## 2 Related Work  
Most of the early researches on spatial databases assumed the stationary objects and focused on developing efficient spatial access methods (e.g., the _R-tree_[22] and its variants [23, 24]) and evaluation of _snapshot queries_, which retrieves the results of queries only once at a specific time point. Later on, the focus has been extended to indexing moving objects. Assuming that the trajectories of moving objects are known a priori or predictable, [PERSON] _et al._[25] proposed the _Time-Parameterized R-tree_ (_TPR-tree_) for indexing moving objects, where the location of each moving object is transformed into a linear function of time. [PERSON] _et al._[26] proposed the improved version of the TPR-tree, called the _TPR\({}^{*}\)-tree_, which uses the exactly same data structure as the TPR-tree but applies new insertion and deletion algorithms. Some index structures were also presented such as the _STRIPES_[27] and the _B\({}^{x}\)-tree_[28], a variant of the _B\({}^{+}\)-tree_, to improve the performance of the TPR-tree family. However, the known-trajectory assumption does not hold for most real-life application scenarios (e.g., the velocity and direction of a typical customer on the road are frequently changed), which leads those index structures to become prohibitively expensive to update. To deal with a large number of moving objects that move arbitrarily, [PERSON] _et al._[29] proposed a generalized bottom-up update strategy for the R-tree, while [PERSON] _et al._[30, 31] proposed two buffer-based index structures, called the R-tree with semibulk loading (R\({}^{\text{sb}}\)-tree) and the R-tree with Update Buffering (R\({}^{\text{ub}}\)-tree), both of which utilize an in-memory buffer structure. Motivated by LBSs, another research direction has recently focused on continuous query monitoring over moving objects. Many methods for continuous range query monitoring have been proposed, which can be broadly classified into two categories depending on whether queries also move or not. The first category focuses on stationary or quasi-stationary queries over moving objects [1, 2, 6, 12, 17, 19, 20], and the second category deals with moving queries over moving objects [3, 5, 6, 14, 15, 16]. Because our work belongs to the first category, we elaborate on the review of the representative methods in the first category and briefly review the methods in the second category. Indexing queries, instead of indexing frequently moving objects with arbitrary velocities and directions, has been considered to be an attractive strategy, which reduces the update cost of index structures because continuous monitoring queries remain active for a long period of time and are stationary (or quasi-stationary). [PERSON] _et al_. [17] suggested to use the R-tree to index queries, while [PERSON] _et al_. [12] used the in-memory grid index. [PERSON] _et al_. [20] proposed a new query indexing method, namely _containment encoded square_ (_CES_) based grid indexing. All of these methods assumed that moving objects blindly report their location-updates to the server whenever they move. The server, meanwhile, continually (i) receives the location-update stream; (ii) determines the queries that are affected by the movements of the objects; and (iii) updates their results if necessary. However, constant location-updates generated by a huge number of moving objects may incur significant communication bottleneck and greatly increase the overhead for determining the affected queries and keeping their results up to date at the server. In addition, because the transmission of a location-update message over a wireless connection takes a substantial amount of energy, the handheld device carried by each moving object exhausts its battery life quickly. To help each moving object reduce the number of sending location-updates, the safe region method (SR) was proposed in [6, 17]. [PERSON] _et al_. [2] and [PERSON] _et al_.