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Improved Method for Nearest Neighbor Search using Keywords


Call for Papers Engineering Journal, May 2019

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Improved Method for Nearest Neighbor Search using Keywords

B. Saravanan

PG Scholar, Department of Computer Engineering,

P.S.R Engineering College, Sivakasi

Abstract:- Spatial query is a special type of database query supported by geo databases and spatial databases. The queries differ from non-spatial SQL queries. Today, many modern applications call for novel forms of queries that aim to find objects satisfying both a spatial predicate, and a predicate on their associated texts. For example, instead of considering all the restaurants, a nearest neighbor query would instead ask for the restaurant that is the closest among those whose menus contain steak, spaghetti, brandy all at the same time. Currently the best solution to such queries is based on the IR2-tree, which, as shown in this paper, has a few deficiencies that seriously impact its efficiency. Motivated by this, we develop a new access method called the spatial inverted index that extends the conventional inverted index to cope with multidimensional data, and comes with algorithms that can answer nearest neighbor queries with keywords in real time. As verified by experiments, the proposed techniques outperform the IR2- tree in query response time significantly, often by a factor of orders of magnitude.

Index Terms Information retrieval, spatial index, keyword search

  1. INTRODUCTION

    A spatial database manages multidimensional objects (such as points, rectangles, etc.), and provides fast access to those objects based on different selection criteria. The importance of spatial databases is reflected by the convenience of modeling entities of reality in a geometric manner. For example, locations of restaurants, hotels, hospitals and so on are often represented as points in a map, while larger extents such as parks, lakes, and landscapes often as a combination of rectangles. Many functionalities of a spatial database are useful in various ways in specific contexts. For instance, in a geography information system, range search can be deployed to find all restaurants in a certain area, while nearest neighbor retrieval can discover the restaurant closest to a given address.

    Nearest neighbor search (NNS), also known as proximity search, similarity search or closest point search, is an optimization problem for finding closest points.

    A search algorithm is an algorithm for finding an item with specified properties among a collection of items.

    A spatial index is a type of extended index that allows you to index a spatial column. A spatial column is a table column that contains data of a spatial data type, such as geometry

    The IR2-tree, however, also inherits a drawback of signature files: false hits. That is, a signature file, due to its conservative nature, may still direct the search to some objects, even though they do not have all the keywords. The penalty thus caused is the need to verify an object whose satisfying a query or not cannot be resolved using only its signature, but requires loading its full text description, which is expensive due to the resulting random accesses. It is noteworthy that the false hit problem is not specific only to signature files, but also exists in other methods for approximate set membership tests with compact storage. Therefore, the problem cannot be remedied by simply replacing signature file with any of those methods.

    Fig. 1. (a) Shows the locations of points and

    (b) Gives their associated texts.

  2. PROBLEM DEFINITION

    Let P be a set of multidimensional points. As our goal is to combine keyword search with the existing location-finding services on facilities such as hospitals, restaurants, hotels, etc., we will focus on dimensionality 2, but our technique can be extended to arbitrary dimen-sionalites with no technical obstacle. We will assume that the points in P have integer coordinates, such that each coordinate ranges in [0, t], where t is a large integer. This is not as restrictive as it may seem, because even if one would like to insist on real- valued coordinates, the set of different coordinates representable under a space limit is still finite and enumerable; therefore, we could as well convert everything to integers with proper scaling.

    Each point p P is associated with a set of words, which is denoted as Wp and termed the document of p. For example, if p stands for a restaurant, Wp can be its menu, or if p is a hotel,

    Wp can be the description of its services and facilities, or if

    p is a hospital, Wp can be the list of its out-patient specialties. It is clear that Wp may potentially contain numerous words.

    Traditional nearest neighbor search returns the data point closest to a query point. Following [12], we ex-tend the problem to include predicates on objects texts. Formally, in our context, a nearest neighbor (NN) query specifies a point q and a set Wq of keywords (we refer to Wq as the document of the query). It returns the point in Pq that is the nearest to q, where Pq is defined as

    Pq = {p P | Wq Wp} (1)

    In other words, Pq is the set of objects in P whose documents contain all the keywords in Wq . In the case where Pq is empty, the query returns nothing. The prob-lem definition can be generalized to k nearest neighbor (kNN)

    In the context of [12], SC works in the same way as the classic technique of bloom filter. In preprocessing, it builds a bit signature of length l from W by hashing each word in W to a string of l bits, and then taking the disjunction of all bit strings. To illustrate, denote by h(w) the bit string of a word w. First, all the l bits of h(w) are initialized to 0. Then, SC repeats the following m times: randomly choose a bit and set it to 1. Very importantly, randomization must use w as its seed to ensure that the same w always ends up with an identical h(w). Furthermore, the m choices are mutually independent, and may even happen to be the same bit. The concrete values of l and m affect the space cost and false hit probability, as will be discussed later.

    Figure 2 gives an example to illustrate the above process, assuming l = 5 and m = 2. For example, in the bit string h(a) of a, the 3rd and 5th (counting from left)

    search, which finds the k points in Pq closest to q; if Pq has

    less than k points, the entire Pq should be returned.

    For example, assume that P consists of 8 points whose locations are as shown in Figure 1a (the black dots), and their documents are given in Figure 1b. Consider a query point q at the white dot of Figure 1a with the set of keywords Wq = {c, d}. Nearest neighbor search finds p6, noticing that all points closer to q than p6 are missing either the query keyword c or d. If k = 2 nearest neighbors are wanted, p8 is also returned in addition. The result is still

    {p6, p8} even if k increases to 3 or higher, because only 2 objects have the keywords c and d at the same time.

    We consider that the dataset does not fit in memory, and needs to be indexed by efficient access methods in order to minimize the number of I/Os in answering a query.

  3. RELATED WORK

Section 3.1 reviews the information retrieval R-tree which is the state of the art for answering the nearest neighbor queries defined in Section 2. Section 3.2 explains an alternative solution based on the inverted index. Finally, Section 3.3 discusses other relevant work in spatial keyword search.

    1. The IR2-tree

      As mentioned before, the IR2-tree combines the R-tree with signature files. Next, we will review what is signature file, before explaining the details of IR2-trees. Our discussin assumes the knowledge of R-trees and the best- first algorithm [14] for NN search, both of which are well- known techniques in spatial databases.

      Signature file in general refers to a hashing-based framework, whose instantiation in is known as superimposed coding (SC), which is shown to be more effective than other instantiations [11]. It is designed to perform membership tests: determine whether a query word w exists in a set W of words. SC is conservative, in the sense that if it says no, then w is definitely not in W . If, on the other hand, SC returns yes, the true answer can be either way, in which case the whole W must be scanned to avoid a false hit.

      Word hashed bit string

      A 00101

      B 01001

      C 00011

      D 00110

      E 10010

      Fig. 2. Example of bit string computation with l = 5 and m = 2 bits are set to 1.

      As mentioned earlier, the bit signature of a set W of words simply ORs the bit strings of all the members of W. For instance, the signature of a set {a, b} equals01101, while that of {b, d} equals 01111.

      Given a query keyword w, SC performs the member- ship test in W by checking whether all the 1s of h(w) appear at the same positions in the signature of W . If not, it is guaranteed that w cannot belong to W . Otherwise, the test cannot be resolved using only the signature, and a scan of W follows. A false hit occurs if the scan reveals that W actually does not contain w.

      For example, assume that we want to test whether word c is a member of set {a, b} using only the sets signature 01101. Since the 4th bit of h(c) = 00011 is 1 but that of 01101 is 0, SC immediately reports no. As another example, consider the membership test of c in {b, d} whose signature is 01111.

      This time, SC returns yes because 01111 has 1s at all the bits where h(c) is set to 1; as a result, a full scan of the set is required to verify that this is a false hit.

      The IR2-tree is an R-tree where each (leaf or non leaf) entry E is augmented with a signature that summarizes the union of the texts of the objects in the sub tree of E. Figure 3 demonstrates an example based on the dataset of Figure 1 and the hash values in Figure 2. The string 01111 in the leaf entry p2, for example, is the signature of Wp2 = {b, d} (which is the document of p2; see Figure 1b). The string 11111 in the non leaf entry E3 is the signature of Wp2

      Wp6 , namely, the set of all words describing p2 and p6. Notice that, in general, the signature of a non leaf entry E can be conveniently obtained simply as the disjunction of all the signatures in the child node of E. A non leaf signature may allow a query algorithm to realize that a certain word cannot exist in the sub tree.

      For example, as the 2nd bit of h(b) is 1, we know that no object in the sub trees of E4 and E6 can have word b in its texts notice that the signatures of E4 and E6 have 0 as their 2nd bits. In general, the signatures in an IR2-tree may have different lengths at various levels.

      On conventional R-trees, the best-first algorithm [14] is a well-known solution to NN search. It is straight forward to adapt it to IR2-trees. Specifically, given a query point q and a keyword set Wq , the adapted algorithm accesses the entries of an IR2-tree in ascending order of the distances of their MBRs to q (the MBR of a leaf entry is just the point itself), pruning those entries whose

      Fig. 3. Example of an IR2-tree.

      1. Shows the MBRs of the underlying R-tree

      2. gives the signatures of the entries.

        signatures indicate the absence of at least one word of Wq in their subtrees. Whenever a leaf entry, say of point p, cannot be pruned, a random I/O is performed to retrieve its text description Wp. If Wq is a subset of Wp, the algorithm terminates with p as the answer; otherwise, it continues until no more entry remains to be processed. In Figure 3, assume that the query point q has a keyword set Wq = {c, d}. It can be verified that the algorithm must read all the nodes of the tree, and fetch the documents of p2, p4, and p6 (in this order). The final answer is p6, while p2 and p4 are false hits.

    2. Solutions based on inverted indexes

      Inverted indexes (I-index) have proved to be an effective access method for keyword-based document retrieval. In the spatial context, nothing prevents us from treating the text description Wp of a point p as a document, and then, building an I-index. Figure 4 illustrates the index for the dataset of Figure 1. Each word in the vocabulary has an inverted list, enumerating the ids of the points that have the word in their documents.

      Note that the list of each word maintains a sorted order of point ids, which provides considerable convenience in query processing by allowing an efficient merge step. For example, assume that we want to find the points that have words c and d. This is essentially to compute the intersection of the two words inverted lists. As both lists are sorted in the same order, we can do so by merging them, whose I/O and CPU times are both linear to the total length of the lists.

      Recall that, in NN processing with IR2-tree, a point retrieved from the index must be verified (i.e., having its text description loaded and checked). Verification is also necessary with I-index, but for exactly the opposite reason. For IR2-tree, verification is because we do not have the detailed texts of a point, while for I-index, it is because we do not have the coordinates. Specifically, given an NN query q with keyword set Wq , the query algorithm of I- index first retrieves (by merging) the set Pq of all points that have all the keywords of Wq ,

      According to the experiments of [12], when Wq has only a single word, the performance of I-index is very bad, which is expected because everything in the inverted list of that word must be verified. Interestingly, as the size of Wq increases, the performance gap between I-index and IR2- tree keeps narrowing such that I-index even starts to outperform IR2-tree at |Wq | = 4. This is not as surprising as it may seem. As |Wq | grows large, not many objects need to be verified because the number of objects carrying all the query keywords drops rapidly. On the other hand, at this point an advantage of I-index starts to pay off. That is, scanning an inverted list is relatively cheap because it involves only sequential I/Os1, as opposed to the random nature of accessing the nodes of an IR2-tree.

    3. Other relevant work

      Our treatment of nearest neighbor search falls in the general topic of spatial keyword search, which has also given rise to several alternative problems. A complete survey of all those problems goes beyond the scope of this paper. Below we mention several representatives, but interested readers can refer to [4] for a nice survey.

      Cong et al. [10] considered a form of keyword-based nearest neighbor queries that is similar to our formu-lation, but differs in how objects texts play a role in determining the query result. Specifically, aiming at an IR flavor, the approach of [10] computes the relevance between the documents of an object p and a query q. This relevance score is then integrated with the Euclidean distance between p and q to calculate an overall similarity of p to q. The few objects with the highest similarity are returned. In this way, an object may still be in the query result, even though its document does not contain all the query keywords. In our method, same as [12], object texts are utilized in evaluating a Boolean predicate, i.e., if any query keyword is missing in an objects document, it must not be returned. Neither approach subsumes the other, and both make sense in different applications. As an application in our favor, consider the scenario where we want to find a close restaurant serving steak, spaghetti and brandy, and do not accept any restaurant that does not serve any of these three items. In this case, a restaurants document either fully satisfies our requirement, or does not satisfy at all. There is no partal satisfaction, as is the rationale behind the approach of [10].

      In geographic web search, each webpage is assigned a geographic region that is pertinent to the webpages contents. In web search, such regions are taken into account so that higher rankings are given to the pages in the same area as the location of the computer issuing the query (as can be inferred from the computers IP address) [8], [13], [21]. The underpinning problem that needs to be solved is different from keyword-based nearest neighbor search, but can be regarded as the combination of keyword search and range queries.

      Zhang et al. [20] dealt with the so-called m-closest key- words problem. Specifically, let P be a set of points each of which carries a single keyword. Given a set Wq of query keywords (note: no query point q is needed), the goal is to find m = |Wq | points from P such that (i) each point has a distinct keyword in Wq , and (ii) the maximum mutual distance of these points is minimized (among all subsets of m points in P fulfilling the previous condi-tion). In other words, the problem has a collaborative nature in that the resulting m points should cover the query keywords together. This is fundamentally different from our work where there is no sense of collaboration at all, and instead the quality of each individual point with respect to a query can be quantified into a concrete value. Cao et al. [6] proposed collective spatial keyword querying, which is based on similar ideas, but aims at optimizing different objective functions.

      In [5], Cong et al. proposed the concept of prestige- based spatial keyword search. The central idea is to evaluate the similarity of an object p to a query by taking also into account the objects in the neighborhood of p. Lu et al. [17] recently combined the notion of keyword search with reverse nearest neighbor queries.

      Although keyword search has only started to receive attention in spatial databases, it is already thoroughly studied in relational databases, where the objective is to enable a querying interface that is similar to that of search engines, and can be easily used by naive users without knowledge about SQL. Well known systems with such mechanisms include DBXplorer [1], Discover [15], Banks [3], and so on. Interested readers may refer to [9] for additional references into that literature.

  1. DRAWBACKS OF THE IR2 -TREE

    The IR2-tree is the first access method for answering NN queries with keywords. As with many pioneering solutions, the IR2-tree also has a few drawbacks that affect its efficiency. The most serious one of all is that the number of false hits can be really large when the object of the final result is faraway from the query point, or the result is simply empty. In these cases, the query algorithm would need to load the documents of many objects, incurring expensive overhead as each loading necessitates a random access.

    To explain the details, we need to first discuss some properties of SC (the variant of signature file used in the IR2-tree). Recall that, at first glance, SC has two parameters: the length l of a signature, and the number m of bits chosen to set to 1 in hashing a word. There is, in fact, really just a single parameter l, because the optimal m (which minimizes the probability of a false hit) has been solved by Stiassny [18]:

    mopt = l · ln(2)/g (2)

    where g is the number of distinct words in the set W on which the signature is being created. Even with such an optimal choice of m, Faloutsos and Christodoulakis [11] show that the false hit probability equals

    Pf alse = (1/2)mopt . (3)

    Put in a different way, given any word w that does not belong to W , SC will still report yes with probability Pf alse, and demand a full scan of W .

    It is easy to see that Pf alse can be made smaller by adopting a larger l (note that g is fixed as it is decided by W ). In particular, asymptotically speaking, to make sure Pf alse is at least a constant, l must be (g), i.e., the signature should have (1) bit for every distinct word of W . Indeed, for the IR2-tree, Felipe et al. [12] adopt a value of l that is approximately equivalent to 4g in their experiments (g here is the average number of distinct words a data point has in its text description). It thus follows that

    Pf alse = (1/2)4 LN(2) = 0.15. (4)

    The above result takes a heavy toll on the efficiency of the IR2-tree. For simplicity, let us first assume that the

    query keyword set Wq has only a single keyword w (i.e.,

    |Wq | = 1). Without loss of generality, let p be the object of the query result, and S be the set of data points that are closer to the query point q than p. In other words, none of the points in S has w in their text documents (otherwise, p cannot have been the final result). By Equation 4, roughly 15% of the points in S cannot be pruned using their signatures, and thus, will become false hits. This also means that the NN algorithm is expected to perform at least 0.15|S| random I/Os.

    So far we have considered |Wq | = 1, but the discussion extends to arbitrary |Wq | in a straightforward manner. It is easy to observe (based on Equation 4) that, in general, the false hit probability satisfies

    Pf alse 0.15|Wq |. (5)

    When |Wq | > 1, there is another negative fact that adds to the deficiency of the IR2-tree: for a greater |Wq |, the expected size of S increases dramatically, because fewer and fewer objects will contain all the query keywords. The effect is so severe that the number of random accesses, given by Pf alse|S|, may escalate as |Wq | grows (even with the decrease of Pf alse). In fact, as long as |Wq | > 1, S can easily be the entire dataset when the user tries out an uncommon combination of keywords that does not exist in any object. In this case, the number of random I/Os would be so prohibitive that the IR2-tree would not be able to give real time responses.

  2. MERGING AND DISTANCE BROWSING

    Since verification is the performance bottleneck, we should try to avoid it. There is a simple way to do so in an I-index: one only needs to store the coordinates of each point together with each of its appearances in the inverted lists. The presence of coordinates in the inverted lists naturally motivates the creation of an R-tree on each list indexing the points therein (a structure reminiscent of the one in [21]). Next, we discuss how to perform keyword-based nearest neighbor search with such a combined structure.

    The R-trees allow us to remedy awkwardness in the way NN queries are processed with an I-index. Recall that, to answer a query, currently we have to first get all the points carrying all the query words in Wq by merging several lists (one for each word in Wq). This appears to be unreasonable if the point, say p, of the final result lies fairly close to the query point q. It would be great if we could discover p very soon in all the relevant lists so that the algorithm can terminate right away. This would become a reality if we could browse the lists synchronously by distances as opposed to by ids. In particular, as long as we could access the points of all lists in ascending order of their distances to q (breaking ties by ids), such a p would be easily discovered as its copies in all the lists would definitely emerge consecutively in our access order. So all we have to do is to keep counting how many copies of the same point

    have popped up continuously, and terminate by reporting the point once the count reaches |Wq |. At

    any moment, it is enough to remember only one count, because whenever a new point emerges, it is safe to forget about the previous one.

    As an example, assume that we want to perform NN search whose query point q is as shown in Figure 1, and whose Wq equals {c, d}. Hence, we will be using the lists of words c and d in Figure 4. Instead of expanding these lists by ids, the new access order is by distance to q, namely, p2, p3, p6, p6, p5, p8, p8. The processing finishes as soon as the second p6 comes out, withot reading the remaining points. Apparently, if k nearest neighbors are wanted, termination happens after having reported k points in the same fashion.

    Distance browsing is easy with R-trees. In fact, the best- first algorithm is exactly designed to output data points in ascending order of their distances to q. How-ever, we must coordinate the execution of best-first on |Wq | R-trees to obtain a global access order. This can be easily achieved by, for example, at each step taking a peek at the next point to be returned from each tree, and output the one that should come next globally. This algorithm is expected to work well if the query keyword set Wq is small. For sizable Wq , the large number of random accesses it performs may overwhelm all the gains over the sequential algorithm with merging.

    A serious drawback of the R-tree approach is its space cost. Notice that a point needs to be duplicated once for every word in its text description, resulting in very expensive space consumption. In the next section, we will overcome the problem by designing a variant of the inverted index that supports compressed coordinate embedding.

  3. SPATIAL INVERTED LIST

The spatial inverted list (SI-index) is essentially a com- pressed version of an I-index with embedded coordi-nates as described in Section 5. Query processing with an SI- index can be done either by merging, or together with R- trees in a distance browsing manner. Furthermore, the compression eliminates the defect of a conventional I-index such that an SI-index consumes much less space.

    1. The compression scheme

      Compression is already widely used to reduce the size of an inverted index in the conventional context where each inverted list contains only ids. In that case, an effective approach is to record the gaps between consecutive ids, as opposed to the precise ids. For example, given a set S of integers {2, 3, 6, 8}, the gap-keeping approach will store

      {2, 1, 3, 2} instead, where the i-th value (i 2) is the difference between the i-th and (i 1)-th values in the original S. As the original S can be precisely reconstructed, no information is lost. The only overhead is that decompression incurs extra computation cost, but such cost is negligible compared to the overhead of I/Os. Note that gap-keeping will be much less beneficial if the

      integers of S are not in a sorted order. This is because the space saving comes from the hope that gaps would be much smaller (than the original values) and hence could be represented with fewer bits. This would not be true had S not been sorted.

      Compressing an SI-index is less straightforward. The difference here is that each element of a list, a.k.a. a point p, is a triplet (idp, xp, yp), including both the id and coordinates of p. As gap-keeping requires a sorted order, it can be applied on only one attribute of the triplet. For example, if we decide to sort the list by ids, gap-keeping on ids may lead to good space saving, but its application on the x- and y-coordinates would not have much effect.

      To attack this problem, let us first leave out the ids and focus on the coordinates. Even though each point has 2 coordinates, we can convert them into only one so that gap- keeping can be applied effectively. The tool needed is a space filling curve (SFC) such as Hilbert- or Z-curve. SFC converts a multidimensional point to a 1D value such that if two points are close in the original space, their 1D values also tend to be similar. As dimensionality has been brought to 1, gap-keeping works nicely after sorting the (converted) 1D values.

      For example, based on the Z-curve2, the resulting values, called Z-values, of the points in Figure 1a are demonstrated in Figure 5 in ascending order. With gap-keeping, we will store these 8 points as the sequence 12, 3, 8, 1, 7, 9, 2, 7. Note that as the Z-values of all points can be accurately restored, the exact coordinates can be restored as well.

      Let us put the ids back into consideration. Now that we have successfully dealt with the two coordinates with a 2D SFC, it would be natural to think about using a 3D SFC to cope with ids too. As far as space reduction is concerned, this 3D approach may not a bad solution. The problem is that it will destroy the locality of the points in their original space. Specifically, the converted values would no longer preserve the spatial proximity of the points, because ids in general have nothing to do with coordinates.

    2. Building R-trees

      Remember that an SI-index is no more than a com-pressed version of an ordinary inverted index with coordinates embedded, and hence, can be queried in the same way as described in Section 3.2, i.e., by merging several inverted lists. In the sequel, we will explore the option of indexing each inverted list with an R-tree. As explained in Section 3.2, these trees allow us to process a query by distance browsing, which is efficient when the query keyword set Wq is small.

      As before, merging demands that points of all lists should be ordered following the same principle. This is not a problem because our design in the previous subsection has laid down such a principle: ascending order of Z- values. Moreover, this ordering has a cru-cial property that conventional id-based ordering lacks: preservation of spatial proximity. The property makes it possible to build good R-trees without destroying the Z-value ordering of any list. Specifically, we can (carefully) group consecutive points of a list into MBRs, and incorporate all MBRs into an R-tree. The proximity-preserving nature of the Z-curve will ensure that the MBRs are reasonably small when the dimensionality is low. For example, assume that an inverted list includes all the points in Figure 5, sorted in the order shown. To build an R-tree, we may cut the list into 4 blocks {p6, p2},{p8, p4}, {p7, p1}, and {p3, p5}. Treating each block as a leaf node results in an R-tree identical to the one in Figure 3a. Linking all blocks from left to right preserves the ascending order of the points Z-values

      Creating an R-tree from a space filling curve has been considered by Kamel and Faloutsos [16]. Different from their work, we will look at the problem in a more rigorous manner, and attempt to obtain the optimal solution. Formally, the underlying problem is as follows. There is an inverted list L with, say, r points p1, p2, …, pr , sorted in ascending order of Z-values. We want to divide L into a number of disjoint blocks such that (i) the number of points in each block is between B and 2B 1, where B is the block size, and (ii) the points of a block must be consecutive in the original ordering of L. The goal is to make the resulting MBRs of the blocks as small as possible.

      How small an MBR is can be quantified in a number of ways. For example, we can take its area, perimeter, or a function of both. Our solution, presented below, can be applied to any quantifying metric, but our discussion will use area as an example. The cost of a dividing scheme of L is thus defined as the sum of the areas of the optimal division of the subsequence pi, pi+1, …, pj . The aim of the above problem is thus to find C[1, r]. We also denote by A[i, j] the area of the MBR enclosing pi, pi+1, …, pj .

      Now we will discuss the properties of C[i, j]. There are j

      • i + 1 points from pi to pj . So C[i, j] is undefined if j i + 1 < B, because we will never create a block with less than B points. Furthermore, in the case where j i + 1 [B, 2B

      • 1], C[i, j] can be immediately solved as the area of the MBR enclosing all the j i + 1 points. Hence, next we will focus on the case j i + 1 2B.

      Notice that when we try to divide the set of points {pi, pi+1, …, pj }, there are at most B 1 ways to decide which points should be in the same block together with the first point pi. Specifically, a block of size B must include, besides pi, also pi+1, pi+2, all the way to pi+B1. If the block size goes to B + 1, then the additonal point will have to be pi+B ; similarly, to get a block size of B + 2, we must also put in pi+B+1 and so on, until the block size reaches the maximum 2B 1. Regardless of the block size, the

      remaining points (that are not together with p1) constitute a smaller set on which the division problem needs to be solved recursively. The total number of choices may be less than B 1 because care must be taken to make sure that the number of those remaining points is at least B. In any case, C[i, j] equals the lowest cost of all the permissible choices, or formally:

      algorithm that runs in O(Br2 ) time: it suffices to derive C[i, j] in ascending order of the value of j i, namely, starting with those with j i = 2B, followed by those with ji = 2B+1, and so on until finishing at ji = r1. We can significantly improve the computation time to O(Br), by the observation that j can be fixed to r throughout the computation in order to obtain C[1, r] eventually.

      We have finished explaining how to build the leaf nodes of an R-tree on an inverted list. As each leaf is a block, all the leaves can be stored in a blocked SI-index as described in Section 6.1. Building the nonleaf levels is trivial, because they are invisible to the merging-based query algorithms, and hence, do not need to preserve any common ordering. We are free to apply any of the existing R-tree construction algorithms. It is noteworthy that the nonleaf levels add only a small amount to the overall space overhead because, in an R-tree, the number of nonleaf nodes is by far lower than that of leaf nodes.

  1. EXPERIMENTS

    In the sequel, we will experimentally evaluate the prac-tical efficiency of our solutions to NN search with key-words, and compare them against the existing methods.

    Competitors. The proposed SI-index comes with two query algorithms based on merging and distance brows-ing respectively. We will refer to the former as SI-m and the other as SI-b. Our evaluation also covers the state-of-the-art IR2-tree; in particular, our IR2-tree implemen-tation is the fast variant developed in [12], which uses longer signatures for higher levels of tree. Furthermore, we also include the method, named index file R-tree (IFR) henceforth, which, as discussed in Section 5, indexes each inverted list (with coordinates embedded) using an R-tree, and applies distance browsing for query process-ing. IFR can be regarded as an uncompressed version of SI-b.

    Data. Our experiments are based on both synthetic and real data. The dimensionality is always 2, with each axis consisting of integers from 0 to 16383. The synthetic category has two datasets: Uniform and Skew, which differ in the distribution of data points, and in whether there is a correlation between the spatial distribution and objects text documents. Specifically, each dataset has 1 million points. Their locations are uniformly distributed in Uniform, whereas in Skew, they follow the Zipf distribution3.For both datasets, the vocabulary has 200

    words, and each word appears in the text documents of 50k points. The difference is that the association of words with points is completely random in Uniform, while in Skew, there is a pattern of word-locality: points that are spatially close have almost identical text documents.

    Our real dataset, referred to as Census below, is a combination of a spatial dataset published by the U.S. Census Bureau4, and the web pages from Wikipedia5. The spatial dataset contains 20847 points, each of which represents a county subdivision. We use the name of the subdivision to search for its page at Wikipedia, and collect the words there as the text description of the corresponding data point. All the points, as well as their text documents, constitute the dataset Census. The main statistics of all of our datasets are summarized in Table 1.

    Parameters. The page size is always 4096 bytes. All the SI-indexes have a block size of 200 (see Section 6.1 for the meaning of a block). The parameters of IR2-tree are set in exactly the same way as in [12]. Specifically, the tree on Uniform has 3 levels, whose signatures (from leaves to the root) have respectively 48, 768, and 840 bits each. The corresponding lengths for Skew are 48, 856, and 864. The tree on Census has 2 levels, whose lengths are 2000 and 47608, respectively.

    Queries:

    As in [12], we consider NN search with the AND semantic. There are two query parameters: (i) the number k of neighbors requested, and (ii) the number |Wq | of keywords. Each workload has 100 queries that have the same parameters, and are generated indepen-dently as follows. First, the query location is uniformly distributed in the data space. Second, the set Wq of keywords is a random subset (with the designated size |Wq |) of the text description of a point randomly sampled from the underlying dataset. We will measure the query cost as the total I/O time (in our system, on average, every sequential page access takes about 1 milli-second, and a random access is around 10 times slower).

    Results on query efficiency:

    Let us start with the query performance with respect to the number of keywords |Wq |. For this purpose, we will fix the parameter k to 10, i.e., each query retrieves 10 neighbors. For each competing method, we will report its average query time in processing a workload. The results are shown in Figure 6, where (a), (b), (c) are about datasets Uniform, Skew, and Census, respectively. In each case, we

    present the I/O time of IR2-tree separately in a table, because it is significantly more expensive than the other solu-tions. The experiment on Uniform inspects |Wq | up to 4, because almost all queries with greater |Wq | return no result at all.

    The fastest method is either SI-m or SI-b in all cases. In particular, SI-m is especially efficient on Census where each inverted list is relatively small (this is hinted from the column the number objects per word in Table 1), and hence, index-based search is not as effective as simple scans. The behavior of the two algorithms on Uniform very well confirms the intuition that distance browsing is more suitable when |Wq | is small, but is outperformed by merging when Wq is sizable. On Skew, SI-b is significantly better than SI-m due to the word-locality pattern. As for IFR, its behavior in general follows that of SI-b because they differ only in whether compression is performed. The superiority of SI-b stems from its larger node capacity.

    IR2-tree, on the other hand, fails to give real time answers, and is often slower than our solutions by a factor of orders of magnitude, particularly on Uniform and Census where word-locality does not exist. As ana-lyzed in Section 3.1, the deficiency of IR2-tree is mainly caused by the need to verify a vast number of false hits. To illustrate this, Figure 7 plots the average false hit number per query (in the experiments of Figure 6) as a function of |Wq |. We see an exponential escalation of the number on Uniform and Census, which explains the drastic explosion of the query cost on those datasets. Interesting is that the number of false hits fluctuates6 a little on Skew, which explains the fluctuation in the cost of IR2-tree in Figure 6b.

    Next, we move on to study the other query parameter k (the number of neighbors returned). The experiments for this purpose are based on queries with |Wq | = 3. As before, the average query time of each method in handling a workload is reported. Figures 8a, 8b, 8c give the results on Uniform, Skew, and Census, respectively. Once again, the costs of IR2-tree are separated into tables. In these experiments, the best approach is still either SI-m or SI-b. As expected, the cost of SI-m is not affected by k, while those of the other solutions all increase monoton-ically.

    The relative superiority of alternative methods, in general, is similar to that exhibited in Figure 6. Perhaps worth pointing out is that, for all distributions, distance browsing appears to be the most efficient approach when k is small.

    Results on pace consumption:

    We will complete our experiments by reporting the space cost of each method on each dataset. While four methods are examined in the experiments on query time, there are only three as far as space is concerned. Remember that SI-m and SI-b actually deploy the same SI-index and hence, have the same space cost. In the following, we will refer to them collectively as SI-index.

    Summary:

    The SI-index, accompanied by the proposed query algorithms, has presented itself as an excellent tradeoff between space and query efficiency. Compared to IFR, it consumes significantly less space, and yet, answers queries much faster. Compared to IR2-tree, its superiority is overwhelming since its query time is typically lower by a factor of orders of magnitude.

  2. CONCLUSIONS

We have seen plenty of applications calling for a search engine that is able to efficiently support novel forms of spatial queries that are integrated with keyword search. The existing solutions to such queries either incur prohibitive space consumption or are unable to give real time answers. In this paper, we have remedied the situation by developing an access method called the spatial inverted index (SI- index). Not only that the SI-index is fairly space economical, but also it has the ability to perform keyword- augmented nearest neighbor search in time that is at the order of dozens of milli-seconds. Furthermore, as the SI- index is based on the conventional technology of inverted index, it is readily incorporable in a commercial search engine that applies massive parallelism, implying its immediate industrial merits.

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