Hello baxy77bax,

you need to rethink your requirements. If you want to have the minimum number of moves, you only have two possibilities:

• slide from aa to ba and then to bb
• slide from aa to ab and then to bb

For instance, is it legal to try to do a tour of the matrix, to pick up as many high value cells as possible? If the priority is "max sum", followed by "min terms", then a "rook's tour" of the positive valued cells is in order.

Another idea, can you revisit a cell? What's to keep you from circling through 4 positive value cells forever?

Can you go "offline", by taking a longer tour than necessary?

a) It is legal to do a tour of the matrix and pick up as many high value cells as possible. :)

b) this is not legal... each cell can only be visited once (but if this was associated to some fintech problem where total sum is your account balance, I guess this would be a good way to boost it up :D )

c)Yes you can go offline if the number of moves is higher that the total sum of the path toward the final value (cell), but this only holds if a path already exists. If it is the initial search then there is no reference and the condition does not hold (at least one path (solution) needs to exist )

thnx I did not catch those conditions but now that you mentioned it i see how they are crucial for the problem

by helping you I helped myself. Thank you !!!

What does slide mean? If you slide A->D, skipping B and C, what is the path length?3 or 1? Also, does sliding mean that the points on the squares you slided over are not counted to your sum? Only the final square you landed on is counted?

slide A -> D means skipping B and C and only values on A and D are counted, B and C are not counted and the path length in case you slide A-> D is 2 and when you go A->B->C->D is 4.

direction in which I am thinking is that maybe I could compute max and min rang query matrix and then from both sides start and end somehow figure out the outer boundaries .... but as QM said i see no way to avoid full all-pair traversal of the matrix ... also i am thinking in direction of maybe starting the taversal from the highest value in the matrix and working my way to reach start and end point but i am not sure if this is a good way to go...

OK. But something does not add up: You can slide over negative values, essentially ignoring them. They have no effect whatsoever! right? Oh except when they are on a turning.

Here are some more questions to rubber-duck it

Do you know your matrix fully in advance? Or do you have to query it as you move along? Because it may be too expensive to evaluate or store the full matrix, for example evaluating a function in 10-decimal accuracy.

Do you have to find the best solution or can you use a heuristic like genetic algorithms which may find a good solution in short time by skipping some (possibly better) solutions? If so, one simple heuristic is to start first with those squares with large points and slide through them. Or plan a path which includes most valueable squares and no negatives. Completely ignoring the small ones and hoping for the best! More concretely, here is a top-down rather than bottom-up approach: first sort your squares wrt value and this will determine your visitation priority. Then construct a path-builder to suggest paths given these top-priority, must-visit squares.

Btw, it reminds me of Gradient Descent (or Hill Climbing). It still suffers from local minima but there are some better alternatives out there fo exploring a neighbourhood and avoiding local minima ...

In any case I thought perhaps you can cache path segments you visit so that you do not have to re-sum.

bw, bliako