IEEE
TRANSACTIONS
OF
SYSTEMS
SCIENCE
AND
CYBERNETICS,
VOL.
ssc-4,
NO.
2,
JULY
1968
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E.
Falk,
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multipliers
and
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1967.
[6]
0.
L.
Mangasarian
and
J.
Ponstein,
"Minimax
and
duality
in
nonlinear
programming,"
J.
Math.
Anal.
Appl.,
vol.
11,
pp.
504-
518,
1965.
[7]
J.
Stoer,
"Duality
in
nonlinear
programming
and
the
minimax
theorem,"
Numerische
Mat
hematik,
vol.
5,
pp.
371-379,
1963.
[8]
R.
T.
Rockafellar,
"Duality
and
stability
in
extremum
prob-
lems
involving
convex
functions,"
Pacific
J.
Math.,
vol.
21,
pp.
167-187,
1967.
[9]
P.
Wolfe,
"A
duality
theorem
for
nonlinear
programming,"
Q.
A
ppl.
Math.,
vol.
19,
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239-244,
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[10]
R.
T.
Rockafellar,
"Nonlinear
programming,"
presented
at
the
American
Mathematical
Society
Summer
Seminar
on
the
Math-
ematics
of
the
Decision
Sciences,
Stanford
University,
Stanford,
Calif.,
July
August
1967.
[1ll
D.
G.
Luenberger,
"Convex
programming
and
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in
normal
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IEEE
Systems
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and
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Mass.,
October
11-13,
1967).
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M.
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theory
of
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with
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J.
SIAM,
vol.
14,
pp.
641-665,
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1966.
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W.
Fenchel,
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cones,
sets,
and
functions,"
mimeo-
graphed
notes,
Princeton
University,
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N.
J.,
September
1963.
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M.
J.
D.
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1963.
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E.
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J.
E.
Falk,
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Analysis
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Va.
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1967.
A
Formal
Basis
for
the
Heuristic
Determination
of
Minimum
Cost
Paths
PETER
E.
HART,
MEMBER,
IEEE,
NILS
J.
NILSSON,
MEMBER,
IEEE,
AND
BERTRAM
RAPHAEL
Abstract-Although
the
problem
of
determining
the
minimum
cost
path
through
a
graph
arises
naturally
in
a
number
of
interesting
applications,
there
has
been
no
underlying
theory
to
guide
the
development
of
efficient
search
procedures.
Moreover,
there
is
no
adequate
conceptual
framework
within
which
the
various
ad
hoc
search
strategies
proposed
to
date
can
be
compared.
This
paper
describes
how
heuristic
information
from
the
problem
domain
can
be
incorporated
into
a
formal
mathematical
theory
of
graph
searching
and
demonstrates
an
optimality
property
of
a
class
of
search
strate-
gies.
I.
INTRODUCTION
A.
The
Problem
of
Finding
Paths
Through
Graphs
MANY
PROBLEIVIS
of
engineering
and
scientific
importance
can
be
related
to
the
general
problem
of
finding
a
path
through
a
graph.
Examples
of
such
prob-
lems
include
routing
of
telephone
traffic,
navigation
through
a
maze,
layout
of
printed
circuit
boards,
and
Manuscript
received
November
24,
1967.
The
authors
are
with
the
Artificial
Intelligence
Group
of
the
Applied
Physics
Laboratory,
Stanford
Research
Institute,
Menlo
Park,
Calif.
mechanical
theorem-proving
and
problem-solving.
These
problems
have
usually
been
approached
in
one
of
two
ways,
which
we
shall
call
the
mathematical
approach
and
the
heuristic
approach.
1)
The
Inathematical
approach
typically
deals
with
the
properties
of
abstract
graphs
and
with
algorithms
that
prescribe
an
orderly
examination
of
nodes
of
a
graph
to
establish
a
minimum
cost
path.
For
example,
Pollock
and
Wiebensonf11
review
several
algorithms
which
are
guaran-
teed
to
find
such
a
path
for
any
graph.
Busacker
and
Saaty[2'
also
discuss
several
algorithms,
one
of
which
uses
the
concept
of
dynamic
programming.
[3]
The
mathematical
approach
is
generally
more
concerned
with
the
ultimate
achievement
of
solutions
than
it
is
with
the
computational
feasibility
of
the
algorithms
developed.
2)
The
heuristic
approach
typically
uses
special
knowl-
edge
about
the
domain
of
the
problem
being
represented
by
a
graph
to
improve
the
computational
efficiency
of
solu-
tions
to
particular
graph-searching
problems.
For
example,
Gelernter's
41
program
used
Euclidean
diagrams
to
direct
the
search
for
geometric
proofs.
Samuel[1
and
others
have
used
ad
hoc
characteristics
of
particular
games
to
reduce
100
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