# Discovering Moving Flock Patterns in Movement Data: A Reeb Graph-Based Approach
## 1 Introduction  
With the proliferation of location-based technologies, such as GNSS (global navigation satellite system), Bluetooth, RFID (radio frequency identification), WiFi, and video tracking, movement data have become increasing prevalent. Consequently, research on moving objects and movement data has grown rapidly in recent years [1, 2]. Common types of data analyzed in movement studies include traffic movement data [3, 4], animal movement data [5, 6], natural phenomenon movement data [7, 8], and sports movement data [9, 10, 11]. Movement patterns can reveal the underlying principles governing objects' movements and interactions, and analyzing these interactions in space and time can provide valuable insights into real-world issues, such as urban dynamics and human social networks [12]. This holds significant importance for practical applications, highlighting the strong demand for approaches to effectively extract movement patterns from movement data. Typical movement patterns include flock patterns [13, 14, 15], convov patterns [16, 17], leadership patterns [18, 19], moving clusters [20], and crews [21]. Note that certain similarities exist between the movement patterns mentioned above. Interested readers are advised to read the relevant references for a clearer distinction. Among these, flock patterns have attracted considerable attention, as they can play a crucial role across various application domains. For example, in sports, coaches can use flock patterns for tactical analysis, while in security, they can help identify suspicious criminal groups. Flock patterns can be broadly described as \"a group of spatially close objects that stay together for a certain period of time\". These patterns can be further categorized into moving flock patterns and stationary-flock patterns based on whether the objects remain in motion or stay stationary over time. Since moving flock patterns capture more information than stationary flock patterns due to the complex temporal evolution of movements, our focus is particularly on developing approaches to discover moving flock patterns. The pioneering work on flock pattern discovery stems from [22; 23], where the concept of \"flock\" was introduced for the first time (to the best of our knowledge). The researchers developed corresponding algorithms based on the REMO (RElative MOtion) concept to discover flock patterns. Subsequently, researchers have worked to refine the definition of flock patterns and develop algorithms to mine them from various perspectives. Typical works can be summarized as follows: [PERSON] et al. [13] extended the definition of flock by taking the minimum lasting time into account. Additionally, they introduced the concept of approximate flock to simplify the algorithmic process and developed corresponding algorithms for discovering approximate flock patterns using the skip-quadtree [24]. Further, [PERSON] and [PERSON] [25] proposed two additional types of flock patterns: fixed-flock pattern and varying-flock pattern. These types mainly differ in the composition of their members, with fixed-flock patterns having constant members and varying-flock patterns having changing members. They also developed algorithms to compute the longest-duration flock patterns from trajectory data. In addition, [PERSON] et al. [14] reported three other types of flock patterns: all maximum flock patterns, largest flock patterns, and longest flock patterns. They developed an efficient parallel GUP-based algorithm to discover these three types of flock patterns. [PERSON] et al. [26] focused on the distance parameter of flock patterns and introduced the concept of \"k-flocks with minimum diameter\". They also developed an algorithm for mining the top-k flock patterns. [PERSON] et al. [27] focused on the particular problem of flock detection in movement data and proposed a fast algorithm to quickly detect flock patterns in large spatiotemporal data. [PERSON] et al.