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Department of Computer Science Technical
Reports
Department of Computer Science
1984
Data Structures for On-Line Updating of Minimum Spanning Trees, Data Structures for On-Line Updating of Minimum Spanning Trees,
with Applications with Applications
Greg N. Frederickson
Purdue University
, gnf@cs.purdue.edu
Report Number:
83-449
Frederickson, Greg N., "Data Structures for On-Line Updating of Minimum Spanning Trees, with
Applications" (1984).
Department of Computer Science Technical Reports.
Paper 368.
https://docs.lib.purdue.edu/cstech/368
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries.
Please contact epubs@purdue.edu for additional information.
\
DATA
STRUCTURES FOR ON-IJNE UPDATING
OF
MINIMUM
SPANNING TREES,
WITH
APPLICATIONS'
Greg N.
Frederickson
Revised May 1984
Depart.ment.
of
Comput.er
Sciences
Purdue
University
West
Lafayette.
IN
47907
• This re9carch
"11.9
supported
in
part by
the
National
Science
Foundation under Grant IlCS-
6201063.
Abstract.
Data
structures
are
presented
for
the
problem
of
maintaining
a.
minimum
spanning
tree
on-line
under
the
operation
of
updating
the
cost
of
some
edge
in
the
graph.
For
the
case
of
a
general
graph,
maintaining
the
data
structure
and
updating
the
tree
are
shown
to
take
O(vm)
time.
where
m
is
the
number
of
edges
in
the
graph.
For
the
case
of
a
planar
graph,
a
data
structure
is
presented
which
supports
an
update
time
of
O«(Iog m )2).
These
structures
contribute
to
improved
solutions
for
the
on-line
connected
components
problem
and
the
problem
of
generating
the
K
smallest
spanning
trees.
Keywords.
connected
components,
data
structure!';,
edge
insertion
and
deletion,
K
smallest
spanning
trees,
minimum
spanning
tree,
on-line
computation,
planar
graphs.
1
1.
Introduction
Consider
the
following
on-line
updat.e
problem:
A
minimum
spanning
tree
is
to
be
maintained
for
an
underlying
graph.
which
is
modified
repeatedly
by
hav-
ing
the
cost
of
an
edge
changed.
How
fast
can
the
new
minimum
spanning
tree
be
computed
after
each
update?
In
this
paper
we
present
novel
graph
decompo~
SiUOD
and
data
structures
techniques
to
deal
with
this
update
problem.
includ-
ing
a
useful
characterization
of
the
topology
of
a
spanning
tree.
Furthermore.
while
dynamic
data
structures
have
been
applied
witb
success
to
various
geometric
problems
[QV.
LW].
our
results
are
among
the
first
[ST.
Hill
in
the
realm
of
graph
problems.
Let
m
be
the
number
of
edges
in
the
graph.
and
n
the
number
of
vertices.
The
current
best
time
to
find
a
minimum
spanning
tree
is
Oem logl08(2+m/n)
n)
[CT, Y].
If
only
straightforward
descriptions
of
the
underlying
graph
and
its
current
minimum
spanning
tree
are
maintained,
then
it
has
been
shown
in
[SP]
that
the
worst-case
time
to
perform
an
edge-cost
update
is
G(m).
The
problem
of
determining
the
replacement
edges
for
all
edges
in
the
spanning
tree
can
be
~olved
in
O(ma(m
,n))
time
[T2],
where
a(·,)
is
a
functional
inverse
of
Ackermann's
function
[TI].
However,
that
solution
is
essentially
static,
so
that
actually
performing
replacements
can
necessitate
considerable
recomputation.
We
show
how
to
maintain
information
about
the
graph
dynamically
so
that
edge
costs
can
be
updated
repeatedly
with
etliciency.
After
each
edge
cost
change,
the
change
in
the
minimum
spanning
tree
is
determined,
and
the
data
structures
are
updated.
We
are
able
to
realize
an
O(vm)
update
time.
More-
over,
if
the
underlying
graph
is
planar,
we
show
how
to
achieve
an
O({log
m)2)
update
time.
Our
structures
require
Oem)
space
and
Oem)
preprocessing
time,
aside
from
the
time
to
find
the
initial
"minimum
spanni.ng
tree.
These
compare
favorably
with
those
developed
recently
in
[HI2],
which
realize
O(n
log
n)
2
update
times.
Our
results
aTe
both
of
practical
and
theoretical
interest.
On
the
one
hand,
a
minimum
spanning
tree
may
be
used
to
connect
the
nodes
of
a
communica-
tions
network.
Variable
demand.
or
transmission
problems,
may
cause
the
cost
of
sowc
some
edge
in
the
network
to
change.
and
the
tree
will
need
to
be
reconfigured
dynamically.
On
the
other
hand,
by
focusing
on
edge
cost
changes.
we
have
formulated
a
natural
version
of
the
problem
of
updating
a
minimum-
cost
base
of
a
matroid
[W].
(In
this
case,
the
matroid
is
a
graphic
matroid.)
Our
work
leads
naturally
into
the
updating
of
minimum-cost
bases
of
certain
simple
ITlatroid
intersections.
These
are
investigated
in
[FS1,
FS2].
in
which
our
data
structures
aTe
used
extensively.
The
problem
of
maintaining
a
mini.mum
span·
ning
tree
when
vertices
are
inserted
and
deleted
bas
been
studied
in
[SP,
CH],
but
the
best
performan~e
lo
date
is
0(71.
2
).
This
suggesls
lhat
because
of
its
connection
to
matroids,
the
edge-updale
problem
is
perhaps
more
natural
than
the
verlex-update
problem.
We
also
show
how
lo
apply
our
data
slruclures
to
a
number
of
relaled
prob-
lems
lo
yield
improved
performllnce
bounds.
We
cast
lhe
problems
of
edge
insertion
and
deletion
inlo
an
edge
update
framework,
and
realize
O(vmt)
update
times,
where
ml
is
the
current
number
of
edges
in
the
graph.
Using
this,
we
improve
on
the
update
time
for
lhe
on·
line
connected
components
problem
in
a
graph
in
which
edges
are
being
inserled
and
deleted.
The
problem
is
to
maintain
a
data
structure
so
that
a
query
asking
if
two
vertices
are
in
the
same
conncct.ed
componcn~.
can
be
answered
in
constant
time.
A
version
involving
deletions
only
was
~Xil.mined
in
[RS1.
for
which
the
t.otal
t.ime
for
TTl.
updaLcs
was
O(mn).
A
more
~encral
version
has
been
discussed
raccnlly
in
L11Il],
for
which
0(71.)
time
per
individual
update
was
realized.
Our
solution
uses
O(vmt)
time
per
update.