How to find a octree node's neighbors when the tree is ordered by Morton code

Question

I have an octree that I want to use to perform pathfinding in 3D space. I'm using Morton codes to order the nodes with the intention of being able to easily find the nodes nearby any given node. It appears to work great; any subset of the nodes I grab are clustered relatively close together. Where I'm struggling is finding an efficient way of getting a node's direct neighbors.

An earlier version of the tree had it subdivide the tree until the leaves were all equal sized nodes at the size I wanted. This made it easy to find neighbors as there was a consistent offset between each node, so I could just recalculate the Morton code at the position I expected to find a node, and then retrieve the node based on the code. But I'm trying to be more efficient with the tree by not breaking it down as much in areas where there isn't as much geometry (e.g. where I would get 8 empty children). As a result, I no longer know how many neighbors a given node may have, nor the size of any of the neighbors (other than it being a power of 2).

My initial thought was to search nearby nodes in either direction in the array, which would work, but I'd have to cast too wide a net to be guaranteed to get all neighbors, and I'd have spent time evaluating a ton of nodes that I don't need. A quick test indicates that not even 50 nodes in either direction in the array is guaranteed to get me all neighbors.

The tree is static, so I could perhaps do a one time generation of a node's neighbors using the above brute force method and store the neighbors; but I fear this may increase the initial loading time of the program beyond acceptable levels. Is there an algorithm that can find the neighbors in real-time?

Edit: Reworded the question to clarify what I'm looking for and what I've tried.

Answer 1

Yes.

Don't brute-force, go straight to the tile you want.

Provided you have:

• A Tile toMorton(Point3D location) function which provides a Tile given its real-world location
• A Point3D fromMorton(Tile t) function that provide the real-world coordinates of a Tile (both available from Wikipaedia I believe)

These are simple, real-time operations.

The naive approach

Simply convert to real-wrold, offset there, and back to Morton space:

Tile originalTile = ...
Point3D originalLocation = fromMorton(original);
Point3D offsetLocation = new Point3D(offsetLocation.x + 1, offsetLocation.y, offsetLocation.y);
Tile offsetTile = toMorton(offsetLocation)

This requires just a few arithmetic operations, so should already be suitable for real-time (more so than brute-forcing).

The Optimised Approach

A quick google search on "morton code neighborhood" led me to a very well written blog entry from volumeseoffun.

I'll provide the best bits here:

for a given (x,y,z) which has a Morton position p; if we want to peek at (x+1,y,z) then the amount by which p must increase, Δp, is dependent only on x and is independent of y and z. This same logic holds for peeking in y and z.

This means we do not need to toMorton / fromMorton all three coordinates

Furthermore (although neither the blog nor its source back it up) it seems that said offset can be precomputed via a lookup table. This gives additional speedup for the realtime capability you're looking for. To me it makes sense given that Morton algo is well-defined and repeatable, so I trust this assumption.

Simply pre-compute the offset table, then from a start tile, use that table to offset yor tile to its real-world neighbour:

int[] offsetX = [ ... compute once ... ]
int[] offsetY = [ ... compute once ... ]
int[] offsetZ = [ ... compute once ... ]
Tile[] allTiles = ... // Init 3D world
public Tile neighbour(Tile original, Direction dir){
     switch(dir){
     case DIR_X_PLUS: return allTiles[original.chunkID + offsetX[original.chunkID];
     case DIR_X_MINUS: return allTiles[original.chunkID - offsetX[original.chunkID];
     case DIR_Y_PLUS: return allTiles[original.chunkID + offsetY[original.chunkID];
     [... etc...]
     default: return original;
     }
}

Hope this helps.