this
of u can be reduced
Then
propogate
is crucial
a
that
component
running in this
of some ver-
of degree let
case
components
is not
u be a vertex
vertices
have polynomial which
an edge
degree
be-
and in this
the degree
Otherwise
let
Fu be the
two
as the
+ 1, we believe
k – 1.
at
> k – 1.0
case as well
two
way.
we remove
of G,
5.1 is powerful
is at least Suppose
that
withmaximal
degree
we do not
of degree
k, along
show
one vertex
Theorem
the
we
here
the number
k and
T. Observe
of Z’, we are done.
(k – 1), we make
continue.
Suppose
these
in S U B is at least
Therefore
for
way
By a simple
spanning
to the Steiner
5.1 points
B.
Avertex
Though
a tree
only
such
reducing
degree Let
If
of degree
one and
for
–1)–2(lS[+lBl–1)
of vertices
that
algorithms
produce
S
the tree T
the
easy
has at least
generalizes
directed
of
the condition
F,
is
in
degree
We observe easily
of
is by connecting
lS[k+/B/(k
k – 1 in S U B.
least
tree degree
be an arbitrary
in S and
it
has at least
spanning
B
different
of tree
–1)/(lSl+/Bl).
degree
the
are no edges in G connecting
Therefore
average
the
Let
breaking
subtrees
at least
subtrees.
k – 1
satisfies
be the
Let
between
argument,
contains
k.
tex
from
are no edges
any edge between
of F can be used to reduce
ment,
the
degree
components
Otherwise
RAGHAVACHARI
If there
5.1 can be applied
k = A*.
harmonic We will
of maximal
F generated.
the different
of k proves
and hence
tree.
G satisfies
a spanning
through
counting
A*
tween
vertices
different
clusters
T
of degree k – 1 in T.
edges
the
be a spanning Let
spanning
As there
can make
T
the graph,
no
when
+ 1.
G.
Suppose
there
A“
of degree
of vertices
removed
tree
of
last
to reduce
theorem
Let
degree
set of vertices
a forest
the
forest
vertex
up the values
size
the
Theorem
the
O(n log n) phases.
that,
5.1.
the
phases
possible
The
we try
is a set B of degree
k of a graph
subset
As
of the following
THEOREM
k.
successful,
(except
Summing
there such
If
phase
to reducing
of T is at most
degree
one.
phase.
if it is not
size of S, then
of degree
O (n/k)
are at most
later
vertices
S by
are at most
there
of all vertices
In each phase
one in each
series corresponding that
(u, v).
set of vertices
works
reduce
one),
from
AND
this
size
one
of
bad
We mark and
cycle along
make with
a all
In all cases, we either degree
of some vertex
in
APPROXIMATING
sk or find
a blocking
Theorem
MINIMUM
set with
DEGREE
which
SPANNING
we can apply
5.1.
LEMMA
made
THE
Conclusions
6
We have Any
5.1.
non-blocking
by the vertices
vertex
within
u marked
good can be
the subgraph
generated
of the good component
the
The
number
When less
proof
of
proceeds
unions
the algorithm than
or
good.
By
blocking.
made
begins,
equal
to
only
–
as good
the
cycle
generated
by
two
good
components.
and
reduce
it
the
only
vertices
techniques spanning
of
Hence
our
within
good
needed
can
to make
the vertices
it was on
added
between edge
u by
one
and
make
algorithm made
maintains
that
any
update
is within
stops,
find
A*+l.
polynomial ing
the
Let
of degree only
S be Sk and
k – 1.
when
there
components. sets
Note
B
that
the
are no edges
Hence
S and
B be the bad vertices
the
satisfy
5.1 and we get the
tree
the
algorithm
between
vertices
only. would
to find
a LOT.
additive
T along
with
conditions
desired
result.
tree
is of degree
Q(~). open
there
of
are at most
we try
to find
vertices ever
Theorem
of
improvements
be marked
Lemma
algorithms
do in this
algorithm
which the
these
among
can
be
optimal these
a vertex
as good,
of improvements 5.2 shows gree
that
of the
optimal
which when
degree.
be implemented disjoint
tree
Each in
components.
gorithm
runs
number
of edges II
in O(mn and
find
algorithm
to can
the
A. Agrawal,
log na(n)),
to handle
the can
well
problem?
to
Is there
the
How
Is there
a tree
of IVP-complete
algorithm time
using
Collide:
within some
of de-
problems one
from
relationship
the
[4]
the
where inverse
entire
al-
m is the Ackerman
Every
J. Math.
planar
map
21 (1977),
P. Klein, through
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Approximation
A. Agrawal, P. Klein lems Approximated: and Interval Graph
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when-
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that k which
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nearly
set union-find
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the
resulting
propagate
w of degree propagates
the
is a LOT
can obtain
list
is small.
four-colorable,
earlier,
In each phase
assures
we can indeed
which The
Is there where
References
observed
which 5.1
can be extended
parallel
+ 1?
there
an
is to ask whether
directed
A*
time
optimal.
a better
case or the
gree
example
is within
an example
Steiner
good
El
As
polynomial a LOT
degree
of Theorem
O (n log n) phases.
sk.
we find
1.4.
than
question
+ 1 algorithm
by us-
on high of any
the
to
We obtained
aware
3 and
Is there
natural
how
algorithms
that
We have
time?
know
time.
of log n from
bound?
[1] K. Appel Proof
more
We showed
term
degree
are not
Is it possi-
in polynomial
we do not
y condition
require
complexities
clear.
approximation
We
prob-
to implement
exact
or a LOST
optimalit
which
stops the
The
are not
the
related
it is possible
earlier,
time local
an NC Proof.
and
one of these in polynomial
other
k s
a LOT
based
for approximating
efficiently.
As we mentioned
A*
algorithm
algorithms
that
algorithms
tree
of some of our algorithms
optimal
Cl
the
We believe
a tighter
non-blocking
non-blocking
When
5.2.
u of degree
that
of its component.
LEMMA
non-
add
Note
a vertex
as
can
be
components.
marked are
degree
ble to find
of degree
was when
We
which
are
edge
degree
non-blocking.
vertices
a vertex
an
on
algorithm.
vertices
that
(k – 1) was marked
the
2)
these
time
induction
by
only
(k
definition,
The
by
iterative
minimum lems.
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demonstrated
on combinatorial
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