Quadtrees - Hierarchical Grids

Let P be a planar map. To be more concrete, let P be a partition of the unit square into triangles (i.e., a mesh). Since any simple polygon can be triangulated, P can represent any planar map, and its partition of the unit square into different regions, with different properties. For the sake of simplicity, assume that every vertex in P shares at most, say, nine triangles. Let us assume that we want to preprocess P for point-location queries. Of course, there are data-structures that can do it with O(n log n) preprocessing time, linear space, and logarithmic query time. Instead, let us consider the following simple solution (which in the worst case, can be much worse). Build a tree T , where the root corresponds to the unit square. Every node v ∈ T corresponds to a cell v (i.e., a square), and it has four children. The four children correspond to the four squares formed by splitting v into four equal size squares, by horizontal and vertical cuts. The construction is recursive, and we start from v = rootT . As long as the current node intersects more than, say, nine triangles, we create its children nodes, and we call recursively on each child, with the list of input triangles that intersect its square. We stop at a leaf, if its “conflict-list” (i.e., list of triangles it intersects) is of size at most nine. We store this conflict-list in the leaf. Given a query point q, in the unit square, we can compute the triangle of P containing q, by traversing down T from the root, repeatedly going into the child of the current node, whose square contains q. We stop at soon as we reach a leaf, and then we scan the leaf conflict-list, and check which of the triangles contains q. Of course, in the worst case, if the triangles are long and skinny, this quadtree might have unbounded complexity. However, for reasonable inputs (say, the triangles are fat), then the quadtree would have linear complexity in the input size (see Exercise 5.1) .The big advantage of quadtrees of course, is their simplicity. In a lot of cases, quadtree would be a sufficient solution, and seeing how to solve a problem using a quadtree might be a first insight into a problem.