Comment by marcodiego | Hacker News Reader

Comment by marcodiego | Hacker News Reader

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Maybe there is a small list of "fundamental simple quadrilaterals" and any other simple quadrilaterals can be obtained by transforming elements from this small list. Maybe it is possible to find an expression to quickly calculate the number of simple quadrilaterals from the initial fundamental list.

Or, maybe Q(m, n) can be calculated as a function of Q(m - 1, n), Q(m, n - 1) and Q(m - 1, n - 1). But calculating Q(m, n) may not be that important, maybe finding its factors is! So, all you have to do is to find the factors of Q(m, n) and then calculate a rest of division. So, maybe it is possible to find Q(m, n) mod x as a function of Q(m - 1, n), Q(m, n - 1) and Q(m - 1, n - 1). If this can be found, the problem becomes trivial.

Note that if we don't have to care about lines crossing, calculating Q(m + 1, n), Q(m, n + 1) and Q(m + 1, n + 1) as a function of Q(m, n) maybe easy. All that needs to be done is to find an expression for these that take line crossing into account, then use this expression to calculate a rest of division.

This is broadly how I was thinking about it, but I don't have an intuition for why there would be a recurrence relation involving the factors of Q(m, n). What made you think of attacking it that way?

Because it may allow us to use dynamic programming to calculate Q(m, n) and because some new quadrilaterals can be easily generated simply by moving edges and vertices to new spaces when m or n is increased. So, it seems natural for me to try to write Q(m + 1, n) as a function of Q(m, n).

Also, note that Q(m, n) = Q(n, m), so if we can calculate Q(m + 1, n) as a function of Q(m, n) we can also do it for Q(m, n + 1). Calculating Q(m + 1, n) as a function of Q(m, n) doesn't seem complicated if it weren't by the rules "no straight angles and does not self-intersect". Maybe it can be done with some combinatorics, but seems beyond my skill. Also, expressing it in terms of combinations may also simplify calculation of rest of division.

If such relation can be found, I think the problem may be easily solved.

This made me think of... U(m, n) as the number of unique forms of Q(m, n). By unique forms, I mean forms that can't be obtained by simply translating previously found forms. Maybe it is easy to calculate Q(m, n) as a function of U(m, n) and maybe calculating U(m, n) as a function of U(m - 1, n) is a bit easier than calculating Q(m - 1, n).

Note: Q(m + 1, n) =

  each valid quad of Q(m, n) moving an edge to another possible one in the new line + 
  each valid quad of Q(m, n) moving a vertex to the new line +
  the same thing moving each unique valid quad of Q(m, n) to the new line.

The number of new possible edges on the new line is n * (n - 1) / 2 . The problem is: moving each edge of each quad that can be generated in Q(m, n) to the new possible edges may generate quads that break the rules.

The number of new possible vertices on the new line is (m + 1) . The problem is: moving each vertex of each quad that can be generated in Q(m, n) to the new possible vertices may generate quads that break the rules.

Finding a way to calculate both terms looks like good progress.

It might be easier to define f(m, n, v) where f(m, n, v) is the number of v-vertex shapes that can fit within a m-n square with exactly one vertex in the top left hand corner. Then Q(m, n) = \sum_{i \in m, j \in n} f(m, n, 4). f also lends itself to a much easier recurrence.

For me, it seems easy if we don't have to care about the imposed restrictions: no crossing and no straight angles.

Hmmmm.... New ideas coming... Maybe we don't have to care about crossings: any crossing can be solved by reordering the vertices, so, all we have to care about is the number unique vertices positions!

I still don't know how to avoid straight angles though.

If you didn’t care about the restrictions, then the answer is N*M choose 4.

I think I've made some progress... When one line is added to the problem, it allows us to put 1 or 2 vertices there, so...

Consider U(m, n) as the number of new unique forms possible for Q(m, n). I'm not sure, but it looks like:

  U(1, 1) = 1

  U(m + 1, n) = U(m, n) * (n + (n+1)*n/2) - X

  U(m, n + 1) = U(m, n) * (m + (m+1)*m/2) - X

Where X is the number of invalid forms generated adding one or two vertices in the new line because of straight angles in diagonals. I'm considering only diagonals, because adding only one or two vertices in the new line will not align 3 vertices horizontally or vertically if all "previous forms" were valid.

If we can find X, then, I think Q is given by:

  Q(m, n) = m*n*U(1, 1) + (m-1)*(n-1)*U(2, 2) + (m-2)*(n-2)*U(3, 3) +...+ 1*1*U(m, n)

> If we can find X

Almost certainly we need to add terms into the DP state to compute X; unfortunately adding just the number of vertices doesn’t work as some triangles have two edges in which a point can be inserted to create a quadrilateral, while others only have one.

The difficulty is finding those terms, though…

What do you mean by "the DP state"?

Also, I think my previous formula is wrong. Maybe it is the sum of:

  (m - i + 1)*(n - j + 1)*U(i, j)

With i, j in [1..m].

But maybe it is still wrong because U(i, j) will re-count combinations from U(i - 1, j) and U(i, j - 1)...

Argh! This is complicated! I think I'll have to revisit it if I ever have a good idea.

By DP state, I mean the arguments to the recursive function (for U, the state is the tuple (i, j)).

For example, if you wanted to use DP to compute all-pairs-shortest-paths (Floyd-Warshall), you add a third parameter to the DP state ((f(u, v, k)).

Sure! Dynamic programming, right?

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