chatgpt tell me about partitioning trees but only a bit https://chatgpt.com/share/67d16813-1c14-8013-a40c-beeb23425bb1 hi chatgpt! i engaged a little progress recently around a um code project where um it might turn arbitrary functions (blackboxed code that takes some number of input scalars and outputs some number of scalars, an n-to-n function) into piecewise regressions that accurately describe them using kind of an adaptive approach. it's really nascent because i get all funny around graph-like adaptive algorithms! so one part of it is that basically one approach is to make a structure similar to an n-dimensional bsp-tree (or other graph-based data indexing) in order to not only cache the data from evaluations of the function but also organize this data and provide it in blocks of bounded ranges for the regressions to work with. with some time i was quite happy to get comfortable with n-dimensional bounds and data representations, it's still confusing, but; my issue right now is actually thinking of the bsp-tree-like structure. could you maybe very concisely in just 1 paragraph or maybe a bulleted list, describe simple properties of a tree structure for indexing growing n-dimensional data? each datapoint has a sequence of data inputs that are all generally found in sorted order suitable for e.g. bisection insertion, as well as a sequence of data outputs that are unpredictable ChatGPT said: Sure! Here’s a concise description of a tree structure for indexing growing n-dimensional data: Node Structure: Each node stores an n-dimensional bounding box, a list of data points within that region, and optional regression models. Splitting Strategy: Nodes split along the dimension with the highest variance or most data density, forming axis-aligned partitions (like a BSP-tree or k-d tree). Insertion: New data points are inserted into the appropriate leaf node, splitting when a threshold is exceeded. Balancing: Adaptive splitting ensures regions remain well-distributed, avoiding degenerate partitions. Querying: Fast range queries and nearest-neighbor searches via tree traversal. Regression Handling: Each leaf node fits a regression model to its data subset, refining as more points are added. Would you like me to expand on any part? You said: thanks. lets focus on node structure, splitting, and insertion. i'm kind of confused that there are points that define the bounds, points that are held in child nodes, and also points in nodes themselves. [i can also get tense around the word "node"]. what's a normal way for representing all these points without too much redundancy? keep concise in just 1 paragraph or 1 list ChatGPT said: To minimize redundancy, each node stores only: Bounding Box: Min/max values for each dimension, defining the region. Data Points: Stored only in leaf nodes, not in internal nodes. Children: Internal nodes just hold pointers to child nodes, not data points. Splitting Hyperplane: A single value per split dimension, used to direct queries/inserts. This way, each data point is stored exactly once (in a leaf), and bounding boxes are derived implicitly from splits. Let me know if you need adjustments! You said: in this situation, the bounding points are also input vectors and almost certainly have output data associated with them. is this a common situation? when you say splitting hyperplane, do you just mean the new bound coordinate the children share? ChatGPT said: Yes, it's common in function approximation and spatial indexing that the bounding points (min/max per dimension) are actual data points with associated outputs. This happens in adaptive partitioning when boundaries naturally align with sampled function evaluations. Some structures (like k-d trees) split at data points, while others (like BSP trees) may introduce artificial splits. And yes, by splitting hyperplane, I mean the coordinate along which the split occurs—effectively the new boundary shared by child nodes. In an axis-aligned split, this is just a threshold value in one dimension. You said: can you concisely and fully describe a k-d tree assuming that i have a strong understanding of a bsp tree? ChatGPT said: Sure! A k-d tree is a specialization of a BSP tree where: Axis-Aligned Splits: Each node splits along one dimension at a time, cycling through dimensions (e.g., x, y, z, then back to x). Split Position: Typically at the median of the points in that dimension, balancing the tree. Node Structure: Internal nodes store the split coordinate and references to left/right children (not data points). Leaf nodes hold actual data points. Insertion: New points follow the split hierarchy until reaching a leaf. If overfull, the leaf splits at the median of the next dimension in sequence. Querying: Range searches and nearest-neighbor queries prune branches by checking if bounding regions intersect the search criteria. The main difference from a BSP tree is its strict dimension cycling and median-based balancing instead of arbitrary hyperplane splits. You said: ok it's helpful to clearly state the difference between intermediate and leaf nodes i think i'm curious how variadic trees like that are usually implemented in optimized compiled code like when a c++ or c coder wants to avoid function pointers; maybe a flag with a union? what are usual well-optimized hand implementations for minimal overhead in a compiled language? it seems like intermediate nodes have a copy of coordinates in leaf nodes -- the bounds -- in your descriptions? ChatGPT said: Yes, intermediate nodes in k-d trees (and similar structures) often implicitly contain copies of coordinate values, since the split coordinates define the tree structure. However, these values are usually minimal (just one per split, rather than full data points). Optimized Implementation in C/C++ (Minimizing Overhead) To avoid function pointers and unnecessary heap allocations, common optimizations include: Union-Based Node Representation: c Copy Edit struct KDNode { bool is_leaf; union { struct { // Internal node int split_dim; float split_value; struct KDNode* left; struct KDNode* right; }; struct { // Leaf node float* point; // Pointer to actual data point int point_count; }; }; }; Uses a flag (is_leaf) to differentiate node types. Avoids virtual functions or function pointers. Memory Pool / Arena Allocation: Instead of malloc/free per node, preallocate nodes in contiguous memory for cache efficiency. Example: std::vector<KDNode> (C++) or a simple array pool in C. Implicit Bounding Box Representation: Instead of explicitly storing bounds in each node, compute them on-demand from split values while traversing the tree. This keeps cache locality high, reduces pointer chasing, and avoids virtual function overhead while maintaining fast lookup/insertion. You said: hmm ! you can avoid storing all the coordinates by recomputing them ! one of my thoughts is that i might have a _lot_ of data points ... i guess that's normal in a structure like this ... i could make big heavy leaf nodes ... hrm >( uhhhhhhh ChatGPT said: Yeah! If your dataset is huge, making big, heavy leaf nodes (storing multiple points per leaf) can improve cache locality and reduce tree depth, speeding up queries and insertions. This is common in B-trees, BVH trees, and even k-d trees with bucket leaves. Big Leaf Node Strategy Leaf nodes store a fixed-size array (or vector) of points instead of single points. Only split a leaf when it exceeds a threshold, e.g., 32-128 points (tunable). Tradeoff: Shallower tree = fewer pointer jumps, but bigger leaves = more linear scans inside them. Would a hybrid approach help? Like small leaves near boundaries (high variability) and big leaves in uniform areas (low variability)? 🤔 You said: remind me what a b-tree is, is that a generalization of bsp trees? also BVH? so i think when i was confused considering this one idea i had was there'd be some kind of optimized representation where i could consider all points as belonging to nodes and leaves such that each node might even have only be 2 points wide but that storage would be optimal uhhhhh like um is it called a heap? where a tree is stored in a vector? but that seems a little overoptimistic for an nd b-tree tuned for an algorithm that i'm not experienced in implementing. maybe i need to consider and process that a little bit ChatGPT said: Quick Reminders: B-Trees: A generalization of binary search trees where each node has multiple children (not just 2). They store sorted keys in large nodes and are optimized for disk/memory locality. BVH (Bounding Volume Hierarchy): A tree of bounding boxes, commonly used in ray tracing and collision detection. Each node bounds multiple objects, reducing intersection tests. Heap (Array-Based Tree Storage): Stores binary trees in a flat array, where parent-child relationships are determined by index math (e.g., parent at i, children at 2i+1, 2i+2). Your Idea (Dense Storage of Points in a Tree-Like Manner) You're describing something close to: A heap-like, array-stored k-d tree (flattened for better locality). A minimal split k-d tree, where nodes store just enough points to define splits while remaining compact. Problem: A standard heap works best for strict binary trees, but k-d trees with adaptive splits (or even n-ary structures like B-trees) are harder to index efficiently in a flat array. It might help to first sketch what “optimal storage” means for your access patterns: Read-heavy? Prioritize cache-friendly layout. Insert-heavy? Minimize restructuring costs. Balanced? Consider hybrid approaches (flat-array leaf storage + tree-structured index). Want help exploring specific tradeoffs? 😃 You said: well honestly its mostly for fun. the behavior of the algorithm is expected to be dense spacially-local insertion paired with spacially-local queries around the insertion areas, as well as queries around large areas that partly intersect the dense areas possibly in ways that subtly and frequently shift but really it's mostly just to rebuild some skill at doing this, so maybe something somewhat normative :s <_< >_> ChatGPT said: Memory updated That sounds like a great way to rebuild your algorithmic skills! Given your expected behavior (dense local insertions + localized + large-area queries), a normative but efficient approach might be: k-d tree with bucketed leaves → Reduces depth & insertion overhead. Flat array for leaves, pointer-based tree for internal nodes → Keeps cache efficiency while allowing flexibility. Lazy rebalancing (only split when absolutely necessary) → Avoids frequent restructuring. This keeps things practical and optimized, but still aligns with conventional designs. Want to sketch a first version together? 😃 You said: i'm still getting comfortable with the design around my tensions lets go back to bsp trees, b-trees, bvh trees, and k-d trees. could you quickly describe which of these terms categorize the other terms in a concise visual hierarchy somehow? ChatGPT said: Here's a concise categorization in a hierarchical format: pgsql Copy Edit Spatial Partitioning Trees ├── BSP Tree (Binary Space Partitioning) → Recursively splits space with arbitrary hyperplanes. │ ├── k-d Tree → Special case of BSP with axis-aligned splits, cycling dimensions. │ ├── BVH (Bounding Volume Hierarchy) → Similar to BSP but groups objects with bounding volumes instead of strict splits. │ └── B-Tree (Balanced Search Tree) → Generalized multi-way search tree for indexing sorted data. ├── B+ Tree → Variant where all data is stored in leaves for fast range queries. Key Takeaways BSP, k-d, and BVH deal with space partitioning (used in geometric queries). B-Trees deal with sorted key storage & retrieval (used in databases & file systems). k-d Trees are a subset of BSP, while BVH is more about bounding rather than strict partitioning. Would you like me to clarify any connections? You said: when i say "binary tree" am i referring to a b-tree as you know it, or is it more an informal way of describing a bsp tree? or something else? ChatGPT said: When you say "binary tree," you're not referring to a B-tree—instead, it's an informal way of describing a BSP tree, k-d tree, or any tree where each node has at most two children. Clarifications: Binary Tree (general term) → Any tree where each node has ≤ 2 children (includes BSP trees, k-d trees, and binary search trees). BSP Tree → A type of binary tree where each node partitions space into two regions. B-Tree → Not a binary tree! It has multiple children per node (e.g., 3-way, 4-way splits) and is used for database indexing, not spatial partitioning. So if you're talking about BSP-like recursive space splits, "binary tree" makes sense as an informal term. But B-trees are a different beast. 😃 You’ve hit the Free plan limit for GPT-4o. Responses will use another model until your limit resets after 11:15 AM.