Technion - Computer Science Department - Tehnical ... - CS, Technion
Technion - Computer Science Department - Tehnical Report CS0049 - 1975
TEOINION - Israel Institute of Technolo,y Computer Science Department
ON THE OPTIMALITY OF A· by
David Gelperin Technical Report No.49
May 1975
Technion - Computer Science Department - Tehnical Report CS0049 - 1975
- I
IN!F-ROOOCT ION
AW
Difficulty with the newer
optimality theorem in
[3, p,64J
to an investigation of under what conditions and compared to what optimal,
After examining
A*
led is
the proof of the newer theorem and demonstrating
its inadequacy, a flaw in the proof of the older theorem [lJ is discussed, We then provide new proofs for modified statements, 3-6
though
3-9
of
[3J
Familiarity with
[lJ, and Sections
is assumed,
AN UNPROOF In the proof of Theorem
3-2
[3J, it is argued that if an arbitrary
admissible algorithm A never expanded some node A 'knows'
that
~(n)
estimate of
f(h)
~
= f(s)
f(s)
directly from the definitions
the value of relation)
before termination, then
and that this knowledge permits a lower bound
- g(n),
any algorithm at any time,
n
of
Note that the relation fen)
and
f(s)
follows
and therefore is known to
If this relation means to imply a knowledge of
f(s) (which is not implied in general by a knowledge of the
then
A, in general, knows this only after a path has peen identified
as optimal and estimation at that stage is irrelevant, a knowledge of the value of is
fen) ~ f(s)
"more informed"
f(s)
If it does not imply
and the argument is correct, then no algorithm
than another because all are equally informed at a non
empty subset of the open
nod~s,
Technion - Computer Science Department - Tehnical Report CS0049 - 1975
- 2
A FLAW
In the proof of Theorem 2 [1], the existence of some graph for which h
hA*(n)
is defined by
is supported by the definition
= inf OE:0
achieved for some
0
E: e,
ho(n)
~
the set
h.
n
Over the set of all
n
of
an assumption that the infimum is
is necessarily consistent with the equality is finite.
n,e
.".
= hen) hen)
G
graphs, this assumption
0.
Therefore, in order for the
A
definition of h to be internally consistent and thereby well defined,
A*
the entire set of
must
TEOINION - Israel Institute of Technolo,y Computer Science Department
ON THE OPTIMALITY OF A· by
David Gelperin Technical Report No.49
May 1975
Technion - Computer Science Department - Tehnical Report CS0049 - 1975
- I
IN!F-ROOOCT ION
AW
Difficulty with the newer
optimality theorem in
[3, p,64J
to an investigation of under what conditions and compared to what optimal,
After examining
A*
led is
the proof of the newer theorem and demonstrating
its inadequacy, a flaw in the proof of the older theorem [lJ is discussed, We then provide new proofs for modified statements, 3-6
though
3-9
of
[3J
Familiarity with
[lJ, and Sections
is assumed,
AN UNPROOF In the proof of Theorem
3-2
[3J, it is argued that if an arbitrary
admissible algorithm A never expanded some node A 'knows'
that
~(n)
estimate of
f(h)
~
= f(s)
f(s)
directly from the definitions
the value of relation)
before termination, then
and that this knowledge permits a lower bound
- g(n),
any algorithm at any time,
n
of
Note that the relation fen)
and
f(s)
follows
and therefore is known to
If this relation means to imply a knowledge of
f(s) (which is not implied in general by a knowledge of the
then
A, in general, knows this only after a path has peen identified
as optimal and estimation at that stage is irrelevant, a knowledge of the value of is
fen) ~ f(s)
"more informed"
f(s)
If it does not imply
and the argument is correct, then no algorithm
than another because all are equally informed at a non
empty subset of the open
nod~s,
Technion - Computer Science Department - Tehnical Report CS0049 - 1975
- 2
A FLAW
In the proof of Theorem 2 [1], the existence of some graph for which h
hA*(n)
is defined by
is supported by the definition
= inf OE:0
achieved for some
0
E: e,
ho(n)
~
the set
h.
n
Over the set of all
n
of
an assumption that the infimum is
is necessarily consistent with the equality is finite.
n,e
.".
= hen) hen)
G
graphs, this assumption
0.
Therefore, in order for the
A
definition of h to be internally consistent and thereby well defined,
A*
the entire set of
must
