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1 Hong Kong Baptist University HKBU Institutional Repository HKBU Staff Publication 18 Towards why-not spatial keyword top-k ueries: A direction-aware approach Lei Chen Hong Kong Baptist University, psuanle@gail.co Yafei Li Zhengzhou University, yafeics@outlook.co Jianliang Xu Hong Kong Baptist University, xujl@hkbu.edu.hk Christian S. Jensen Aalborg University, csj@cs.aau.dk This docuent is the authors' final version of the published article. Link to published article: APA Citation Chen, L., Li, Y., Xu, J., & Jensen, C. (18). Towards why-not spatial keyword top-k ueries: A direction-aware approach. IEEE Transactions on Knowledge and Data Engineering, 3 (4), This Journal Article is brought to you for free and open access by HKBU Institutional Repository. It has been accepted for inclusion in HKBU Staff Publication by an authorized adinistrator of HKBU Institutional Repository. For ore inforation, please contact repository@hkbu.edu.hk.

2 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 1 Towards Why-Not Spatial Keyword Top-k Queries: A Direction-Aware Approach Lei Chen, Yafei Li, Jianliang Xu, Senior Meber, IEEE, and Christian S. Jensen, Fellow, IEEE Abstract With the continued proliferation of location-based services, a growing nuber of web-accessible data objects are geo-tagged and have text descriptions. An iportant uery over such web objects is the direction-aware spatial keyword uery that ais to retrieve the top-k objects that best atch uery paraeters in ters of spatial distance and textual siilarity in a given uery direction. In soe cases, it can be difficult for users to specify appropriate uery paraeters. After getting a uery result, users ay find soe desired objects are unexpectedly issing and ay therefore uestion the entire result. Enabling why-not uestions in this setting ay aid users to retrieve better results, thus iproving the overall utility of the uery functionality. This paper studies the direction-aware why-not spatial keyword top-k uery proble. We propose efficient uery refineent techniues to revive issing objects by inially odifying users direction-aware ueries. We prove that the best refined uery directions lie in a finite solution space for a special case and reduce the search for the optial refineent to a linear prograing proble for the general case. Extensive experiental studies deonstrate that the proposed techniues outperfor a baseline ethod by two orders of agnitude and are robust in a broad range of settings. Index Ters Why-not uestions, spatial keyword top-k ueries, uery refineent. 1 INTRODUCTION WEB objects are becoing increasingly content-rich. In particular, the continued proliferation of locationbased services (LBS) and the increasing nuber of web objects with both textual keywords and spatial location inforation cobine to give proinence to spatial keyword ueries [3], [7], [15]. Aong the, considering the directionaware search reuireents fro any LBS users, a directionaware spatial keyword top-k uery [1], [5], [6] takes a user location, a set of keywords, and a search direction as arguents and retrieves the k objects in the search direction that are ranked highest according to a ranking function that considers both spatial proxiity and textual siilarity. It is relevant to take into account the uery direction in a nuber of scenarios. For exaple, a user walking to a superarket ay want to find an ATM in his/her walking direction, or a user on a high-way ay want to find a gas station or restaurant in his/her general travel direction (i.e., the right front region in right-driving countries); in role-playing gaes with a first-person perspective, players ay want to search the battlefield inforation such as weapon stores and edical stations in the angle of view. However, there ay be cases where users are not fully aware of the appropriate direction to feed to a spatial keyword top-k uery; it ay be difficult for a user to specify the direction that best captures the intent of her uery. After a user issues a (direction-aware) spatial keyword top- L. Chen, Y. Li and J. Xu are with the Departent of Coputer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong. L. Chen is also with Huawei Noah s Ark Lab, Hong Kong, and Y. Li is also with the School of Inforation Engineering, Zhengzhou University, Zhengzhou, China. E-ail: {lchen, yafeili, xujl}@cop.hkbu.edu.hk. C. S. Jensen is with the Departent of Coputer Science, Aalborg University, Denark. E-ail: csj@cs.aau.dk. k uery and receives the result, the user ay find the uery result is not as expected and that soe desirable objects are unexpectedly issing fro the result. This ay lead the user to uestion the overall result and to wonder whether other unknown objects that are relevant to the uery are also issing. To enhance the utility of the uery functionality, it is relevant to provide explanations about desired but issing objects and to autoatically suggest a refined uery that includes the desired objects in its result. The otivation and significance of this functionality is illustrated by two exaples. Exaple 1. After a busy day of sightseeing, Bob wants to have dinner nearby before walking back to the hotel, which is on the downhill side. He issues a uery to find the top-3 nearby Sushi restaurants. Surprisingly, he finds that the result contains only restaurants that are on the uphill side; and a restaurant on the downhill way to the hotel that he visited yesterday is not in the result. Bob uestions the overall result. Are the returned restaurants really the best, or do better options exist? Should the restaurants be searched in the general direction of his way to the hotel? How can he add a search direction so that the issing restaurant and possibly other good options appear in the result? Exaple. In preparation for attending an overseas conference, Clair issues a uery to find the top-3 hotels that are near the conference location and are in the direction of the old town. She is surprised that the result contains only local hotels that are unknown to her and that a well-known international hotel in the vicinity is not in the result. Clair wonders whether the exclusion occurs because the search direction is not set properly and how the uery direction can be odified so that the expected hotel, as well as potentially other good hotels, appear in the result?

3 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING A coon pattern in the above scenarios is that the user wants to know why an expected object does not appear in a result. This type of functionality relates to the uery uality and is called why-not functionality [6]. Three typical solution odels exist: (1) anipulation identification [6], which identifies uery operators that prevent issing objects fro being included in a result; () database odification [], [3], which updates the original database so that the uery can revive issing objects; and (3) uery refineent [9], [11], [1], [36], which revises the original uery so that issing objects can enter the result. We adopt the uery refineent odel to answer why-not uestions on direction-aware spatial keyword top-k ueries. The previous works adjust the preferences between spatial proxiity and textual relevance [9] or suggest ore accurate uery keywords [11] to get the inclusion of issing objects in the uery result. In this paper, we address the proble fro a new perspective, i.e., odifying the uery direction as otivated by the above exaples. We ai to inially odify users initial ueries to reintroduce the expected but issing objects into the uery result. To this end, we consider the proble in both the special case, where the initial uery is a traditional spatial keyword top-k uery without a uery direction, and the general case, where a uery direction is specified initially. To achieve an efficient solution to this proble, we prove that the best refined uery direction lies in a finite solution space for the special case, and we reduce the search for the best refined ueries into solving linear prograing probles for the general case. Furtherore, we extend proposed algoriths to support why-not uestions with ultiple issing objects. The ain contributions of this paper are suarized as follows: We identify a novel direction-aware why-not spatial keyword top-k uery and forulate it as a uery refineent proble. To the best of our knowledge, we are the first to study this proble. We provide a detailed proble analysis and propose efficient uery refineent algoriths to answer direction-aware why-not uestions by reducing solution spaces for both the special case and the general case. We extend the proposed algoriths to support whynot uestions with ultiple issing objects. We perfor extensive experients on real-life datasets to evaluate the perforance of the proposed algoriths. The results indicate that the algoriths are efficient in a broad range of settings. In particular, the proposed solution is two orders of agnitude faster than a baseline ethod. The rest of the paper is organized as follows. Section reviews related work. Section 3 introduces preliinaries and defines the direction-aware why-not spatial keyword uery proble. Sections 4 and 5 present the proble analysis and solutions in the two different cases. We extend the algoriths to support ueries with ultiple issing objects in Section 6. The experiental studies are covered in Section 7. Finally, we conclude in Section 8. RELATED WORK To the best of our knowledge, no prior studies consider why-not uestions on spatial keyword top-k ueries fro the perspective of uery direction refineent. In the following, we survey studies on spatial keyword ueries and why-not ueries separately, and we relate the studies to the setting of this paper..1 Spatial Keyword Query Processing A spatial keyword uery retrieves the ost relevant spatial web objects with respect to both spatial proxiity and textual relevance. A nuber of indexing and uery processing techniues have been proposed for this uery. The IR-tree [15], [7], [37] is a widely used index structure that integrates R-trees and inverted files. To uickly prune the search space, it supports siultaneous estiation of spatial distance and textual siilarity during index access. The IR -tree [16] is another hybrid index that cobines R-trees with superiposed text signatures. This index is applicable when the keywords serve as a Boolean filter. Rocha-Junior and Nørvåg [34] study the spatial keyword uery on road networks. To rank the objects considering both network distance and textual siilarity, inverted files are eployed to index the docuents of the web objects lying on a segent of the road network. A coprehensive coparison of existing spatial keyword indexing techniues is available [7]. Different variants of the spatial keyword uery have been studied. Chen et al. [13] consider a uery that retrieves the web objects which contain the uery keywords and whose page footprints intersect with a uery footprint. Zhang et al. [4], [41] investigate an -closest keywords (CK) uery to retrieve the spatially closest objects that atch user-specified keywords. To efficiently evaluate this uery, the br*-tree and virtual br*-tree are proposed to augent each node with a bitap and a set of MBRs for the keywords. Cao et al. [4] introduce a uery that finds the top-k spatial web objects ranked according to both prestigebased relevance and location proxiity. Another study [5] proposes a uery that retrieves a group of nearby spatial web objects whose keywords cover the uery keywords and that have the lowest inter-object distances. Further, Fan et al. [18] study the spatial keyword siilarity search in regions of interest. Li et al. [8] explore a spatial approxiate string uery that is a range uery augented with a string siilarity predicate. Bouros et al. [] identify the pairs of objects fro a spatio-textual database that are both spatially close and textually siilar. Another study [38] integrates the social influence into traditional spatial keyword search to iprove answer uality. More recently, Lee et al. [31] study the processing and optiizations for ain eory spatial keyword ueries. Choudhury et al. [14] ai to find an optial location and a set of keywords that axiize the size of bichroatic reverse spatial textual k nearest neighbors. Shi et al. [35] study location-based keyword search on RDF data. Lin et al. [9] and Xie et al. [39] ai to find the uery keyword sets and the uery locations, respectively, to ake a target object in the result of a spatial keyword top-k uery. Direction-aware spatial keyword ueries have been investigated [5], [6], the ai being to find the spatially

4 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 3 closest objects in a given uery direction that cover all uery keywords. However, none of the above studies address the why-not spatial keyword uery proble.. Why-Not Query Processing To iprove the usability of database systes, the concept of explaining a null answer to a database uery was presented by Motro [3], [33], and the why-not proble was first introduced by Chapan and Jagadish [6]. Existing approaches towards answering why-not uestions can be classified into three ain categories. Chapan and Jagadish [6] use anipulation identification to identify uery operators that eliinate users desired but issing objects on Select-Project-Join (SPJ) ueries. Other studies [], [3] study the why-not uestions on SPJ ueries and SPJUA (SPJ + Union + Aggregation) ueries by adopting a database odification approach, which updates the original database so that the issing objects becoe part of uery results. Tran and Chan [36] propose to retrieve issing objects through uery refineent, which ais to propt users how to revise their uery paraeters to revive their expected objects in the uery results. He and Lo [1] eploy the uery refineent to answer why-not uestions on top-k preference ueries. They ai to ake issing objects enter the result by inially odifying the original uery, where a penalty function easuring the aount of odifications is adopted. More recently, the uery refineent odel has been applied to answer why-not uestions on different ueries and data settings, including social iage search [1], reverse skyline ueries [4], reverse top-k ueries [19], and etric probabilistic range ueries [8]. In previous work [9], [11], we study the why-not spatial keyword ueries fro different perspectives. In one study [9], we propose techniues that help users adjust the preferences between the spatial distance and textual siilarity; in another [11], we provide users with ore precise uery keywords. These two why-not functionalities have been integrated into a spatial keyword uery syste [1]. However, these studies do not consider uery direction refineent and are unable to suggest ore accurate uery directions that better capture users uery intent, which is the focus of this paper. 3 PRELIMINARIES AND PROBLEM FORMULATION In this section, we foralize the proble of direction-aware why-not spatial keyword top-k ueries. 3.1 Spatial Keyword Top-k Queries Let D denote a database of spatial objects. Each object o D is associated with a pair (loc, doc), where o.loc is the object s spatial location and o.doc is a set of keywords that describes the object. A spatial keyword top-k uery takes four paraeters (loc, doc, w, k). Here.loc is the uery location,.doc is a set of uery keywords,.k denotes the nuber of objects to retrieve, and. w = w s, w t, where w s, w t 1 and w s + w t = 1, denotes the user s preferences between spatial proxiity and textual relevance. The uery retrieves the topk objects fro D ranked according to a scoring function that aggregates the spatial distance and textual siilarity into an overall scoring value. For broad applicability, we adopt a widely used ranking function [15]: ST (o, ) = w s (1 SDist(o, )) + w t TSi(o, ), (1) where SDist(o, ) denotes the Euclidean distance between o.loc and.loc, and TSi(o, ) denotes the textual siilarity between o.doc and.doc. The spatial distance SDist(o, ) is noralized into the range [, 1] by dividing the axiu possible distance between two objects in D. The textual siilarity TSi(o, ) can be coputed using an inforation retrieval odel [17], such as the language odel, cosine siilarity, Jaccard siilarity, or BM5, and is also assued to be noralized into the range [, 1]. Without loss of generality, we adopt the language odel [9], [15] in this paper. The larger score coputed by En. 1 denotes the higher relevance an object to the uery. Given a uery, the rank of an object o is given in ters of En. (1) as follows: R(o, ) = {o D ST (o, ) > ST (o, )} + 1 () With this definition of rank, the spatial keyword top-k uery is defined as follows: Definition 1. Spatial Keyword Top-k Query. A spatial keyword top-k uery returns a set R of k objects fro D, where o R ( o D R (ST (o, ) ST (o, ))), or in ters of object ranking, o R ( o D R (R(o, ) R(o, ))). While the spatial keyword top-k uery has been extensively studied, direction-aware search, which is deanded in any LBS scenarios, has recently received ore attention [], [5], [6]. In a direction-aware spatial keyword topk uery, an object can be a result only if it is located in a certain direction of a uery location. A direction is defined in ters of rays eanating fro the uery location. Without loss of generality, we assue that the objects are apped to a Cartesian coordinate syste. We delineate a direction by the angles between two rays and the positive direction of the x-axis. The direction d in a uery is a range (α, β), denoting that the uery is interested only in objects in the direction (α, β). 1 That is, a uery s direction is an angular space. As such, a direction-aware spatial keyword top-k uery d is a 5-tuple (loc, doc, w, k, d). Definition. Direction-Aware Spatial Keyword Top-k Query. Let D d denote the objects in D that are located in the angular region d = (α, β), and let R(o,, d) denote the rank of an object o under a uery d. A direction-aware spatial keyword top-k uery d returns a set R of k objects fro D d, where o R ( o D d R (R(o,, d) R(o,, d))). In other words, instead of considering the whole database D, a direction-aware spatial keyword uery considers only the objects in a directional range as candidates 1. A direction (α, β) is the angular space passed through by rotation fro α to β counterclockwise. We use open or closed intervals to denote whether a boundary direction is included. For the sake of presentation, we convert all directions to the space where π < α π and α β α + π. We say a direction θ (α, β), if n Z, θ + nπ (α, β).

5 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 4 o 5 o 1 o 6 β α o 4 (a) π 3 o o 3 π 6 ST (o, ) o 1.7 o.3 o 3.58 o 4.65 o 5.43 o 6.83 (b) Fig. 1. Top k and Direction-Aware Top-k Spatial Keyword Query for the uery result. We note that the traditional spatial keyword top-k uery can be treated as a direction-aware uery with a direction of [, π). Exaple 3. Fig. 1 shows an exaple of the top-k and the direction-aware top-k spatial keyword ueries, where (a) shows the locations of the uery and objects, while (b) lists the ranking score of each object. Consider a top- uery. A traditional uery returns the objects with the highest scores aong all the objects, i.e., o 6 and o 1. However, if the uery has a direction, say, ( π 6, π 3 ), the uery considers only the objects in this direction and returns the top- objects aong the, i.e., o 1 and o Direction-Aware Why-Not Spatial Keyword Query When issuing a direction-aware spatial keyword uery = {loc, doc, w, k, d }, it ight be difficult for a user to specify a direction d that best captures the uery intent. Due to an iproper setting of the uery direction, after the user receives a uery result, the user ay find that one or ore desired objects are unexpectedly issing. These issing objects iply the user s actual direction reuireent. Then, the user ay pose a follow-up why-not uestion with a set M = { 1,,..., j } of desired but issing objects, asking why these expected objects are issing and seeking a refined uery = {loc, doc, w, k, d } that includes the issing objects in its result. As siply odifying the direction in the initial uery ay not be able to revive the issing objects, the enlargeent of k is also considered [9], [11], [19], [1], [4]. Many possible odifications of these two paraeters ay yield a ualified uery retrieving the issing objects. We prefer the one that odifies the initial uery inially. To foralize this, we adopt a penalty odel [11], [19], [1] that uantifies the odification as a weighted su of the changes of paraeters k and d. The penalty of a refined uery against the initial uery is defined as follows: Penalty(, ) = λ k d + (1 λ), (3) k ax d ax where λ [, 1] is a user preference on odifying k versus d. Here, k ax and d ax denote the axiu possible odifications of k and d, respectively. They are used to noralize k and d into the range [, 1]. Since their settings would vary in different cases, we leave their definitions to the coverage of the corresponding cases in Sections 4 and 5. We have k = ax(, k k ), since k can reain as k if a refined k is saller than k. As the uery direction d is an angular space between the start angle α and the end angle β, we easure the odification fro d = (α, β ) to d = (α, β ) in ters of how uch d is rotated and how uch the size of d is changed, i.e., r and s. Forally, d is defined as follows: d = γ r + (1 γ) s = γ α + β α + β +(1 γ) (β α ) (β α ) The rotation of the direction r is deterined by the difference between the angular bisectors, i.e., +β α and α+β. Finally, γ [, 1] is used to balance the changes to the rotation and the size of the direction. Based on the above, the direction-aware why-not spatial keyword uery is defined as follows: Definition 3. Direction-Aware Why-Not Spatial Keyword Top-k Query. Given a set D of spatial objects, a issing object set M, an original direction-aware spatial keyword uery = (loc, doc, w, k, d ), the direction-aware why-not spatial keyword top-k uery returns the refined uery = (loc, doc, w, k, d ), with the lowest penalty according to En. (3) and the result of which includes all objects in M. 3.3 Baseline Algorith We consider refining the direction d and the result cardinality k to achieve the inclusion of the issing objects. Only refined pairs (k, d ) that satisfy Lea 1 are candidates for the best refined uery. Lea 1. Given an initial uery and a set M of issing objects, a pair of a odified direction and a refined result cardinality (d, k ) can be a candidate for the best refined uery if and only if (i) i M (θ i d ), where θ i denotes i s angle; and (ii) k = R(M,, d ); or R(M,, d ) k k, where R(M,, d ) = ax i M R( i,, d ). Proof. The proof is straightforward and hence oitted. According to Lea 1, given a refined direction d, we can always set k = R(M,, d ) to achieve the iniu penalty. In other words, if we fix paraeter d, k can be set accordingly. This observation inspires a baseline solution as follows: (i) enuerate all possible refined directions; (ii) for each candidate direction, process a direction-aware spatial keyword top-k uery to deterine the ranks of the issing objects; (iii) copute the penalty of each candidate direction and return the one with the iniu penalty. Two key challenges exist in this baseline solution. First, the nuber of possible directions is generally infinite, aking enueration ipossible. One way to overcoe this issue is to saple part of the candidate refined directions [19], [1]. Nevertheless, it is hard to guarantee the solution uality, and the baseline ay fail to find the optial solution. Second, the baseline needs to invoke a spatialkeyword uery for each enuerated direction, where high coputation and I/O costs are incurred. As such, the baseline algorith ight be inefficient and inapplicable to the general case. In the following sections, we develop ore efficient solutions based on a careful proble analysis. We ai to invoke the spatial keyword uery only once during (4)

6 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 5 the why-not uery processing. We consider a single issing object in Sections 4 and 5, and we extend the algoriths to support ultiple issing objects in Section 6. 4 ANSWERING WHY-NOT QUESTIONS: A SPE- CIAL CASE In this section, we assue a single issing object and study the direction-aware why-not proble for the special case, where the initial uery is a traditional uery with no uery direction. This case ay occur when users do not at first realize their uery direction reuireents; or they do not indicate a uery direction, possibly because they would like the syste to add the direction as a direction is ore difficult to specify than it is to point out soe expected result object(s). 4.1 Case Analysis Recall that the traditional spatial keyword top-k uery can be treated as a uery with a direction of [, π). Actually, it can be represented by a direction-aware uery with a direction in {d d = [α, α + π), π < α π}. In other words, the bisector of the direction can be any ray around the uery location. Therefore, for a refined direction (α, β ), we can always find an α that akes α +β = α+β. Thus, the odification in rotating the initial direction can be treated as, and d can be rewritten as follows: d = π (β α ) (5) Thus, the axiu possible odification of the direction is d ax = π, which is obtained when α = β. Moreover, k ax can be estiated as R(,, d ) k, where R(,, d ) denotes the rank of the issing object under the initial uery, as a very naive ethod to revive the issing object is to increase k until is included in the result without adjusting the direction. As such, the penalty function becoes: Penalty(, k ) = λ + (1 λ) π (β α ) R(,, d ) k π (6) To achieve the iniu penalty, for a given k, the largest direction that could rank the issing object within top-k is preferred. 4. Ranking Updates and Direction Modifications To answer the direction-aware why-not uery, we ake two observations. First, after the initial uery is issued, the ranking score of each object is a constant. Second, the uery direction works as a filter, where only the objects in the uery direction are candidates for the result. Therefore, the rank of the issing object in the refined uery is deterined by the nuber of objects in the refined uery direction that ranks better. As entioned, the uery direction d of the spatial keyword uery has a start angle α and an end angle β. To deterine a refined direction, we need to identify its start and end angles. Given a start angle α of a refined direction, the following theore holds. Theore 1. Consider an initial uery, a issing object, and a refined start angle α. Let θ denote the angle of β o 5 β 1 o4 o 3 α o o 1 o 6 o 6 (a) β o 5 o 4 o 3 (b) Fig.. A candidate end (start) direction for a given α (β ) α 1 o o 1 w.r.t.. If a direction (α, β ) is a candidate for the best refined uery, it holds that: (i) θ (α, β ); and (ii) there exists an object o with angle β such that ST (o, ) > ST (, ). Proof. The first condition is obviously necessary. It ensures that the issing object is in the uery direction. We prove the second condition by contradiction. Assue that β is a candidate end angle and that no object with angle β scores higher than the issing object under the initial uery. Then there are only two cases: there exists objects that rank higher than in (β, α + π]; or no object ranks higher than in (β, α + π]. In the forer case, let o be the first object that ranks higher than in (β, α + π], and let θ o denote its angle w.r.t.. Let β be any angle in (β, θ o ). Since o (β, θ o ) (ST (o, ) ST (, )), the rank of the issing object would be the sae in the directions (α, β ) and (α, β ). Thus, k is identical for the end angles β and β. However, as (α, β ) (α, β ), d β > d β. According to En. (6), for a given k, a larger direction is preferred. The penalty for the end angle β exceeds that of β. Thus, β cannot be a candidate end angle for start angle α, which contradicts our assuption. Siilarly, in the latter case, any angle in (β, α + π] would have a saller penalty than β so that β cannot be a candidate end angle. Thus, Theore 1 holds. Siilarly, for a given refined end angle β, the refined start angle α should be set to β π, or should be the angle of an object that ranks higher than the issing object under the initial uery. Exaple 4. In Fig., the objects that have ranking scores higher than the issing object under the initial uery are arked as black points and the others are arked in grey. In Fig. (a), for start angle α, when increasing the end angle β fro β 1 to β, the rank of in the direction (α, β ) reains unchanged. But as the size of the direction increases, according to En. (6), the penalty for the refined direction (α, β ) keeps decreasing. Conseuently, no end angle between β 1 and β can yield the best refined direction. The case is siilar for the candidate start angles when a refined end angle β is given (See Fig. (b)). These observations yield the following proposition for the candidate refined directions: Proposition 1. A refined direction (α, β ) can result in the best refined uery if: (i) θ (α, β ); and (ii) both the start angle α and the end angle β have an object that α

7 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 6 θ[5] θ[4] θ[6] θ[1] Candidate directions for k = 3: θ[] θ[3] 1: (θ[1], θ[4]) : (θ[], θ[5]) 3: (θ[3], θ[6]) of θ[i + 1] and θ[(r(,, d ) 1) (r i 1)], respectively, to obtain its sallest penalty (Lines 1 13). The end angle β is further coputed to satisfy the constraint that α β α +π (Line 13). We then copute their penalties and exaine whether they are better than the currently seen best refined uery (Lines 14 16). Fig. 3. Candidate directions for a given refined k ranks higher than the issing object under the initial uery. Proposition 1 liits the candidate directions to a finite set and thus enables enueration. We also note that the rank of the issing object in a direction (α, β ) is deterined by the nuber of objects that have higher ranking scores than, hereafter called s doinators, in the direction. These doinator objects are deterined when the initial uery is issued. Thus, instead of processing a spatial keyword uery to copute the rank of the issing object for each candidate direction, we can first identify s doinators and then deterine s rank by counting how any of the are in each candidate direction. 4.3 Answering Why-Not Based on the above discussions, we present the proposed algorith for answering direction-aware why-not uestions in the special case. The pseudo-code is given in Algorith 1. We first copute the rank of the issing object under the initial uery, i.e., R(,, d ), and we record s doinators (Line 1). This can be done by slightly odifying the underlying spatial keyword top-k algorith (e.g., [37]) by changing the stop condition fro retrieving k objects to retrieving the issing object. We then invoke the function CalDirection to calculate the angles of the issing object and its doinators w.r.t. the initial uery (Lines 4). The function CalDirection confines the angles of the objects within the range ( π, π]. For ease of presentation, we assue no s doinators locate at the uery location. The algorith can be adapted easily to support this case by treating all such doinators to be in any direction. Next, s doinators are sorted in clockwise order according to their angles w.r.t. the issing object s angle, i.e., in ascending order of (θ θ[i]+π)%π (Line 5). Afterwards, we initialize the currently seen best refined uery with the basic one that siply odifies k to R(,, d ) (Line 6). We then enuerate each possible refined k in increasing order (Line 7). At k, if erely odifying k to k results in a penalty larger than the iniu obtained so far, the process is terinated (Lines 8 1). The range of a possibly refined k is fro k to R(,, d ) 1, since for k < k, the candidate directions for k would be contained in that of k and hence cannot achieve a saller penalty. For each possibly refined k, we check the candidate directions that rank as a top-k object. These candidate directions are enuerated in ascending order of the nuber of s doinators located to the right of (Line 11). For an enuerated nuber i, we set the start angle α and the end angle β as the extree cases. We use % to denote the odular operation throughout the paper. Algorith 1 Answering Why-Not Questions: Special Case INPUT: Original uery = (loc, doc, w, k, d = [ π, π)), Missing object, Penalty option λ OUTPUT: Best refined uery = (loc, doc, w, k, d ) 1: copute R(,, d ) and record s doinators in set S : θ CalDirection(, ) 3: for each o i S 4: θ[i] CalDirection(, o i ) 5: sort θ[i] in clockwise order w.r.t. θ 6: d d, k R(,, d ), p c λ 7: for r k to R(,, d ) 1 do 8: k r k 9: k if λ R(,,d ) k p c then 1: return (loc, doc, w, k, d ) 11: for i to r 1 do 1: α θ[i + 1] 13: β α + ((θ[(r(,, d ) 1) (r i 1)] θ[i + 1] + π)%π) 14: copute the penalty p for the current candidate direction according to En. (6) 15: if p < p c then 16: k r, d (α, β ), p c p 17: return (loc, doc, w, k, d ) Function CalDirection(, o) INPUT: Original uery, object o OUTPUT: θ o // the direction of o w.r.t. 18: X o.x.x, Y o.y.y 19: if X = then : if Y < then return π/ 1: else return π/ : θ o Arctan(Y/X) 3: if X < and Y then 4: θ o θ o + π 5: if X < and Y < then 6: θ o θ o π 7: return θ o Exaple 5. Take Fig. 3 as an exaple. Assue that the initial uery is a traditional spatial keyword top-3 uery. Six objects doinate w.r.t., i.e., R(,, d ) = 7. This akes issing fro the top-3 result. We first identify these s doinators and copute their angles. To easily locate the candidate directions, we sort s doinators clockwise according to their angles w.r.t. s angle. The very basic refined uery is a top-7 uery with no uery direction. We enuerate each possible refined k, which is fro 3 to 6. For each k, we find the candidate directions that rank as topk according to Proposition 1 and copute their penalties to

8 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 7 check whether they are better than the known refined ueries. For exaple, the candidate directions that give a rank of 3 are: (θ[1], θ[4]), (θ[], θ[5]) and (θ[3], θ[6]). We next analyze the tie coplexity of the proposed algorith. Theore. The tie coplexity of Algorith 1 is O(SKT (r)+r ), where r = R(,, d ) denotes the rank of the issing object under the initial uery and SKT (r) denotes the tie coplexity of a spatial keyword top-k uery in retrieving the top-r objects. Proof. The algorith has two phases: (i) copute the initial rank of the issing object; (ii) find the best refined uery. The first phase takes advantage of an existing spatial keyword top-k uerying algorith and has tie coplexity O(SKT (r)). For the second phase, we enuerate each possible refined k fro k to r 1. For each k, we need to verify k candidate directions. Thus, in the worst case, the tie coplexity of this phase is O(r ). Optiizations: We reark that optiizations can be applied to both phases in the processing of a why-not uery. To speed up the coputation of the initial rank of, we buffer the result and internal data structures of the initial uery and proceed to process it if a follow-up why-not uestion is posed. For the second phase, we enuerate each possible refined k in increasing order so that we can stop the enueration early if the penalty of the next k in odifying k exceeds that of the currently seen best refined uery. 5 ANSWERING WHY-NOT QUESTIONS: GENERAL CASE We proceed to study how to answer why-not uestions on general direction-aware spatial keyword ueries. Here, initial ueries are specified with search direction reuireents. In response to a follow-up why-not uestion, the syste provides the user with a ore precise direction so that the refined uery can retrieve ore useful results. 5.1 Case Analysis Recall that we ai to find the refined uery that inially odifies the initial uery and achieves the inclusion of the user s expected but issing object in its result. En. (3) uantifies the penalty of a refined uery when odifying the direction d = (α, β ) and the paraeter k. Unlike the special case in Section 4, the bisector of the initial direction is fixed in the general case, i.e., α+β. According to En. (4), the axiu possible odification of the direction is as follows: d ax = γ π +(1 γ) Max{β α, π (β α )} (7) This is obtained when both rotating the initial direction and changing the size of d reach the axiu values, i.e., π and Max{β α, π (β α )}, respectively. The reason why an expected object is issing fro the result of an initial uery = (loc, doc, w, k, d ) can be discussed in two scenarios: (i) is in s uery direction d, 1 θ α β α β α β θ 3 θ Fig. 4. Cases of the reason about s issing but has a worse rank than k, i.e., R(,, d ) > k ; (ii) is outside s direction d. Fig. 4 illustrates an expected object issing fro a uery result in different cases. Assue that the initial uery is a top- uery and the points in the figure denote the objects that have better ranking scores than w.r.t.. Obviously, case 1 belongs to scenario (i), as is in the uery direction but has a rank of 3. We further divide scenario (ii) into two sub-cases, i.e., and 3, according to the distance of s angle (i.e., θ ) to α and β. We easure the distance as the length of the direction interval fro β to θ or fro θ to α. In, θ is closer to β, since (θ β +π)%π is saller than (α θ +π)%π. In 3, θ is closer to α. Next, we analyze the proble in these cases Case 1 Here the issing object is in the uery direction. In other words, the issing object does have a rank R(,, d ) under the initial uery. Thus, one very basic refined uery that can revive is to keep the initial direction (i.e., d = ) and siply enlarge k to R(,, d ). According to Lea 1, we ust set k = R(,, d ) to obtain the sallest penalty, and its corresponding k = R(,, d ) k. Any other refined ueries with d > and k R(,, d ) k have larger penalties and have no chance to be the best refined uery. Hence, the axiu possible odification to k is k ax = R(,, d ) k. As such, the penalty function for the why-not proble in Case 1 is the following: Penalty(, k ) = λ R(,, d ) k d + (1 λ) γ π + (1 γ) Max{β α, π (β α )} (8) Like the special case in Section 4, the ranking score of each object reains unchanged for the refined uery. The rank of the issing object is deterined by the nuber of objects that doinate in the refined direction. Siilar to the special case, we can first find the objects that have a ranking score larger than in all directions, then deterine the best direction for each possible refined k, and finally return the one with the sallest penalty. Nevertheless, to avoid exploring all uery directions around the uery location, we prove that the best refined direction belongs to a sall search space. Theore 3. Consider an initial uery with direction d = (α, β ) and a issing object in d. Let θ be the angle of w.r.t.. A direction (α, β ) can result in the

9 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 8 best refined uery if: (i) θ (α, β ); and (ii) (α, β ) [θ (β α ), θ + (β α )]. Proof. The first condition is obviously necessary. It ensures that the issing object is in the refined uery direction. Proving the second condition is euivalent to proving that (a) α [θ (β α ), θ ], and (b) β [θ, θ + (β α )]. We prove these two by contradiction. (a) Assue α < θ (β α ) and α is a candidate start angle for the best refined uery. Then, the possible refined end angle β can be considered in two cases: β [θ, β θ ] or β (β θ, α + π]. In the first case, consider a refined start angle α = θ (β α ) and copare the two refined ueries with directions [α, β ] and [α, β ]. Since both [α, β ] and [α, β ] have sizes no less than the size of the initial direction and [α, β ] [α, β ], s [α,β ] < s [α,β ]. And for rotating the initial direction, r [α,β ] r [α,β ] = α +β α+β α +β α+β = α α <. Hence, r [α,β ] < r [α,β ]. Cobining these two facts, we can deduce that d [α,β ] < d [α,β ]. Moreover, since [α, β ] is a subset of [α, β ], the doinators of located in [α, β ] ust also exist in [α, β ], which infers that the refined k for [α, β ] is no larger than that for [α, β ]. Therefore, it follows that k [α,β ] k [α,β ]. As both d [α,β ] d [α,β ] and k [α,β ] k [α,β ], the penalty of the refined uery with the refined direction [α, β ] would be saller. Thus, α < θ (β α ) cannot be a candidate start angle for an end angle β [θ, β θ ]. In the second case, consider a refined direction with α = θ (β α ) and β = β θ. As both [α, β ] and [α, β ] are larger than the initial direction and [α, β ] [α, β ], s [α,β ] < s [α,β ] and k [α,β ] k [α,β ]. In addition, r [α,β ] = θ (β α)+β θ α+β = r [α,β ]. Thus, the refined uery with direction [α, β ] would have a saller penalty. Any direction [α, β ] with α < θ (β α ) and β (β θ, α + π] has no chance to be the best refined direction. The proof of (a) follows. The proof of (b) is siilar and hence oitted in the interest of space. Theore 3 reduces the search space for the refined directions to a saller candidate space, aking the processing of the why-not uery ore efficient. We denote this Candidate Space as CS(d ), i.e., CS(d ) = [θ (β α ), θ + (β α )] Case Siilar to Case 1, the search space for Case can be reduced by Theore 4 to a saller candidate space CS(d ) = [α (θ β ), θ + (β α )]. Theore 4. Consider an initial uery with a direction d = (α, β ) and a issing object outside d. Let θ denote the angle of w.r.t.. If (θ β + π)%π < (α θ θ b d αl β β α α L β α U β U (a) Case 1 (b) Case Fig. 5. The candidate space of refined directions θ + π)%π, a direction (α, β ) can result in the best refined uery if: (i) θ (α, β ); and (ii) (α, β ) [α (θ β ), θ + (β α )]. Proof. We oit the proof since it is siilar to that of Theore 3. Unlike Case 1, in this case the expected but issing object is outside the initial direction. The expected object is filtered out by d and does not have a rank under the initial uery. It is thus ipossible to revive the issing object by siply enlarging k without odifying the initial direction. Nevertheless, Theore 4 iplies that the refined direction belongs to CS(d ). That is, the rank of can only be influenced by the objects in CS(d ) that score better than. The refined k can never exceed the rank of the issing object in the whole candidate direction space. Thus, we set the axiu possible refined k to be the rank of in CS(d ), denoted by R(,, CS(d )). As this rank could be saller than k, to noralize k and avoid the denoinator k ax being, the axiu odification on k is given as follows: k ax = ax{r(,, CS(d )) k, 1} Thus, the penalty function for the why-not proble in Case is represented as follows: Penalty(, k ) = λ ax{r(,, CS(d )) k, 1} d + (1 λ) γ π + (1 γ) Max{β α, π (β α )} (9) Case 3 Case 3 is syetric to Case except that the candidate space for the refined direction in this case is CS(d ) = [θ (β α ), β + (α θ )]. Exaple 6. Fig. 5 illustrates the candidate space for refined directions, i.e., CS(d ) = [α L, β U ], in Case 1 and. For Case 1, the lower bound α L of the start angle is obtained by rotating the initial uery direction clockwise until the end direction β reaches θ, then the corresponding α is α L ; the upper bound β U of the end angle is obtained in a siilar way by rotating d counterclockwise until α reaches θ. For Case, while β U is also obtained siilarly, α L is actually the syetrical angle of s angle θ w.r.t. b d, where b d is the bisector of the initial uery direction d, i.e., α L +θ = α+β.

10 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 9 Algorith Answering Why-not Questions: General Case INPUT: Original uery = (loc, doc, w, k, d ), Missing object, Penalty Option λ, γ OUTPUT: Best refined uery = (loc, doc, w, k, d ) 1: θ CalDirection(, ) : identify which case the proble falls into according to the relationship between θ and d 3: copute the corresponding CS(d ) [α L, β U ] 4: deterine R(,, CS(d )) and record s doinators in set S 5: if Case 1 then 6: k ax R(,, d ) k // R(,, d ) is obtained by counting how any objects in S are in d 7: d d, k R(,, d ), p c λ 8: else 9: k ax ax{r(,, CS(d )) k, 1} 1: d CS(d ), k R(,, CS(d )) 11: p c penalty(, ) 1: for each o i S 13: θ[i] CalDirection(, o i ) 14: sort θ[i] in clockwise order w.r.t. θ 15: θ[] θ, θ[ S + 1] θ 16: for r 1 to R(,, CS(d )) do 17: k ax{r k, } 18: k if λ k ax p c then 19: return (loc, doc, w, k, d ) : for i 1 to r do 1: if θ[i 1] / [α L, θ ] break : α U θ[i 1], θ L θ[i 1] + (θ[ S (r i) + 1] θ[i 1] + π)%π 3: if θ[i] [α L, θ ] then 4: α L θ[i] 5: else 6: 7: α L α L if θ[ S (r i)] [θ, β U ] then 8: β U θ[i 1]+(θ[ S (r i)] θ[i 1]+π)%π 9: else 3: β U θ[i 1] + (β U θ[i 1] + π)%π 31: solve the linear prograing proble with the objective function as d and with the constraints as: (i) α (α L, α U ]; (ii) β [β L, β U ) 3: [α, β ] the point that iniizes d 33: copute the penalty p for the current candidate direction according to the corresponding penalty function 34: if p < p c then 35: k r, d [α, β ], p c p 36: k ax{k, k } 37: return (loc, doc, w, k, d ) 5. Answering Why-Not The above discussions analyze the why-not uery proble in different cases and provide theores that reduce the search space. Based on this, we propose an algorith for solving direction-aware why-not uestions in the general case. The pseudo-code is given in Algorith. First, we identify which case the proble falls into according to β U θ[3] θ[4] θ[] θ θ[1] (a) Range 1 α L β U Fig. 6. Ranges of α and β that rank at θ[3] θ[4] θ θ[1] θ[] α L (b) Range the relationship between θ and d (Line ). Then the candidate search space CS(d ), the axiu possible k, and the initially refined (d, k ) for the corresponding case are coputed accordingly (Lines 3 11). Next, we enuerate each possible refined k and copute the refined uery with the sallest penalty for each of the (Lines 16 35). Finally, we return the best one as the result (Line 37). Unlike the special case in Section 4, the refined start and end angles do not have to be angles of doinators of the issing object. Instead, a candidate refined direction d = [α, β ] that ranks the issing object at a given k belongs to several range pairs of α and β. See Fig 6 as an exaple. With either pair of α (θ[1], θ ], β [θ[4], θ[3]) or α (θ[], θ[1]], β [θ, θ[4]), the issing object has a rank of. More generally, let CS(d ) = [α L, β U ], and let S be the set of s doinators in the candidate search space. We denote the angle of each object o i in S as θ[i] and sort the clockwise w.r.t θ. Let t be the nuber of objects in S and in the direction [α L, θ ]. For a given k, the possible α ranges can be enuerated as (θ[i], θ[i 1]] [α L, θ ], 1 i in{k, t + 1}, where θ[] is set as θ. The corresponding β range for an α range can then be selected accordingly by ensuring that k 1 doinators of exist in [α, β ]. There are at ost k such α and β range pairs for a given k. For exaple, in Fig. 6, there are two such pairs that rank at, as discussed above. To copute the optial refined uery for a given k, we solve a linear prograing proble for each range pair of α and β, with the objective function set as the corresponding penalty function (Line 31). Given an initial uery and a refined k, k is deterined. Miniizing the penalty function is euivalent to iniizing d. Specifically, the linear prograing proble of finding the optial refined direction for a range pair of α and β that ranks the issing object at a given rank k can be odeled as follows: 3 in d = γ α +β α+β +(1 γ) (β α ) (β α ) α (θ[i], θ[i 1]] s.t. β [θ[i 1]+(θ[ S (k i)+1] θ[i 1]+π)%π, θ[i 1]+(θ[ S (k i)] θ[i 1]+π)%π), where i Z and 1 i in{k, t + 1}, and α and β are the start and end angles of the initial uery s direction. Note that the β range is coputed in a way that ensures that it satisfies the constraint β [α, α + π). 3. For clarify of the presentation, here we oit the criteria that α [α L, θ ] and β [θ, β U ].

11 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 1 θ [] θ [1] 1 (a) Fig. 7. An exaple of ultiple issing objects θ [] θ [1] (b) 1 Exaple 7. Fig. 6 shows an exaple, where two range pairs of α and β rank at. The first pair is α (θ[1], θ ] and β [θ[4], θ[3]) (Fig. 6(a)). To ensure that the refined β is in [α, α + π), the range of β is easured fro θ, i.e., [θ +(θ[4] θ +π)%π, θ +(θ[3] θ +π)%π). The optial refined α and β in this range are then coputed by solving the linear prograing proble of iniizing d. It is coputed siilarly for the second range pair of α and β (Fig. 6(b)). By coparing these two results, we then find the best α and β that rank at. We give the tie coplexity of Algorith in the next theore. Theore 5. Let r = R(,, CS(d )) denote the rank of the issing object in the candidate search space, and let DSKT (r) denote the tie coplexity of a directionaware spatial keyword top-k uery in retrieving the top-r objects. Algorith has a tie coplexity of O(DSKT (r) + r ). Proof. The proof is siilar to that of Algorith 1. The difference is that in Algorith, we enuerate the possible k fro 1 to r, and for each k, there exist at ost k range pairs of refined α and β. We need to solve a linear prograing proble for each range pair. Nevertheless, the tie coplexity of solving such a linear prograing proble is constant, as the optial point is always on the vertices of the convex polygon. Thus, the tie coplexity of coputing the refined uery is also O(r ). Optiizations: The optiizations proposed in Section 4 are also applicable to Algorith. 6 HANDLING MULTIPLE MISSING OBJECTS Next, we extend the proposed algoriths to support whynot ueries with a set M of issing objects. Recall that we refine the uery direction d and the result cardinality k to revive the issing objects. As the uery direction works as a filter, only the objects in the direction are considered as candidates for a uery result. To achieve the inclusion of all issing objects, it is a basic reuireent that the refined direction d covers all issing objects, i.e., i M (θ i d ). Another reuireent is that, the refined k should be able to get the inclusion of the issing object with the worst rank in the refined d, i.e., k R(M,, d ), where R(M,, d ) = Max i M R( i,, d ). With these observations, the already proposed algoriths are extended to support ultiple issing objects as follows: (i) find the sufficient directions d s s that cover all the issing objects; (ii) for each sufficient direction d s, enuerate the possible refined k to find the optial refined uery; (iii) deterine the best refined uery by coparing the optial refined ueries for all sufficient directions. Let θ [i] denote the angle of a issing object i in M and sort the objects in increasing order of their angles. Assue θ [] = θ [ M ]. A sufficient direction that covers all the issing objects in M is defined as [θ [i], θ [i 1]], where 1 i M. See Fig. 7(a) for an exaple. Here, the issing object set is M = { 1, }, and the doinators of the worst ranked object in M are arked as black points. Next, d s1 = [θ [1], θ []] and d s = [θ [], θ [1]] are two possible sufficient directions for M. Consider d s1 = [θ [1], θ []]. To find the best refined uery, we enuerate the possible refined k starting fro R(M,, d s1 ), i.e., 3. The process of finding the optial refined direction for a refined k is the sae as that for a single issing object, i.e., coparing the directions with start and end angles as the angles of the doinators in the special case and solving linear prograing probles in the general case. We need to find the optial refined direction for each sufficient direction of M, as users preferences of odifying k vs. d and the distributions of the doinators of the worst ranked issing object vary. See Fig. 7(b) for an exaple. Assue a top- uery is initially issued. If we only consider the sufficient direction [θ [1], θ []], the sallest refined k euals R(M,, [θ [1], θ []]) = 6. In contrast, the sallest refined k for another sufficient direction, i.e., [θ [], θ [1]], is only, which incurs a saller penalty for odifying k. Theore 6. Let r = R(M,, [ π, π)) denote the lowest rank of the issing objects in the whole database and SKT (r) denote the tie coplexity of a spatial keyword top-k uery retrieving the top-r objects. The tie coplexity of the why-not algorith for a set M of issing objects is O(SKT (r) + M r ). Proof. As we need to consider all sufficient directions, a traditional spatial keyword top-k uery needs to be processed to copute the doinators of the objects in M in all directions. Moreover, an optial refined uery is sought for each sufficient direction. If no two issing objects are in the sae direction, there are M sufficient directions for a issing object set M. Thus, the overall tie coplexity is O(SKT (r) + M r ). Optiizations: In addition to the optiizations that we proposed for a single issing object, we enuerate the sufficient directions in increasing order of the ranks of the worst ranked object aong all issing objects in the, i.e., R(M,, d s ), so that we can early stop the enueration if the sallest refined k for the next d s, i.e., R(M,, d s ), has a larger penalty in odifying k than the penalty of the currently seen best refined uery. 7 EMPIRICAL STUDY This section evaluates the effectiveness and efficiency of the proposed algoriths.

12 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING Experiental Setup Syste Setup and Metrics Our experients are all conducted on a PC with an Intel Core i5.7ghz CPU and 8GB eory running Windows 7 OS. The algoriths are ipleented in Java, and the axiu ain eory of the Java Virtual Machine is set to 4GB. The proposed why-not uery techniues are applicable to any direction-aware spatial keyword top-k uery algorith. In our experients, we extend an existing algorith [15] to process direction-aware spatial keyword top-k ueries using the language odel by exaining whether each accessed MBR or object is in the uery direction. The index structure adopted, i.e., the IR-tree, is disk-resident. The page size is set to 4KB and the capacity of a node is set to. Note that the proposed algoriths for processing why-not uestions are independent of the algorith for the direction-aware spatial keyword search. For each set of experients, we randoly generate 1, ueries and report the average CPU tie Datasets We study the perforance of the proposed algoriths using two real datasets, EURO and GN. Both are used widely in spatial keyword related research [5], [9], [11], [3]. Each dataset contains a nuber of objects with a spatial location and a set of keywords. EURO is a dataset of points of interest such as ATMs, hotels, and stores in Europe ( and GN is obtained fro the US Board on Geographic Naes (geonaes.usgs.gov) and contains a set of geographic objects. Table 1 gives ore details about the datasets. TABLE 1 Dataset Inforation Dataset EURO GN Total # of objects 16,33 1,868,81 Total # of distinct words 35,315,47 Avg. # of words per object Paraeters We evaluate the perforance of our algoriths when varying different paraeters. Table lists these paraeters, where the default values are highlighted in bold. By default, the why-not uestion is issued for a issing object that ranks at 1 k +1 under the initial uery without taking into account a uery direction. As such, the issing object ight be located inside or outside the user-specified uery direction. As a default, we fix the weighting factor γ in En. (4) to.5 to balance the changes between the rotation and the size of a refined direction. For each set of experients, the paraeters are set to their default values unless specified otherwise. 7. Experiental Results 7..1 Baseline vs. Algorith 1 We first copare Algorith 1 against the baseline ethod presented in Section 3.3. Note that the baseline ethod is workable for the special case only, since it is ipossible to enuerate an infinite nuber of candidate directions TABLE Paraeter Setting Paraeter Setting k 1, 3, 1, 3, # of keywords, 4, 6, 8, 16 w <.1,.9>,<.3,.7>, <.5,.5>,<.7,.3>,<.9,.1> R(,, [ π, π)) 11, 31, 11, 31, 1 π size of d 6, π 3, π, π, π # of issing objects 1, 3, 1, 3 in the general case. Thus, in this set of experients, the initial uery is a traditional spatial keyword top-1 uery with no uery direction. The candidate directions for the baseline algorith are obtained according to Proposition 1. Fig. 8 shows the runtie of the algoriths under different penalty options, where stands for Prefer Modifying K (λ =.1) ; stands for Prefer Modifying Direction (λ =.9) ; and stands for Never Mind (λ =.5). As we can see, Algorith 1 outperfors the baseline ethod by two orders of agnitude. That is ainly because the baseline ethod needs to invoke a spatial keyword top-k uery to deterine the rank of the issing object for each candidate direction, which incurs high coputation cost. In contrast, Algorith 1 is able to calculate the penalty of a candidate direction in constant tie. Running tie (s) 1 1 Baseline Algorith 1 Different ethods (EURO) (a) Running Tie (EURO) Running tie (s) 1e+6 1 Baseline Algorith 1 Different ethods (GN) (b) Running Tie (GN) Fig. 8. Varying penalty options (Baseline vs. Algorith 1) In the reaining experients, we exaine the perforance of the algorith (i.e., Algorith ) proposed for the general case. We do not include the baseline algorith for coparison since, as explained above, it does not work for the general case. 7.. Varying k In this set of experients, we investigate how different values of the paraeter k under the initial uery affects the perforance of the algorith. In our setting, the rank of the issing object under the initial uery varies with k, i.e., R(,, [ π, π)) = 1 k + 1. For instance, when an initial top-3 uery is issued, the corresponding whynot uestion seeks to revive the object ranked at 31 in the database w.r.t. the initial uery. Fig. 9 plots the results. Recall that the proposed algorith consists of two phases, i.e., i) coputing the initial rank of the issing object and ii) finding the best refined uery. Both of the two phases take ore tie when the issing object has a worse rank. In our setting, the rank of the issing object gets worse when k increases; hence, the runtie of the algorith increases with k. However, the algorith scales well with the increase of k. For instance, when k = 1, the issing

13 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 1 object ranks at 1; the algorith is able to process the why-not uery within 4s and 1.8s on datasets Euro and GN, respectively. Running tie (s) k (EURO) (a) Running Tie (EURO) Fig. 9. Varying k Running tie (s) k (GN) (b) Running Tie (GN) 7..3 Varying the nuber of uery keywords We next evaluate the effect of different nubers of uery keywords. Fig. 1 plots the results. Intuitively, the nuber of uery keywords only affects the first phase of the algorith, i.e., coputing the issing object s rank in the candidate search space. As having ore uery keyword reuires ore tie to copute the textual siilarities between the uery keywords and index tree nodes/objects, the runtie shows an increasing tendency when the nuber of uery keywords increases. Running tie (s) Nuber of keywords (EURO) (a) Running Tie (EURO) Fig. 1. Varying # uery keywords 7..4 Varying w. Running tie (s) Nuber of keywords (GN) (b) Running Tie (GN) The weighting vector w allows users to set their preferences between spatial proxiity and textual relevance when issuing a spatial keyword top-k uery. We evaluate the perforance of the proposed algorith by varying w in this set of experients. Different settings of w also only affect the algorith in the first phase. As we can see fro Fig. 11, the uery tie decreases when w s in w is increased. The reason is that, a saller w s eans a higher weight to textual relevance, which lowers the iportance of spatial proxiity in the ranking function. Conseuently, the pruning ability of the IR-tree decreases and ore tree nodes need to be accessed Varying the size of d Next, we investigate the perforance of the algorith when varying the initial uery direction d. In the experients, ueries with different initial direction sizes, fro π 6 to π, are generated randoly. The average uery tie is shown in Fig. 1. A larger d results in a larger candidate space CS(d ) for the refined uery directions. As the processing of a direction-aware spatial keyword uery consues ore tie Running tie (s) w s (EURO) (a) Running Tie (EURO) Fig. 11. Varying w s in w Running tie (s) w s (GN) (b) Running Tie (GN) for a larger search space, the tie for coputing the rank of the issing object in a larger candidate space increases. This explains why the runtie increases with the size of d. Nevertheless, the algorith scales well as d increases. For exaple, the running tie only increases % when the direction varies fro π 6 to π. Running tie (s) 15 5 π/6 π/3 π/ π π d (EURO) (a) Running Tie (EURO) Fig. 1. Varying the size of d Running tie (s) π/6 π/3 π/ π π d (GN) (b) Running Tie (GN) 7..6 Varying the initial rank of the issing object We also study the perforance of our algorith when issuing why-not uestions for issing objects with different ranks in the initial uery. In the experients, a default top-1 uery is used. We ask five why-not uestions with the issing object being the one ranked at 11, 31, 11, 31, and 1, respectively. Fig. 13 shows the results. As expected, the spatial keyword top-k uery consues ore tie to copute the rank of the issing object that ranks worse under the initial uery. Moreover, a worse ranked issing object results in any ore doinators and thus produces ore candidate directions, which akes the phase of finding the best refined uery take longer as well. These are the reasons for that the runtie increases when the issing object s rank gets worse. It is interesting to observe that the result of this set of experients is uite siilar to that of varying k. This suggests that the initial rank of the issing object affects the perforance of the algorith significantly while k has little effect. Running tie (s) Rank of issing object (EURO) (a) Running Tie (EURO) 14 Fig. 13. Varying the initial rank of the issing object Running tie (s) Rank of issing object (GN) (b) Running Tie (GN)

14 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING Varying the nuber of issing objects We next consider the perforance of the algorith when changing the nuber of issing objects. The initial uery is a top-1 uery with a direction of size π 3. Missing objects are selected randoly aong the objects ranked between 11 and 11 in the whole database w.r.t. the initial uery. Fig. 14 shows that the uery tie increases with ore issing objects. The reason is that having ore issing objects produces a larger candidate search space for the refined directions. Nevertheless, the increase of the uery tie is only oderate. This is because only the issing object with the worst initial rank has an ipact on the perforance. Running tie (s) nuber of issing objects (EURO) (a) Running Tie (EURO) Fig. 14. Varying # issing objects 7..8 Scalability Running tie (s) nuber of issing objects (GN) (b) Running Tie (GN) In the last set of experients, we study the scalability of the proposed algorith. To do so, we randoly select different nubers of objects fro the datasets to test the perforance of the algorith with different dataset sizes. All paraeters for the ueries are set to the default values. Fig. 15 plots the results. In our setting, the initial rank of the issing object does not change when the dataset size increases, which eans that the cost of the phase that finds the best refined uery is unaffected. However, the cost of processing a spatial keyword top-k uery increases with the dataset size. This is the reason why the uery tie of the proposed algorith is sublinear to the increase of dataset size. Running tie (s) K 6K 9K 1K 15K Data size (EURO) (a) Running Tie (EURO) Fig. 15. Varying dataset size Running tie (s) K 6K 9K K 15K Data size (GN) (b) Running Tie (GN) Ipact of Penalty Options: Fro all of the above experiental results, we ay observe that the user s penalty preference (i.e.,,, or ) has very sall ipact on the perforance of the proposed algoriths under the considered paraeter settings. This is because the penalty preference λ only affects the second phase of the algoriths. Since we enuerate the possible refined k settings in an increasing order, a larger λ can help stop the enueration earlier. This explains the overall trend of the uery tie with different penalty option settings: >>. 8 CONCLUSIONS AND FUTURE WORK In this paper, we have studied the proble of answering why-not uestions in the context of direction-aware spatial keyword top-k ueries by refining users uery directions. We ai to inially odify users initial ueries to revive their expected but issing objects. We have tackled the proble in two cases. Based on insightful proble analysis, we proved that the solution space is a finite set of candidates for the special case where no initial uery direction is specified, and we provided a linear prograing solution for the general case. We have also extended the proposed algoriths to support ultiple issing objects. Extensive experients with real datasets deonstrate that the proposed algoriths are scalable and are of superior perforance copared to a baseline ethod under a wide range of syste settings. As for future work, we plan to investigate the refineent of the uery location to ake this line of work ore coplete. We also plan to study the relevant why uestions to explain to users why soe particular objects appear in a uery result. Based on this, we would like to build an integrated fraework that supports the answering of whynot/why uestions on spatial keyword top-k ueries while considering different paraeters, including the refineent of the preference weighting vector, the uery keyword set, the uery direction, and the uery location in a concerted fashion. ACKNOWLEDGMENTS This work is supported by HK-RGC Grants 11615, , and 817. The work of Yafei Li is supported by NSFC Grant REFERENCES [1] S. S. Bhowick, A. Sun, and B. Q. Truong, Why Not, WINE?: Towards answering why-not uestions in social iage search, in Proc. 1st ACM Int. Conf. on Multiedia, pp , 13. [] P. Bouros, S. Ge, and N. Maoulis, Spatio-textual siilarity joins, Proc. VLDB Endowent, vol. 6, no. 1, pp. 1 1, Nov. 1. [3] X. Cao, L. Chen, G. Cong, C. S. Jensen, Q. Qu, A. Skovsgaard, D. Wu, and M. L. Yiu, Spatial Keyword Querying, in 31st International Conference ER, pp. 16 9, 1. [4] X. Cao, G. Cong, and C. S. 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He is currently a researcher at Huawei Noah s Ark Lab, Hong Kong. His research interests include spatial databases, uery processing, and applied achine learning. Yafei Li is an assistant professor in the School of Inforation Engineering, Zhengzhou University, Zhengzhou, China. He holds a visiting position in the database research group ( db) at Hong Kong Baptist University. He received his PhD degree in Coputer Science fro Hong Kong Baptist University in 15. His research interests include obile and spatial data anageent, location-based services, and sart city Jianliang Xu is a professor in the Departent of Coputer Science, Hong Kong Baptist University. He received his BEng degree in coputer science and engineering fro Zhejiang University, Hangzhou, China and his PhD degree in coputer science fro Hong Kong University of Science and Technology. He held visiting positions at Pennsylvania State University and Fudan University. His research interests include big data anageent, obile coputing, data security and privacy. He has published ore than 15 technical papers in these areas. He has served as a progra co-chair/vice chair for a nuber of ajor international conferences including IEEE ICDCS 1, IEEE CPSNA 15, and WAIM 16. He is an Associate Editor of IEEE TKDE and PVLDB 18. Christian S. Jensen is an Obel professor of coputer science at Aalborg University, Denark. He was recently at Aarhus University for three years and at Google Inc. for one year. His research concerns data anageent and dataintensive systes, and its focus is on teporal and spatio-teporal data anageent. He has received several national and international awards for his research. He is an editor-in-chief of the ACM Transactions on Database Systes (TODS) and was an editor-in-chief of The VLDB Journal fro 8 to 14. He is an ACM and an IEEE fellow, and he is a eber of the Acadeia Europaea, the Royal Danish Acadey of Sciences and Letters, the Danish Acadey of Technical Sciences, and the EDBT Endowent, as well as a trustee eeritus of the VLDB Endowent.