]> Simple Paths in a Grid

Simple Paths in a Grid

Given:

A grid of filled and unfilled points $R$ rows tall and $C$ columns wide

With starting point $S (0, 0)$

And finish point $F (R - 1, C - 1)$

And $n$ filled points $X_{i} (R_{i}, C_{i}), R > R_{i + 1} \geq R_{i} \geq 0 and C > C_{i + 1} \geq C_{i} \geq 0, \forall i \in \{1, 2, \dots, n\}, n \geq 0 \in ℤ$

Find:

The number of shortest simple paths $N$ from starting space $S$ to finish space $F$ .

Solution:

Let $P ((r_{1}, c_{1}), (r_{2}, c_{2}))$ be the number of shortest simple paths from $(r_{1}, c_{1})$ to $(r_{2}, c_{2})$ , ignoring filled points.

$P ((r_{1}, c_{1}), (r_{2}, c_{2})) = {\begin{matrix} {}_{((r_{2} - r_{1}) + (c_{2} - c_{1}))}C_{(r_{2} - r_{1})} & , r_{2} \geq r_{1} and c_{2} \geq c_{1} \\ 0 & , r_{2} < r_{1} or c_{2} < c_{1} \end{matrix}$

$N = P (S, F) - \sum_{i = 1}^{n} [P (S, X_{i}) \cdot P (X_{i}, F)] + \sum_{i = 1}^{n - 1} [P (S, X_{i}) \cdot P (X_{i}, X_{i + 1}) \cdot P (X_{i + 1}, F)]$