Parallel Tree Algorithms

Parallel Tree Algorithms are the techniques for constructing, traversing, and computing on tree data structures using multiple processors simultaneously — challenging because trees have inherent parent-child dependencies that limit parallelism, but critical for applications like spatial indexing (BVH for ray tracing), database B-trees, decision tree inference, and hierarchical reduction, where specialized algorithms like parallel BVH construction, bottom-up parallel reduction, and level-synchronous traversal achieve significant speedups.

Why Trees Are Hard to Parallelize

• Arrays: Element i independent of element j → embarrassingly parallel.
• Trees: Child depends on parent's position, depth depends on insertion order.
• Traversal: Visit root → children → grandchildren → inherently sequential per path.
• Key insight: Different PATHS in the tree are independent → exploit inter-path parallelism.

Parallel Tree Construction (BVH)

Bounding Volume Hierarchy (BVH) — used in ray tracing:

 1. Assign Morton codes to all primitives (sort by spatial location)
 2. Parallel sort by Morton code → O(N log N) on GPU
 3. Build radix tree from sorted codes → O(N) parallel
 4. Bottom-up: Compute bounding boxes from leaves → root

 All steps are parallel → GPU BVH construction in milliseconds

• LBVH (Linear BVH): Morton code based → fully parallel construction.
• SAH BVH: Surface Area Heuristic → higher quality but harder to parallelize.
• GPU: Millions of primitives → BVH built in 5-20 ms on A100.

Level-Synchronous Traversal (BFS on Trees)

BFS by level:
 Level 0: Process [root]                      → 1 task
 Level 1: Process [child0, child1]             → 2 tasks
 Level 2: Process [c00, c01, c10, c11]         → 4 tasks
 Level k: Process [all nodes at level k]       → 2^k tasks

 Parallelism grows exponentially with depth!

• Good for: Balanced trees where most nodes are at deeper levels.
• GPU: Launch one thread per node at each level → synchronize between levels.

Parallel Tree Reduction (Bottom-Up)

Leaves:  [3] [5] [2] [8] [1] [4] [7] [6]
              \ /      \ /      \ /      \ /
 Level 1:    [8]      [10]     [5]     [13]    (max of children)
                \ /                \ /
 Level 2:      [10]              [13]
                    \        /
 Level 3:          [13]                        (global max)

• Bottom-up reduction: Start at leaves, combine pairs → root has result.
• O(log N) levels, each level fully parallel → efficient on GPU.
• Used for: Hierarchical bounding box computation, segment trees, aggregation.

Decision Tree Inference (Parallel)

// Parallel: Each thread evaluates one data sample through the tree
__global__ void tree_predict(float *data, int *nodes, int *results, int n) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx < n) {
        int node = 0;  // Start at root
        while (!is_leaf(nodes[node])) {
            float val = data[idx * features + nodes[node].feature];
            node = (val <= nodes[node].threshold) ?
                   nodes[node].left : nodes[node].right;
        }
        results[idx] = nodes[node].prediction;
    }
}
// Data parallelism: Different samples take different paths → all independent

Parallel B-Tree Operations

Performance Examples

Parallel tree algorithms are the bridge between hierarchical data structures and massively parallel hardware — while trees appear inherently sequential due to parent-child dependencies, techniques like Morton-code-based construction, level-synchronous traversal, and data-parallel inference transform tree operations into GPU-friendly parallel workloads, enabling real-time ray tracing, high-throughput database queries, and millisecond-latency decision tree inference at scale.