Question

On the wikipedia entry for k-d trees, an algorithm is presented for doing a nearest neighbor search on a k-d tree. What I don’t understand is the explanation of step 3.2. How do you know there isn’t a closer point just because the difference between the splitting coordinate of the search point and the current node is greater than the difference between the splitting coordinate of the search point and the current best?

Nearest neighbor search Animation of
NN searching with a KD Tree in 2D

The nearest neighbor (NN) algorithm
aims to find the point in the tree
which is nearest to a given input
point. This search can be done
efficiently by using the tree
properties to quickly eliminate large
portions of the search space.
Searching for a nearest neighbor in a
kd-tree proceeds as follows:

1. Starting with the root node, the algorithm moves down the tree
   recursively, in the same way that it
   would if the search point were being
   inserted (i.e. it goes right or left
   depending on whether the point is
   greater or less than the current node
   in the split dimension).
2. Once the algorithm reaches a leaf node, it saves that node point as
   the “current best”
3. The algorithm unwinds the recursion of the tree, performing the
   following steps at each node:
   1. If the current node is closer than the current best, then it
   becomes the current best.
   2. The algorithm checks whether there could be any points on
   the other side of the splitting plane
   that are closer to the search point
   than the current best. In concept,
   this is done by intersecting the
   splitting hyperplane with a
   hypersphere around the search point
   that has a radius equal to the current
   nearest distance. Since the
   hyperplanes are all axis-aligned this
   is implemented as a simple comparison
   to see whether the difference between
   the splitting coordinate of the search
   point and current node is less than
   the distance (overall coordinates)
   from the search point to the current
   best.
   1. If the hypersphere crosses the plane, there could be
   nearer points on the other side of the
   plane, so the algorithm must move down
   the other branch of the tree from the
   current node looking for closer
   points, following the same recursive
   process as the entire search.
   2. If the hypersphere doesn’t intersect the splitting plane,
   then the algorithm continues walking
   up the tree, and the entire branch on
   the other side of that node is
   eliminated.
4. When the algorithm finishes this process for the root node, then the
   search is complete.

Generally the algorithm uses squared
distances for comparison to avoid
computing square roots. Additionally,
it can save computation by holding the
squared current best distance in a
variable for comparison.

Answer 1

Look carefully at the 6th frame of the animation on that page.

As the algorithm is going back up the recursion, it is possible that there is a closer point on the other side of the hyperplane that it’s on. We’ve checked one half, but there could be an even closer point on the other half.

Well, it turns out we can sometimes make a simplification. If it’s impossible for there to be a point on the other half closer than our current best (closest) point, then we can skip that hyperplane half entirely. This simplification is the one shown on the 6th frame.

Figuring out whether this simplification is possible is done by comparing the distance from the hyperplane to our search location. Because the hyperplane is aligned to the axes, the shortest line from it to any other point will a line along one dimension, so we can compare just the coordinate of the dimension that the hyperplane splits.

If it’s farther from the search point to the hyperplane than from the search point to your current closest point, then there’s no reason to search past that splitting coordinate.

Even if my explanation doesn’t help, the graphic will. Good luck on your project!