一个我自己研发的支持k-th路径查询的数据结构-owl tree

由于本人水平有限，数据结构很多地方并没有优化，但是提出了一些思想，为了保证通用性，使用了英文。

# Owl Tree: A Novel Tree-Based Data Structure for Efficient k-Path Operations

## Abstract

We introduce the **Owl Tree**, a novel data structure designed for efficiently supporting dynamic operations on rooted trees under a deterministic \$k\$-based child traversal model. By decomposing the tree into linear structures termed **owl chains**, the Owl Tree enables fast query and update operations via segment trees. It supports a rich set of dynamic operations, including path queries, subtree deletions, node insertions, and full structural reconstructions. Our structure achieves an amortized time complexity of \$\mathcal{O}(Q \log n)\$ for \$Q\$ operations on a tree with \$n\$ nodes. Notably, any modification to the global parameter \$k\$ requires a complete structural rebuild, highlighting a key open problem in incremental maintenance.

---

## 1. Introduction

Tree-based data structures are fundamental in domains such as file systems, compiler construction, network routing, and hierarchical data modeling. A frequently encountered pattern in these contexts involves traversing trees under deterministic rules—such as following the \$k\$-th smallest child of a node—which arises naturally in decision trees, search systems, and knowledge hierarchies.

While classical structures like Heavy-Light Decomposition (HLD) and Link-Cut Trees (LCT) are optimized for arbitrary path queries and updates, they do not leverage the regularity of deterministic traversals. To address this gap, we propose the **Owl Tree**, a structure designed to efficiently handle deterministic \$k\$-path traversals. It organizes the tree into structured linear segments called **owl chains**, each managed by a dedicated segment tree, enabling efficient localized updates and queries.

---

## 2. Problem Formulation

We consider a dynamic rooted tree with the following properties:

* Each node is uniquely identified and associated with a weight.
* A global parameter \$k\$ governs deterministic traversal: from a node \$x\$, we recursively follow the \$k\$-th smallest child (or the largest available if fewer than \$k\$ children exist).
* The data structure supports the following operations efficiently:

  1. **Global parameter modification** (\$k\$-update)
2. **Path sum queries** along the \$k\$-path
3. **Range updates** along the \$k\$-path
4. **Subtree deletions**
5. **Node insertions**
6. **Full structural reconstruction**

---

## 3. Owl Chain Decomposition

The Owl Tree’s core mechanism is the decomposition of deterministic \$k\$-paths into disjoint linear chains, referred to as **owl chains**, which can be processed efficiently via segment trees.

### 3.1 Chain Construction

Each node’s children are sorted by ID. A depth-first traversal assigns nodes to owl chains by following the \$k\$-th smallest child:

```cpp
void dfs(int u, int depth) {
chainID[u] = currentChain;
chainPosition[{currentChain, depth}] = u;
nodeDepth[{currentChain, u}] = depth;
if (children[u].size() < k) return;
int v = children[u][k - 1];
dfs(v, depth + 1);
}
```

Initialization proceeds as:

```cpp
for (int i = 1; i <= n; i++) {
sort(children[i].begin(), children[i].end());
}
for (int i = 1; i <= n; i++) {
if (!chainID[i]) {
currentChain++;
dfs(i, 1);
}
}
```

Each owl chain is then converted into a contiguous array and paired with a segment tree to support fast range operations.

---

## 4. Supported Operations

### 4.1 Path Queries

Given a node, the sum of weights along its \$k\$-path is computed via a prefix sum query on the segment tree corresponding to its chain.

**Time complexity:** \$\mathcal{O}(\log n)\$

### 4.2 Path Updates

To apply a constant addition along a \$k\$-path, a range update is issued on the segment tree managing the relevant chain.

**Time complexity:** \$\mathcal{O}(\log n)\$

### 4.3 Subtree Deletion

Subtrees are deleted by logically marking the root and its descendants as inactive. These nodes are ignored in subsequent operations.

**Time complexity:** \$\mathcal{O}(1)\$

### 4.4 Node Insertion

New nodes are appended to their parent’s child list. To maintain structural correctness, a full reconstruction is triggered.

**Amortized time:** \$\mathcal{O}(n \log n)\$

### 4.5 Global Reconstruction

Rebuilding the Owl Tree involves:

1. Sorting children by ID
2. Recomputing owl chains
3. Rebuilding segment trees
4. Filtering out deleted nodes

**Worst-case complexity:** \$\mathcal{O}(n \log n)\$

### 4.6 \$k\$-Modification Handling

Since the definition of owl chains depends on \$k\$, any change to \$k\$ requires a full structural rebuild.

This forms a major performance bottleneck. **No known incremental technique** currently exists to efficiently update the chain structure after a \$k\$-modification.

---

## 5. Segment Tree Integration

Each owl chain is managed by a dedicated segment tree supporting:

* Point and range updates
* Prefix sum queries
* Logical deletions via masking or lazy propagation

Segment trees are rebuilt during reconstruction and updated incrementally during most operations.

---

## 6. Complexity Analysis

Let \$Q\$ denote the number of operations and \$R\$ the number of reconstructions (triggered by insertions or \$k\$-modifications). Then:

* **Query/Update:** \$\mathcal{O}(\log n)\$
* **Deletion:** \$\mathcal{O}(1)\$
* **Reconstruction:** \$\mathcal{O}(n \log n)\$

The total runtime is bounded by:

$$
\mathcal{O}(Q \log n + R \cdot n \log n)
$$

In practice, reconstructions are infrequent, yielding an amortized complexity of:

$$
\mathcal{O}(Q \log n)
$$

---

## 7. Future Directions

Promising areas for future exploration include:

* **Generalized path queries** beyond deterministic \$k\$-paths
* **Persistent variants** to support historical operations
* **Incremental \$k\$-updates** without full reconstruction
* **Parallelization** of disjoint owl chain operations

---

## 8. Optimization and Extension Strategies

### 8.1 Lazy Deletion and Deferred Rebuild

To avoid performance degradation due to accumulated deletions:

* Trigger partial reconstructions when deletion ratios exceed thresholds
* Use chain-level metadata to efficiently prune deleted segments

### 8.2 Incremental Insertions

Insertions currently trigger full reconstruction. We propose:

* **Tail insertion** if appending to a leaf
* **Localized chain rebuilding** for subtree-local changes
* **Batch reconstruction** based on queued insertions

### 8.3 k-Modification Optimizations (Open Problem)

We propose speculative strategies:

* **Multi-\$k\$ caching** for a small set of historical \$k\$ values
* **Prefix reuse** across owl chains under different \$k\$
* **Delta updates** for owl chain diffs

A fully incremental \$k\$-update remains an open and challenging problem.

### 8.4 Segment Tree Alternatives

Depending on workload:

* **Fenwick Trees** for fast prefix queries
* **Block decomposition** for long but rarely updated chains
* **Skip lists / jump pointers** for sparse queries

### 8.5 Parallelism and Concurrency

Due to owl chain independence:

* Chains can be built in parallel
* Disjoint path queries can run concurrently
* Lock-free segment trees enable concurrent updates

### 8.6 Partial Subtree Rebuild

Use **topmost subtree tracking** to localize reconstructions:

* Metadata identifies the smallest affected subtree
* Chains within that subtree are selectively rebuilt
* Chain merging strategies maintain global consistency

### 8.7 Historical Caching for \$k\$-Modifications

Inspired by working-set theory:

* Maintain a **fixed-size cache** of recently used \$k\$ values
* On change, reuse cached chains if available
* Evict via LRU or random policies

This technique significantly reduces rebuild frequency in workloads with fluctuating but repetitive \$k\$ values.

---

## 9. Conclusion

The Owl Tree presents a flexible and efficient data structure for handling deterministic \$k\$-path operations in rooted trees. Through strategic chain decomposition and segment tree integration, it achieves strong performance for both queries and dynamic updates. While the full incremental handling of \$k\$-modifications remains unresolved, the proposed optimization techniques provide substantial improvements for practical usage. We hope this work inspires further advancements in the study of deterministic traversal structures.

---

## 10. Comparative Experiments

### 10.1 Setup

We compare the Owl Tree against two baseline structures:

1. **HLD** with segment trees
2. **LCT** using splay trees

All experiments are conducted on an Intel Core i7-9700K, 16GB RAM, Ubuntu 20.04, compiled with `g++ -O2`.

We generate random trees with \$n = 10^5\$, \$2 \times 10^5\$, and \$5 \times 10^5\$ nodes. Each node has \$0\$ to \$5\$ children (uniform distribution), and weights are random integers in $\[1, 10^3]\$. We fix \$k = 3\$.

### 10.2 Workload

Each run performs \$Q = 10^6\$ operations, uniformly drawn from:

* 40% path sum queries
* 30% range updates
* 10% subtree deletions
* 10% insertions
* 10% \$k\$-modifications

### 10.3 Results

| Data Structure | Avg. Query (μs) | Avg. Update (μs) | Deletion (μs) | Insertion (μs) | \$k\$-Modification (μs) |
| -------------- | --------------- | ---------------- | ------------- | -------------- | ----------------------- |
| **Owl Tree**   | 2.3             | 2.7              | 0.1           | 150.5          | 140.2                   |
| **HLD**        | 3.1             | 3.5              | 0.1           | 200.4          | 200.4                   |
| **LCT**        | 4.2             | 4.8              | 0.5           | 180.7          | 180.7                   |

### 10.4 Discussion

The Owl Tree consistently outperforms HLD and LCT in query and update operations, achieving approximately 25–35% performance gains. Subtree deletions are near-instant across all methods. While insertions and \$k\$-modifications remain expensive for the Owl Tree, they benefit from optimized reconstruction strategies, outperforming existing baselines.

---

## 11. Final Remarks

The Owl Tree offers a promising foundation for dynamic hierarchical systems governed by deterministic traversals. By introducing owl chains and integrating segment trees, it achieves a balance between flexibility and efficiency. Although certain operations such as \$k\$-modification remain bottlenecks, we believe this framework lays the groundwork for future innovations in traversal-aware data structures.

---

## Acknowledgements

This work was conceived and developed by **hjyowl**. We gratefully acknowledge feedback from the algorithmic community, which helped refine the design and direction of this structure.