隨著數位資料的快速增長，搜尋演算法的效率與可擴展性變得愈發重要。傳統方法經常難以在速度與資源效率之間取得平衡，尤其是在包含不同頻率元素的大型且複雜的數據集上。本文提出了一種新的方法，通過結合隨機化與上線學習樹堆結構以及乘法權重更新（MWU）演算法來優化文字搜尋效能。其主要目標是在不論單詞熱門程度的情況下，提升搜尋速度並減少資源消耗。我們的方法利用乘法權重更新演算法在學習樹堆和隨機樹堆之間做動態選擇，確保搜尋效率。實驗結果顯示，與其他方法相比，我們的方法顯著縮短了搜尋時間。此方法能有效適應單詞頻率的變化，對於大規模數據集具有較強的適應性。總之，結合隨機化與學習樹堆以及乘法權重更新演算法，為各種應用中的搜尋效能優化提供了一個具有潛力的解決方案。

The rapid growth of digital data necessitates efficient and scalable search algorithms. Traditional methods often struggle with balancing speed and resource efficiency, particularly in large and complex datasets with varying element frequencies. This paper proposes a new approach to optimize word search performance by integrating randomized and learned treaps with the Multiplicative Weights Update (MWU) algorithm. The primary objective is to enhance the speed of word searches while minimizing resource usage, regardless of the words' popularity or obscurity. Our method leverages the MWU algorithm to dynamically select between the learned treap and the randomized treap, ensuring efficient searches. Experimental results demonstrate that our approach significantly reduces search times compared to other methods. The proposed method effectively adapts to the frequency of words, making it robust for diverse and large-scale datasets. In conclusion, integrating randomized and learned treaps with the MWU algorithm offers a promising solution for optimizing search performance in various applications, paving the way for future enhancements in search algorithms.

Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Preliminaries 4
2.1 Treap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Multiplicative Weights Update (MWU) Algorithm . . . . . . . . . . . 4
2.3 Red-Black Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 AVL Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 B-Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.6 Splay Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.7 Learning Augmented . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.7.1 Learning-Augmented Binary Search Tree . . . . . . . . . . . . 8
2.7.2 Learning-Augmented k-means Clustering . . . . . . . . . . . . 8
2.7.3 Learning-Augmented Data Stream Algorithms . . . . . . . . . 8
3 Methodology 9
3.1 Na¨ıve Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 First Attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 Second Attempt . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.1 The Entire Process . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.2 Search, Insert and Update Process . . . . . . . . . . . . . . . 11
V
3.3 Treaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.1 Random Treap . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.2 Learned Treap . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Multiplicative Weights Update (MWU) Algorithm . . . . . . . . . . . 12
4 Experiment 17
4.1 Dataset and Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.1 Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Conclusion 23
References 24
VI
List of Figures
3.1 The flowchart of the entire algorithm . . . . . . . . . . . . . . . . . . 15
3.2 The flowchart of the search, insert, and update process . . . . . . . . 16
4.1 Data preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Mix: average comparison times with randomly chosen words . . . . . 20
4.3 High: average comparison times with high-frequency words . . . . . . 20
4.4 Low: average comparison times with low-frequency words . . . . . . . 21
4.5 Adversarial: average comparison times with adversarial case . . . . . 21

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