Dear Jacob,
thanks for your clarification.
But what if in your example you come up with three different incidence
matrices
connecting all pairs of nodes (i,j) , the first representing connection
values
with respect to Q, the second S, the third E, respectively?
To select some overall 'optimal' way for a choosing a path, one would then
also need
a loss (or gain, if you prefer) function that weighs the optimality values
with respect to the individual criteria.
I feel that it is necessary to avoid evaluating n^d ( n-number of paths, d-
dimension
of criteria) items in searching for an optimal path ...
Using several incidence matrices, you would need n*d evaluations.
Best regards,
Axel
But what if in your example you come up with three different incidence
matrices connecting all pairs of nodes (i,j) , the first representing
connection values with respect to Q, the second S, the third E,
respectively?
I think you misunderstood. The three paths weren't one each for each
criterion. There were three distinct paths (or as Gavin suggested as
an alternate example, three distinct songs connecting two letters).
Each path then had three weights, one for each criterion. A Quickness
incidence matrix would still need to store (somehow) 3 values, one for
each of the edges connecting A and B.
I feel that it is necessary to avoid evaluating n^d ( n-number of paths, d-
dimension of criteria) items in searching for an optimal path ...
You don't need to evaluate them, unless you want to consider all
criteria at once. Usually you'd only choose one criterion. A
pathfinder which only cares about one criterion only needs to evaluate
one value from each of n paths. With incidence matrices you'd only
select one matrix (that corresponding to the chosen criterion) and
have n evaluations. Same boat.
The difference isn't in the number of criteria -- that was just an
illustrative concept. I admit my approach to the criteria may have
been simplistic (or even incorrect).
Best regards,
Axel
Thanks for the feedback, Axel 
Jacob Fugal
···
On 5/9/05, Nuralanur@aol.com <Nuralanur@aol.com> wrote:
Aye. My goal for this library is to first make it really clear (and probably slow). It will yield repeatedly to user-supplied blocks/methods to decide what to do.
"In general, would you like me to try breadth-first or depth-first?"
"Hey, I can't get to the end node from the current node, but here are the nodes I can directly get to from here. Given my path so far, do you have any preference for which node I try next?"
"OK, I have 3 edges to get to the node I'm going to try. Which one seems best?"
"Given the route I've got so far (and given any sort of instance variables you've stored as I've been traversing) is it worthwhile to continue trying this particular path, or should I give up on this one?"
"Here are two routes I've found so far. Which one is better?"
The "which edge should I use?" question will help pare things down a good bit, but (in my experience with the quiz) the "is it worthwhile to continue trying this particular path?" question is the one that really gives you the ability to pare down the search tree massively.
Thanks for the continuing questions that help me critically think about this.
···
On May 9, 2005, at 5:30 PM, Jacob Fugal wrote:
I feel that it is necessary to avoid evaluating n^d ( n-number of paths, d-
dimension of criteria) items in searching for an optimal path ...
You don't need to evaluate them, unless you want to consider all
criteria at once. Usually you'd only choose one criterion. A
pathfinder which only cares about one criterion only needs to evaluate
one value from each of n paths. With incidence matrices you'd only
select one matrix (that corresponding to the chosen criterion) and
have n evaluations. Same boat.
--
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