Rapid advances in accuracy and miniaturization of location-aware devices, such as GPS-enabled smartphones, and increased use of social networks services (e.g., check-in updates) have enabled a generation of large volumes of spatial data(e.g., Manyika et al., 2011). In addition to the (location, time) values, that data is often associated with other contextual attributes. Numerous methods for effective processing of various queries of interest in such settings—e.g., range, (k) nearest neighbor, reverse nearest-neighbor, skyline, etc.—have been proposed in the literature(cf., Zhang et al., 2003; Zhou et al., 2011; Issa and Damiani, 2016).

One particular spatial query that has received recent attention is the, so called, Maximizing Range-Sum (MaxRS) (Choi et al., 2014), which can be specified as follows: given a set of weighted spatial-point objects O and a rectangle r with fixed dimensions (i.e., a × b), MaxRS retrieves a location of r that maximizes the sum of the weights of the objects in its interior. Due to diverse applications of interest, variants of MaxRS (e.g., Phan et al., 2014; Amagata and Hara, 2016; Feng et al., 2016; Hussain et al., 2017a,b; Wongse-ammat et al., 2017; Liu et al., 2019, etc.) have been recently addressed by the spatial database and sensor network communities.

What motivates this work is the observation that in many practical scenarios, the members of the given set O of objects can be of different types, e.g., if O is a set of restaurants, then a given o_i ∈ O can belong to a different class from among fast-food, Asian, French, etc. Similarly, a vehicle can be a car, a truck, a motorcycle, and so on. In the settings where data can be classified in different (sub)categories, there might be class-based participation constraints when querying for the optimum region—i.e., a desired/minimum number of objects from particular classes inside r. However, due to updates in spatial databases—i.e., objects appearing and disappearing at different times—one needs to accommodate such dynamics too. Following two examples illustrate the problem:

Example 1: Consider a campaign scenario where a mobile headquarters has limited amount of staff and needs to be positioned for a period of time in a particular area. The US Census Bureau has multiple surveys on geographic distributions of income categories¹ and, for effective outreach purposes, the campaign managers would like to ensure that within the limited reachability from the headquarters, the staff has covered a maximum amount of voters—with the constraint that a minimum amount of representative from different categories are included. This would correspond to the following query:

Q1: “What should be the position of the headquarters at time t so that at least κ_i residents from each income Category_i can be reached, while maximizing the number of voters reached, during that campaign date.”

Example 2: Consider the scenario of X's Loon Project², where there are different types of users—premium (class A), regular (class B), and free (class C), and users can disconnect or reconnect anytime. In this context, consider the following query:

Q2: “What should be the position of an Internet-providing balloon at time t to ensure that there are at least Θ_i users from each Class_i inside the balloon-coverage and the number of users in its coverage is maximized?”.

It is not hard to adapt Q1 and Q2 to many other applications settings:—environmental tracking (e.g., optimizing a range-bounded continuous monitoring of different herds of animals with both highest density and diversity inside the region);—traffic monitoring (e.g., detecting ranges with densest trucks);—video-games (e.g., determining a position of maximal coverage in dynamic scenarios involving change of locations of players and different constraints).

We call such queries Conditional Maximizing Range-Sum (C-MaxRS) queries, a variant of the traditional MaxRS problem. For dynamic settings, where the objects can be inserted and/or deleted, we have Conditional Maximizing Range-Sum with Data Updates (C-MaxRS-DU) query.

An illustration for C-MaxRS query in a setting of 7 users grouped into 3 classes (i.e., A, B, and C), and with a query rectangle size a × b (i.e., height a and width b) is shown in Figure 1. Assume that the participation constraint is that the positioning of r must be such that at least 1 user is included from each of the classes A, B, and C, respectively. There are two rectangles r₁ and r₂, with dimension a × b, that are candidates for the solution. However, upon closer inspection it turns out that although r₂ contains most users (corresponding to the traditional MaxRS solution), it is r₁ that is the sought-for solution for the C-MaxRS problem. Namely, r₂ does not satisfy the participation constraints (see Figure 1i).

Now, suppose that at time t₂, user o₆ disconnects and a new user o₈ joins the system. Then the C-MaxRS solution will need to be changed to r₂ from r₁ (see Figure 1ii).

Our key idea for efficient C-MaxRS processing is to partition the space and apply effective pruning rules for each partition to quickly update the result(s). The basic processing scheme follows the technique of spatial subdivision from Feng et al. (2016), dividing the space into a certain number of slices, whose local maximum points construct the candidate solution point set. In each slice, the subspace was divided into slabs which helps in reducing the solution space. To handle dynamic data stream scenarios, i.e., appearances and disappearances of objects, we propose two algorithms, C-MaxRS⁺ and C-MaxRS⁻, respectively, which works as a backbone for solving the constrained maximum range sum queries in the dynamic insertions/deletions settings (C-MaxRS-DU). Our novelty is in incorporating heuristics to reduce redundant calculations for the newly appeared or disappeared points, relying on two trees: a quadtree and a balanced binary search tree. Experiments over a wide range of parameters show that our approach outperforms the baseline algorithm by a factor of three to four, for both Gaussian and Uniform distribution of datasets.

The above idea for the C-MaxRS-DU algorithm takes an event-based approach, in the sense that C-MaxRS is evaluated (maintained) every time an event occurs, i.e., new point appears (e⁺) or an old point disappears (e⁻). This approach works efficiently when events are distributed fairly uniformly in the temporal domain and occur at different time instants that are enough apart for reevaluation to complete. However, the recent technological advancements and the availability of hand-held devices have enabled a large increase (or decrease) of the number of active/mobile users in multitude of location-aware applications in relatively short time-spans. In the context of Examples 1 and 2, this would correspond to the following scenarios:

Example 1: If the area involves businesses, then one would want to exploit the fact that many individuals may: (a) come (or leave) their place of work in the morning (or evening); (b) enter (or leave) restaurants during lunch-time; etc.

Example 2: In the settings of X's Loon Project, there can be multiple users disconnecting from the service simultaneously (within a short time span), or new users may request connections.

There are many other scenarios from different domains—e.g., Facebook has on average 2 billion daily active users—approximately 23,000 users per second. These Facebook users can be divided into many groups (classes), and C-MaxRS can be used to retrieve the most interesting regions (with respect to particular requirements) among the active daily users. In this scenario, a large number of users can become online (e⁺), or go offline (e⁻) at almost-same time instant. Similarly, flocks of different kinds of animals may be approaching the water/food source; the containment of the diseases across the population and regions may vary; etc.

To address the efficiency of processing in such settings, we propose a novel technique, namely C-MaxRS-Bursty. The key idea of our approach is as follows: instead of processing every single update, we assume that the update streams are gathered for a period of time. Then, we create a modified slice-based index for the entire batch of the new events, and then snap the new data over the existing slice structure in a single pass. Finally, we perform the pruning conditions for each slice only once in an aggregated manner. Experimental results show that C-MaxRS-Bursty outperforms our one-at-a-time approach, C-MaxRS-DU, by a speed-up factor of 5–10.

The main contributions of this work can be summarized as follows:

A preliminary version of this paper has appeared in Mostafiz et al. (2017), where we focused on non-weighted version of the C-MaxRS problem, i.e., we only count the number of objects inside the query window. We proposed two algorithms, C-MaxRS⁺ and C-MaxRS⁻ to efficiently solve C-MaxRS for data updates appearing and disappearing one at a time. The current article provides the following modifications and extensions to Mostafiz et al. (2017): (1) we provide the modified version of our algorithms from Mostafiz et al. (2017) to explicitly incorporate weighted version of the C-MaxRS problem, where each object and/or class can have different weights denoting its importance in the MaxRS computation. As it turns out (and demonstrated in the corresponding experiments) the weighted variant enables an increased pruning power; (2) we extend the work to consider novel settings of bulk updates handling of objects' appearance and disappearance and propose techniques for efficient computation of the C-MaxRS in those settings; (3) we conducted an extensive set of additional experiments to evaluate the benefits of our approaches.

In the rest of this paper, section 2 positions the work with respect to the existing literature, and section 3 formalizes the C-MaxRS problem. Section 4 describes the necessary properties of the conditional count functions and lays out the basic solution. Section 5 presents the details of our pruning strategies, along with the data structures and algorithms for incorporating dynamic data, while section 6 presents an extension of the C-MaxRS problem to include weights of objects (or, classes). Section 7 discusses the challenges of processing bursty inputs, and offers additional data structures and algorithms to deal with them. Section 8 presents the quantitative experimental analysis and Section 9 summarizes and outlines directions for future work.

The Range Aggregation and Maximum Range Sum (MaxRS) queries, and their variants have been extensively studied in the recent years (e.g., Lazaridis and Mehrotra, 2001; Tao and Papadias, 2004; Cho and Chung, 2007; Sheng and Tao, 2011; Choi et al., 2014). A Range Aggregation Query, returning the aggregate result from a set of points, was solved for both 1-dimensional space—i.e., calculating result from set of values in given interval by Tao et al. (2014) and for 2 dimensional point space, i.e., calculating result from a given rectangle with fixed location by Papadias et al. (2001). To calculate the aggregate result, an Aggregate Index, storing the summarized result for specific region referenced by that index is used in Cho and Chung (2007). Different data structures are introduced to store the aggregate index—e.g., Lazaridis and Mehrotra (2001) proposed Multi-Resolution Aggregate tree (MRA-tree) to reduce the complexity. Although closely related, the MaxRS problem itself differs from these range aggregation queries.

The MaxRS problem was first addressed by researchers in the computational geometry community—e.g., Imai and Asano (1983) used a technique that finds connected components and a maximum clique of an intersection graph of rectangles in the plane. A solution based on plane sweep strategy was presented in Nandy and Bhattacharya (1995), where the input point-objects were “dualized” into rectangles (centered at the points and with dimensions equivalent to the query rectangle r). Then an interval tree was used to record the regions (a.k.a. windows) with highest number of intersecting (dual) rectangles along the sweep—denoting the possible locations for placing the (center of the) query rectangle, yielding O(nlogn) time complexity (n = number of points). However these solutions are not scalable, and Choi et al. (2014) proposed scalable extensions suited for LBS-applications—e.g., retrieve best location for a new franchise store with a specified delivery range. Subsequently, different variants of the MaxRS problem have been investigated:—constraining to underlying road networks (Phan et al., 2014; Zhou and Wang, 2016);—processing MaxRS queries in wireless sensor networks (Hussain et al., 2015; Wongse-ammat et al., 2017);—considering rotating MaxRS problem (Chen et al., 2015), where rectangles do not need to be axes parallel, i.e., allowing much more flexibility. A rather complementary work, tackling the problem of approximate solution to the MaxRS query was presented in Tao et al. (2013), using randomized sampling to bound the error with higher probability, with increasing number of objects in question. A more recent work, Liu et al. (2019) has proposed a novel solution PMaxRS to deal with the inherent location uncertainty of objects, and used smart candidate generation process (pruning) and sampling-based approximation algorithm (refinement) to efficiently solve the problem.

Monitoring MaxRS for dynamic settings, where objects can be inserted and/or deleted was first addressed in Amagata and Hara (2016). To efficiently detect the new locations for placing the query rectangle, Amagata and Hara (2016) exploited the aggregate graph aG2 in a grid index and devised a branch-and-bound algorithm (cf. Narendra and Fukunaga, 1977) over that aG2 graph for efficient approximation. We note that our work is complementary to Amagata and Hara (2016), in the sense that we addressed the settings of having different classes of objects and participation constraints based on them—whereas Amagata and Hara (2016) solves the basic MaxRS problem. Moreover, Amagata and Hara (2016) considered a sliding-window based model in the problem settings (i.e., if m new objects appear, then m old objects disappear in a time-window T), which is completely different to our event-based model. Additionally, we used contrasting approaches (and different data structures) in this work—dividing the 2D space into slices and slabs.

An interesting variant of MaxRS is addressed in Feng et al. (2016)—the, so called, Best Region Search problem, which generalizes the MaxRS problem in the sense that the goal of placing the query rectangle is to maximize a broader class of aggregate functions³. Our work adapts the concepts from Feng et al. (2016) (slices and pruning)—however, we tackle a different context: class-based participation constraints and dynamic/streaming data updates and, toward that, we also incorporated additional data structures (see section 5). As a summary, our methodology (as well as the actual implementation) is based on the idea of event driven approach for monitoring appearing and disappearing cases of objects, and we included a self-balancing binary tree (i.e., AVL-tree) to reduce the processing time that is needed for computing the MaxRS as per the event queue needs.

The issue of real-time query processing and indexing over spatio-temporal streaming data have been addressed extensively in prior literature, e.g., Hart et al., 2005; Mokbel et al., 2005; Dallachiesa et al., 2015, etc. For real-time computation, it is necessary to restrict the set of inspected data points at any time using techniques such as punctuation (embedded annotations), synopses (data summaries), windows (e.g., sliding windows—only items received in past t minutes), etc. In Mokbel et al. (2005), the authors implemented a continuous query processor designed specifically for highly dynamic environments. The proposed system utilized the idea of predicate-based sliding windows, and employed an incremental evaluation paradigm by continuously updating the query answer over a window. Dallachiesa et al. (2015) proposed both exact and approximate algorithms to manage count-based uncertain sliding windows for uncertain data streams (e.g., tuples can have both value and existential uncertainty). In contrast to these traditional window-based settings, we process C-MaxRS query in an event-based manner using all the data points received so far. This is necessary to maintain accurate answers for C-MaxRS over the whole dataset, i.e., trading off real-time processing power for accuracy.

On the other hand, both tree-based (cf. Hart et al., 2005) and grid-based (cf. Amini et al., 2011) indexing schemes have been proposed previously to deal with traditional streaming data. Dynamic Cascade Tree (DCT) is used in Hart et al. (2005) to index spatio-temporal query regions, ensuring optimized query processing for Remotely- Sensed Imagery (RSI) streaming data. Additionally, researchers such as Amini et al. (2011) have devised many hybrid clustering algorithms for data streams, using both density-based methods and grid-based indexing. In these density-based clustering algorithms, each point in a data-stream maps to a grid and grids are subsequently clustered based on their density. In our approach, we used slice-based (a specialized version of grid) indexing schemes to compute the range and class constrained optimal density clustering of data points (i.e., C-MaxRS).

Finally, as mentioned in section 1, a preliminary version of this work has been presented in Mostafiz et al. (2017). However, we note that the techniques for processing continuous monitoring queries over data streams (i.e., dynamic settings) must be adaptive, as data updates are often bursty and input characteristics may vary over time. Many previous works have demonstrated the tendency of bursty streams in various applications, and proposed general solutions such as Kleinberg (2003), Babcock et al. (2004), and Cervino et al. (2012), etc. For example, Babcock et al. (2004) utilized “load shedding” technique for aggregation queries over data updates, i.e., gracefully degrading performance when load is unmanageable; while Cervino et al. (2012) offered distributed stream processing systems to handle unpredictable changes in update rates. In this work, we address specifically the “algorithmic” part of the problem, i.e., presenting an optimal processing technique for C-MaxRS during bursty inputs. We conclude this section with a note that our proposed technique is implementation-independent, and can be augmented by existing distributed and parallel schemes seamlessly (cf. section 7).

We now introduce the C-MaxRS problem and extend the definition for appearance of new objects, and disappearance of existing ones. Additionally, we discuss the concept of submodular monotone functions.

C-MaxRS & C-MaxRS-DU: Let us define a set of POIClass K = {k₁, k₂, …, k_m}, where each k_i ∈ K refers to a class (alternatively, tag and/or type) of the objects, a.k.a. points of interest (POI). In this setting, each object o_i ∈ O is represented as a (location, class) tuple at any time instant t. We denote a set X= {x₁, x₂, …, x_m} as MinConditionSet, where |X| =|K| and each x_i ∈ ℤ+ denotes the desired lower bound of the number of objects of class k_i in the interior of the query rectangle r—i.e., the optimal region must have at least x_i number of objects of class k_i. Let us assume l_i as the number of objects of type k_i in the interior of r centered at a point p. A utility function f(O):P(O)→ℕ0, mapping a subset of spatial objects to a non-negative integer is defined as below,

Additionally, we mark O_rp as the set of spatial objects in the interior of rectangle r centered at any point p. Formally, Conditional-MaxRS (C-MaxRS): Given a rectangular spatial field 𝔽, a set of objects of interest O (bounded by 𝔽), a query rectangle r (of size a × b), a set of POIClass K = {k₁, k₂, …, k_m} and a MinConditionSet X = {x₁, x₂, …, x_m}, the C-MaxRS query returns an optimal location (point) p^* for r such that:

where O_rp ⊆ O.

Note that, in the case that there is no placement p for which all the conditions of MinConditionSet is met, the query will return an empty answer—indicating to the user to either increase the size of R or decrease the lower bounds for some classes.

In a spatial data stream environment, old points of interest may disappear and new ones may appear at any time instant. We can deal with this in two-ways:

Although faster algorithms can be developed in time-based settings, the solutions provided would be inherently erroneous for time between t and t + δ. On the other hand, event-based processing ensures that a correct answer-set is maintained all the time. Thus, we deal with the streaming data in event-based manner, for which we denote e⁺ as the new point appearance and e⁻ as the old point disappearance event. We note that, most of the settings for basic C-MaxRS remains same, except that the set of objects O is altered at each event. We define the set of points of interest in this data stream for any event e_i over an object o_ei as:

Formally, Conditional-MaxRS for Data Stream/Updates (C-MaxRS-DU) definition is an extension of the above definition of C-MaxRS, for which we additionally have a sequence of events E={e₁, e₂, e₃, …} where each e_i denotes the appearance or disappearance of a point of interest.

Submodular Monotone Function: Feng et al. (2016) devised solutions to a variant of the MaxRS problem (best region search) where the utility function for the given POIs is a submodular monotone function—which is defined as: [Submodular Monotone Function] If Ω is a finite set, a submodular function is a set function f:P(Ω)→ℝ if ∀_{X, Y}⊂Ω, with X ⊆ Y and x ∈ Ω\Y we have (1) f(X∪{x}) − f(X) ≥ f(Y∪{x}) − f(Y) and (2) f(X) ≤ f(Y).

In the above definition, (1) represents the condition of submodularity, while (2) presents the condition of monotonicity of the function. In section 4, we will discuss these properties of our introduced utility function f(O):P(O)→ℕ0.

Discussion: Note that, for the sake of simplicity, initially we have considered only the counts of POIs when defining the utility function or conditions in X. In section 6, we show that they can be extended to incorporate different non-negative weights for objects with only minor modifications. Similarly, although in our provided examples, for brevity, we've only depicted one class per object, the techniques proposed in this work extends to the objects of multiple classes (or tags), e.g., objects can be considered as (location, classes) tuple.

In this section, we first convert the C-MaxRS problem to its dual variant and then discuss important properties of the conditional weight function f(.), showing how we can utilize them to devise an efficient solution to process C-MaxRS.

We now proceed with introducing novel techniques to deal with more realistic scenarios, i.e., data arriving in streams with the possibility of objects appearing and disappearing at different time instants. Using the approach of the basic C-MaxRS problem presented in previous section as a foundation, we augment the solution with compact data-structures and pruning strategies that enable effective handling of data streams environment.

In the discussions so far, we only considered the counting variant of the C-MaxRS problem, i.e., the weights of each participating object are all equal to 1. While we have noted the portability of the results, in this section, we explicitly show how the algorithms and pruning schemes proposed thus far should be modified to cater to the case when the objects can have different weights. Firstly, we appropriately revise the definitions of f, g, and C-MaxRS-DU to allow different weights, and show that it does not affect the monotonicity and non-submodularity of f and g. Subsequently, we outline the modifications for the pruning schemes for the weighted version. While there are no major changes incurred in the fundamental algorithmic aspects, we note that weights may have impact on the pruning effects, as illustrated in section 8.

In many spatial applications, the data streaming rate often varies wildly depending on various external factors—e.g., the time of the day, the need of the users, etc. A peculiar phenomenon in such cases is the, so called, bursty streaming updates—which is, the streaming rate becomes unusually high and a large number of objects appearing or disappearing in a very short interval. In such scenarios, instead of processing every single update, we assume that the update streams are gathered for a period of time. The C-MaxRS-DU algorithm is based on sequential processing of events, and thus, its efficiency is particularly sensitive to the bursty input scenario. In this section, we first briefly discuss the challenges of processing bulk of events via Algorithm 2 and 3, and argue that a different technique is necessary. Subsequently, we propose additional data-structures and a new algorithm, C-MaxRS-Bursty, to maintain C-MaxRS during bursty streaming updates scenarios. Finally, we briefly discuss how our proposed scheme can be utilized in a distributed manner, for the purpose of further improvements in scalability.

In this section, we evaluate the performance of our algorithms. Since there are no existing solutions, to evaluate our solutions to the C-MaxRS-DU problem, we extended the best known MaxRS solution to cater to C-MaxRS-DU (see section 5.2—i.e., processing the C-MaxRS at each event without any pruning) and used it as a baseline. For bursty streams, we compare the performance of C-MaxRS-Bursty and C-MaxRS-DU, i.e., C-MaxRS-DU becomes the baseline then.

Dataset: Due to user privacy concerns and data sharing restrictions, very few (if any) authentic large categorical streaming data (with accurate time information) is publicly available. Thus, we used synthetic datasets in our experiments to simulate spatial data streams. Data points are generated by using both Uniform and Gaussian distributions in a two-dimensional data space of size 1, 000 × 1, 000m = 1km². To simulate the behavior of spatial data streams from these static data points, we use exponential distribution with mean inter-arrival time of 10s and mean service time of 10s. Initially, we assume that 60% of all data points have already arrived in the system, and use this dataset for static part of evaluation. The remaining 40% of the data points arrive in the system by following exponential distribution as stated earlier. Any data point that is currently in the system, can depart after being served by the system. For experiments related to C-MaxRS-Bursty, we select γ number of events (either in Gaussian or uniform distribution) at any time instant to emulate bursty inputs.

Parameters: The list of parameters with their ranges, default values and symbols are shown in Table 1.

Settings: We have used Python 3.5 programming language to implement our algorithms. All the experiments were conducted in a PC equipped with intel core i5 6500 processor and 16 GB of RAM. We measure the average processing time of monitoring C-MaxRS in various settings. We also compute the performance of Static C-MaxRS computation. In the default settings, the processing time for Static C-MaxRS is 85.86 s. Note that, we exclude the processing time for static C-MaxRS computation in further analysis as this part is similar for both baseline and our approach.

In this paper, we have proposed a new variant of MaxRS query, namely Conditional Maximizing Range-Sum (C-MaxRS) query in spatial data streaming updates for both non-weighted and weighted objects. Initially, we simply adapted the traditional MaxRS settings to incorporate conditional constraints of different class of objects. However, to handle data updates (i.e., appearance and disappearance of objects) with class-awareness, we needed additional spatial data structures, quadtree and a variant of self-balancing binary tree (e.g., we used AVL-tree), which enabled our algorithm to efficiently compute the changes in the result for different partitions (or slices) of the dataspace. To further improve the overall time-efficiency, we developed two pruning rules: one to handle the appearance of an object and the other to handle disappearance of an object while updating C-MaxRS results. Additionally, to accommodate a different kind of applications settings where a bursty stream of data updates occur in a short time interval, we have proposed a novel technique, C-MaxRS-Bursty to efficiently compute the C-MaxRS results via bulk updates handling. We considered a large parameters space and conducted extensive set of experiments. In sequential spatial data stream scenario, our approach, C-MaxRS-DU yields three to four times improvements (on average) in terms of processing time, when compared to the baseline algorithm. We have also observed that in a bursty scenario, our approach C-MaxRS-Bursty outperforms our one-at-a-time approach, C-MaxRS-DU, by 5–10 times.

There are several immediate extensions to our work. Firstly, we would like to investigate the trade-offs arising when there is a constraint between the time-instant of a particular update and the update of the answer. This, in some sense, may require a new approach where the bulk update algorithms and data structures proposed in this work will need to be adapted to handle dynamic invocations (e.g., when the buffer of new data reaches certain capacity). Complementary to this, we plan to investigate the C-MaxRS in more traditional streaming settings—i.e., when there is a constraint on the memory and the arrival rate is explicitly taken in consideration. In such cases, relying on data sketches may be inevitable (similar to Cormode, 2017). Lastly, we are investigating the variations of C-MaxRS where different kinds of mobility may need to be incorporated—for both the users (cf. Hussain et al., 2017a) and the query rectangle (e.g., in the Loon Project settings), as well as the mutual dependencies of both.

The datasets and the code used in the experiments are publicly available at: https://users.cs.northwestern.edu/~mmh683/project-works/Conditional-MaxRS-Streams/.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.