qwLSH: Cache-conscious Indexing for Processing Similarity Search Query Workloads in High-Dimensional Spaces

TL;DR

qwLSH is a novel indexing technique that improves cache efficiency and query processing speed for similarity search workloads in high-dimensional spaces by using intelligent cache division and cost models.

Contribution

The paper introduces qwLSH, a cache-conscious index structure that optimizes similarity search workload processing in high-dimensional spaces using novel cost models.

Findings

qwLSH outperforms existing techniques in query workload speed.

It effectively utilizes cache by dividing cache based on novel cost models.

Experimental results validate the efficiency of qwLSH.

Abstract

Similarity search queries in high-dimensional spaces are an important type of queries in many domains such as image processing, machine learning, etc. Since exact similarity search indexing techniques suffer from the well-known curse of dimensionality in high-dimensional spaces, approximate search techniques are often utilized instead. Locality Sensitive Hashing (LSH) has been shown to be an effective approximate search method for solving similarity search queries in high-dimensional spaces. Often times, queries in real-world settings arrive as part of a query workload. LSH and its variants are particularly designed to solve single queries effectively. They suffer from one major drawback while executing query workloads: they do not take into consideration important data characteristics for effective cache utilization while designing the index structures. In this paper, we present qwLSH, an index structure for efficiently processing similarity search query workloads in high-dimensional spaces. We intelligently divide a given cache during processing of a query workload by using novel cost models. Experimental results show that, given a query workload, qwLSH is able to perform faster than existing techniques due to its unique cost models and strategies.

Equations

$$P(r)={\int }_0^w\frac{1}{r}\frac{2}{\sqrt{2\pi }}{e}^{\frac{-t^2}{2r^2}}(1-\frac{t}{w})dt.$$

$$cost(C_I)={\alpha }_{card}\times \sum _{i=1}^m(cost(C_{I_i}^q))$$

$$cost(C_D)={\alpha }_{dim}\times \sum _{j=1}^w(cost(C_{D_j}^q))$$

$$\displaystyle{\sum_{g=1}^{|\mathcal{Q}|}}\bigg{(}\alpha_{card}\times\sum_{i=1}^{m}(cost(C_{I_{i}}^{g}))$$

$$size(C_I)=size(C)-size(C_D).$$

Full text

Nearest-neighbor Search, Locality Sensitive Hashing, High-dimensional Search

1. Introduction

The similarity search problem in high-dimensional spaces is a well-known problem with wide-ranging applications in domains such as information retrieval, artificial intelligence, machine learning, etc. Exact tree-based spatial indexing techniques, such as KD-tree, R-tree, SR-tree, etc., are effective for improving searching in low-dimensional spaces, but as the dimensions increase ($\sim>$10), they all suffer from the popular curse of dimensionality problem (where they are often outperformed even by a brute force linear scan) (Datar:2004:LHS:997817.997857, ). One approach to addressing this curse of dimensionality problem is to search for approximate solutions instead of exact solutions. In many applications, where 100% accuracy is unnecessary, good enough results are acceptable. Approximate solutions sacrifice accuracy for a much faster performance. Formally, the goal of the approximate version of the similarity search problem, also called c-approximate Nearest Neighbor search, is to return the objects that are within the distance $c\times R$ from the query object, where $c>1$ is an approximate ratio and $R$ is the distance from the query to the true nearest neighbor.

1.1. Locality Sensitive Hashing

Locality Sensitive Hashing, first proposed in (Indyk:1998:ANN:276698.276876, ), is one of the most popular solutions for approximate searching in high-dimensional spaces. The purpose of Locality Sensitive Hashing (LSH) is to map high-dimensional data to lower dimensions while preserving the distances in the original space. The lower dimensional space is generated through a series of random projections. In this lower dimensional space, data objects are mapped to individual buckets based on a hash function, with the intuition that nearby data points in the original space are mapped to the same hash buckets in the lower dimensional space with a higher probability than mapping dissimilar or far away points to the same buckets. This may lead to misses and false positives. Given a distance metric and a corresponding LSH family (detailed in Section 3), LSH data structures control their precision and recall by using multiple independently chosen hash functions organized into several hash layers. Since the original work (Indyk:1998:ANN:276698.276876, ) was proposed, there has been considerable amount of research done on improving Locality Sensitive Hashing (Datar:2004:LHS:997817.997857, ; Bawa:2005:LFS:1060745.1060840, ; Liu:2014:SEI:2732939.2732947, ; Gao:2015:SHC:2783258.2783284, ; Gan:2012:LHS:2213836.2213898, ; Sun:2014:SSC:2735461.2735462, ; Tao:2010:EAN:1806907.1806912, ; Zheng:2016:LAN:2882903.2882930, ; Huang:2015:QLH:2850469.2850470, ).

1.2. Motivation

Often times, in real-world settings, queries arrive as part of a query workload. Several research works have shown the benefits of designing index structures particularly for efficient handling of query workloads (Curino:2010:SWA:1920841.1920853, ; Aly:2015:AAQ:2831360.2831361, ; Tzoumas:2009:WIC:1687627.1687761, ; DBLP:journals/pvldb/NagarkarCB15, ; DBLP:conf/edbt/NagarkarC14, ; Quamar:2013:SSW:2452376.2452427, ). Most of these works focus on partitioning the data space such that regions that are queried with high frequency are partitioned with more granularity. e.g., in 2D geographical spaces, points of interest in downtown areas are queried with much more frequency than places in the suburbs. Similarly, consider the following two scenarios for high-dimensional data: (1) Genomic DNA data, often represented as high-dimensional vectors, require similarity search queries to find similar genomes given a query genome (Buhler2001, ; Berlin2015, ; Rasheed2013, ). Certain common genome sequences are often queried more times than the rest (Rasheed2013, ). (2) Similarly, earthquake detection or satellite image data, both high-dimensional data (Yoon2015, ; Buaba2010, ), often query certain similar regions of the space that are of more interest. These queries can be viewed as part of a query workload. While LSH has been shown to be effective in processing single queries in high-dimensional spaces, it is not particularly designed to handle query workloads efficiently.

Assuming that the index and the data are stored on an external storage, Locality Sensitive Hashing and its variants have two main IO operations: (1) accessing the index (denoted by IndexIO) in order to find the candidates, and (2) once the candidates are found, the candidate points need to be brought from the external storage (denoted by DataIO) into the main memory to remove false positives. We observe that these two operations, however, have different costs:

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the IndexIO cost is dependent on the size of the index, which is in turn dependent on the size of the dataset (also referred to as cardinality of the dataset), and

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the dataIO cost is dependent on the size of each data point, i.e. the number of dimensions of the dataset, and the desired number of objects in the result (top-$k$).

Figures 1 and 2 show our observation. In Figure 1, it can be seen that as the cardinality of the dataset increases (when the dimensionality is fixed), the ratio of IndexIO/TotalIO (where TotalIO = IndexIO + DataIO) increases, because the cost of IndexIO increases. Similarly, in Figure 2, it can be seen that as the dimensionality increases, the ratio of DataIO/TotalIO increases, because the cost of DataIO increases. By using this important observation, the efficiency of an LSH algorithm can be further improved by intelligently utilizing the cache for a given query workload. In this paper, we propose qwLSH, an index structure that improves the performance of query workloads in high-dimensional spaces by improving upon the cache utilization of the system.

1.3. Contributions of this paper

Existing LSH-based index structures are designed to efficiently handle single queries. Given a query workload, they naively process the queries in the query workload individually, independent of each other. There is a need for index structures that can improve the performance of a query workload by taking into consideration the important characteristics of the workload. In this paper, we present our proposed index structure, qwLSH, that can efficiently process similarity search query workloads in high-dimensional spaces. The following are the contributions of this paper:

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Given the important observation, that different data have different query processing needs, we intelligently divide a given cache for efficient processing of a query workload

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We present novel cost models for intelligently dividing a cache given a query workload

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We present different cache utilization strategies for LSH-based query workloads

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Finally, we use real datasets to show the efficiency of our proposed index structure, qwLSH, by comparing against state-of-the-art algorithms.

To the best of our knowledge, there are no existing works that particularly tackle the problem of solving query workloads in high-dimensional spaces in an efficient manner. In this paper, we present the design and analysis of qwLSH that was particularly designed to efficiently execute query workloads in high-dimensional spaces. The paper is organized as follows: in Section 2, we present the related works. In Section 3, we present the key concepts and preliminaries necessary to understand the problem domain and qwLSH. In Sections 4 and 5, we formalize the problem statement and explain the design of qwLSH respectively. We present our experimental analysis in Section 6 and finally conclude in Section 7.

2. Related Work

Similarity search queries in high-dimensional spaces have wide-ranging applications in various domains such as multimedia retrieval, artificial intelligence, etc. Exact tree-based index structures (such as R-trees, KD-trees, etc.) work efficiently for low-dimensional data, but as the dimensions increase ($\sim>10$), their performance degrades and are often outperformed by brute-force linear scans (a well-known problem called the curse of dimensionality) (Indyk:1998:ANN:276698.276876, ). Approximate techniques were proposed where the performance of the queries was drastically improved by trading off some accuracy of the query result. Locality Sensitive Hashing, first proposed in (Indyk:1998:ANN:276698.276876, ), is a popular approximate technique for solving similarity search queries in high-dimensional spaces. The original work was proposed to solve the (R, c)-Near Neighbor problem, which is a decision-based version of the Approximate Nearest Neighbor problem (Datar:2004:LHS:997817.997857, ). While the LSH family (explained further in Section 3) in (Indyk:1998:ANN:276698.276876, ) was originally defined for the Hamming distance, it was later defined for other distance measures such as the Euclidean distance (Datar:2004:LHS:997817.997857, ).

**Effective Variants of LSH: ** LSH has been shown to be useful in various domains such as biomedical sciences (Buhler2001, ; Berlin2015, ; Rasheed2013, ), geological sciences (Yoon2015, ; Buaba2010, ), etc. Several works were subsequently proposed (Lv:2007:MLE:1325851.1325958, ; Bawa:2005:LFS:1060745.1060840, ; Liu:2014:SEI:2732939.2732947, ; Gan:2012:LHS:2213836.2213898, ; Sun:2014:SSC:2735461.2735462, ; Huang:2015:QLH:2850469.2850470, ; Tao:2010:EAN:1806907.1806912, ; Gao:2015:SHC:2783258.2783284, ) to improve upon the original work. In (Bawa:2005:LFS:1060745.1060840, ), the authors created a prefix-tree of hash functions for each hash layer to appropriately decide on the number of hash functions to use during query processing in order to return the desired number of top-$k$ results. In (Lv:2007:MLE:1325851.1325958, ), the authors propose a probing technique to look into neighboring buckets of the query point’s hash bucket, and hence instead of creating more number of layers to get the desired number of top-$k$ results, their technique probes into neighboring buckets. In (Tao:2010:EAN:1806907.1806912, ), the authors represent each of the points in the lower dimensional space using Z-order (also called Morton) codes which are then further retrieved effectively by finding the closeby points based on the distances between their Z-order codes. While these techniques improve upon the original LSH and its most popular implementation, E2LSH111https://www.mit.edu/~andoni/LSH/, they still suffer from having large index sizes and slow processing speeds. Recently, in (Gan:2012:LHS:2213836.2213898, ), the authors presented C2LSH, in which they proposed two novel ideas: a “collision counting” approach that counted the number of times a candidate object is mapped to the same bucket and a “virtual rehashing” approach that automatically incremented the lookup space in each projection without having the need to physically rehash the data. In (Gao:2015:SHC:2783258.2783284, ), the authors analyzed the data distribution and created indexes of different granularity to make their index structure scalable for different data with different distributions. In (Sun:2014:SSC:2735461.2735462, ), a projection-based method that uses only 6 to 10 random projections is presented. While the index size is much smaller than (Gan:2012:LHS:2213836.2213898, ) and (Huang:2015:QLH:2850469.2850470, ), the accuracy is much worse and it is shown to be unstable (Huang:2015:QLH:2850469.2850470, ). In (Liu:2014:SEI:2732939.2732947, ), the authors present SK-LSH, where they propose a novel linear order on the index files stored on the external storage based on their Z-order codes. The idea is that if nearby index files are stored contiguously on the external storage, then the number of I/Os will be reduced. In (Huang:2015:QLH:2850469.2850470, ), the authors build upon the “collision counting” and the “virtual rehashing” approaches presented in (Gan:2012:LHS:2213836.2213898, ) by proposing to create “query-aware” hash functions. QALSH builds B-trees on every hash function, and then given a query object, perform range searches on each of the B-trees. This method has shown to be the most accurate and fast LSH-based technique (Gao:2015:SHC:2783258.2783284, ).

**Query Workloads: ** Many index structures and systems have been particularly designed to execute query workloads efficiently (Curino:2010:SWA:1920841.1920853, ; Aly:2015:AAQ:2831360.2831361, ; Tzoumas:2009:WIC:1687627.1687761, ; DBLP:journals/pvldb/NagarkarCB15, ; DBLP:conf/edbt/NagarkarC14, ; Quamar:2013:SSW:2452376.2452427, ). (Curino:2010:SWA:1920841.1920853, ; Pavlo:2012:SAD:2213836.2213844, ) uses data partitioning techniques that are query workload-aware on 1D data. In (Achakeev:2012:SQL:2396761.2398577, ; Tzoumas:2009:WIC:1687627.1687761, ; Aly:2015:AAQ:2831360.2831361, ; Aly:2016:KWP:2835776.2835841, ), the authors present different query workload-aware partitioning techniques for existing data structures, such as R-trees, on 2D spatial data. In (DBLP:conf/edbt/NagarkarC14, ) and (DBLP:journals/pvldb/NagarkarCB15, ), the authors present unique cost models to bring in the most relevant part of the index structures into the main memory for future use based on past query workload statistics for 1D and 2D space respectively. The above works deal with data partitioning techniques for 1D and 2D data. To the best of our knowledge, there is no existing work that deals with improving the efficiency of query workloads in high-dimensional spaces.

**Cache-Conscious Querying: ** There have been several works that design techniques to reduce cache misses to improve the query performance (10.1007/978-3-319-56111-0_1, ; Rao:2000:MBT:342009.335449, ; Hankins:2003:ENS:781027.781063, ). An experimental survey on cache conscious algorithms can be found at (Yotov:2007:ECC:1248377.1248394, ). In the domain of LSH and its variants, there is one main work that has attempted making LSH cache-conscious (Sundaram:2013:SSS:2556549.2556574, ). The authors present a caching strategy to improve the index construction cost. Their main goal is to speed up indexing (and searching) over streaming Twitter data in a distributed environment. In the caching strategy presented in their work, the authors present a hierarchical 2-level hashing approach to reduce cache misses during the index construction phase. During query processing, the authors utilize the cache by “software prefetching”, i.e., by prefetching succeeding data items into the cache. Our cache utilization strategies are very different than the techniques used in (Sundaram:2013:SSS:2556549.2556574, ). Our strategies are based on the important observation as explained in Section 1.2, and our main goal is to speed up the processing speed of the entire query workload, which is different from (Sundaram:2013:SSS:2556549.2556574, ).

3. Key Concepts and Preliminaries

In this section, we present the key concepts underlying LSH. We rely mainly on the terminologies presented in E2LSH222https://www.mit.edu/~andoni/LSH/, C2LSH (Gan:2012:LHS:2213836.2213898, ), and QALSH (Gao:2015:SHC:2783258.2783284, ).

Let $\mathcal{D}$ be a database of $n$ data objects in a $d$-dimensional Euclidean space $\mathcal{R}^{d}$. Let $\lVert o_{1},o_{2}\rVert$ denote the Euclidean distance between two objects $o_{1}$ and $o_{2}$. Given a query object $q$ in $\mathcal{R}^{d}$, the $c$-ANN search (for an approximation ratio $c>1$) returns all objects $o\in\mathcal{D}$ such that $\lVert o,q\rVert\leq c\times\lVert o^{*},q\rVert$, where $o^{*}$ is the true nearest neighbor of $q$ in $\mathcal{D}$. Similarly, the $c$-$k$-ANN problem finds the top-$k$ objects $o_{i}\in\mathcal{D}$ where $1\leq i\leq k$, and such that $\lVert o_{i},q\rVert\leq c\times\lVert o_{i}^{*},q\rVert$, where $o_{i}^{*}$ is the true $i$th nearest neighbor of $q$.

Definition 3.0 (A Locality Sensitive Hashing Family).

A hash function family $\mathcal{H}$ is said to be $(r,c,P1,P2)$-sensitive if it satisfies all following conditions for any two points $x$ and $y$ in a dataset $\mathcal{D}\subset\mathcal{R}^{d}$:

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if $|x-y|\leq r$, then $Pr[h(x)=h(y)]\geq P_{1}$, and

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if $|x-y|>cr$, then $Pr[h(x)=h(y)]\leq P_{2}$

Here, $c$ is an approximation ratio, $P_{1}$ and $P_{2}$ are probabilities, $r$ is the distance between two points commonly referred to as the radius, and in order for the definition to work, $c>1$ and $P_{1}>P_{2}$. The above definition states that the two points $x$ and $y$ are hashed to the same bucket with a very high probability $\geq P_{1}$ if they are close to each other (i.e. the distance between the two points is less than or equal to $r$), and if they are not close to each other (i.e. the distance between the two points is greater than $cr$), then they will be hashed to the same bucket with a low probability $\leq P_{2}$.

In the original LSH scheme for Euclidean distance, each hash function is defined as $h_{a,b}(v)={\frac{a.v+b}{w}},$ where $a$ is a $d$-dimensional random vector with entries chosen independently from the standard normal distribution and $b$ is a real number chosen uniformly from $[0,w)$, such that $w$ is the width of the hash bucket (Datar:2004:LHS:997817.997857, ). This leads to the following collision probability function (Liu:2014:SEI:2732939.2732947, ):

[TABLE]

Note that the collision probability is governed by the width, $w$, of the hash bucket: if the size is chosen to be much larger than the query radius, then there can be a lot of candidates generated. If the size is chosen to be much smaller than the query radius, then there can be potentially several misses.

3.1. Collision Counting

C2LSH (Gan:2012:LHS:2213836.2213898, ) introduced the technique of “collision counting” because of which it is not necessary to have $l$ hash layers. Instead of hash layers, the index requires $m$ hash functions and a collision count threshold to find the candidate points. From here on, since each hash function can be viewed as a single hash layer, we use the terms hash layers and hash functions interchangeably.

Recent LSH variants (such as C2LSH (Gan:2012:LHS:2213836.2213898, ), QALSH (Huang:2015:QLH:2850469.2850470, ), etc.) have an upper bound on the number of candidate objects returned, i.e., the number of data points that have to be brought from the external storage into the main memory. This bound is often controlled by a user-input, “allowed” false positives, $v$. Intuitively, higher the $v$, less time is needed to find the candidate objects, and vice-versa. As a good trade-off, these works set $v=100$, which is what we continue using in our work as well. Thus, for a maximum $k$ (i.e. the desired number of output results) of 100, the maximum number of data objects that will be read from the disk will be 200 (since $k+v=200$). The basic LSH formulation does not have an upper bound on the number of candidate points that are needed to be accessed from the external storage.

4. Problem Specification

In this paper, our goal is to create an efficient index structure to execute similarity search query workloads in high-dimensional spaces. Let us consider we are given a query workload $\mathcal{Q}$ that consists of $q$ queries, where $\forall q\subset\mathcal{D}$. For each query $q\in\mathcal{Q}$, given $k$ and an approximation ratio $c>1$, we return the top-$k$ results such that $\lVert o_{i},q\rVert\leq c\times\lVert o_{i}^{*},q\rVert$, where $o_{i}^{*}$ is the true $i$th nearest neighbor of $q$.

As mentioned in Section 1.2, our goal is to leverage the important observation that datasets with different cardinalities and dimensionalities have different IO costs, and then design the cache such that it is most effectively utilized for a given query workload.

Given a cache $C$, our goal is to divide the cache into two parts: $C_{I}$ and $C_{D}$, where $C_{I}$ is the part of the cache that stores the index files (i.e. data from the hash buckets) and $C_{D}$ is the part of the cache that stores the data objects, such that $size(C)=size(C_{I})+size(C_{D})$. Assume $cost(C)$ denotes the cost of bringing all necessary files (index files + data objects) into the cache $C$. Similarly, $cost(C_{I})$ denotes the cost of bringing the index files into the cache, and $cost(C_{D})$ denotes the cost of bringing the data objects into the cache, such that $cost(C)=cost(C_{I})+cost(C_{D})$. In order to understand the design of qwLSH, we need to first understand the costs, $cost(C_{I})$ and $cost(C_{D})$, in detail.

$cost(C_{I})$: Let us first consider the cost of bringing all necessary index files into $C_{I}$ for a single query, $q$. Note that, we have $m$ projections in our index. Let us assume that the cost for bringing necessary index files for $i$th projection is: $cost(C_{I_{i}}^{q})$. Also, the amount of data that is read from the index files is a function of the cardinality of the dataset (Figure 1). Let us denote this constant as $\alpha_{card}$. Thus, we have:

[TABLE]

$cost(C_{D})$: Similarly, let us first consider the cost of bringing all necessary data objects into $C_{D}$ for a single query, $q$. Note that, as explained in Section 3, for a single query, the worst case scenario would be to have to read $k+v$ data objects from the external storage. Let us denote this as $w$ (i.e. $w=k+v$). Let us denote the cost of reading a single $j$th data object as $cost(C_{D_{j}}^{q})$. The size of each data object is a function of the dimensionality of the dataset (Figure 2). Let us denote this constant as $\alpha_{dim}$. Thus, we have:

[TABLE]

Note that the above costs are for a single query. When we have more than 1 query in the given query workload $\mathcal{Q}$, assuming that there will be at least 1 index file that will be reused within the queries, $cost(C_{I})$ will depend on the size of $C_{I}$. If a large amount of $C$ is dedicated to $C_{I}$, then the $cost(C_{I})$ will be less and vice-versa. Conversely, $cost(C_{D})$ will depend on the size of $C_{D}$. Thus, given a query workload $\mathcal{Q}$, we want to find $size(C_{I})$ (or $size(C_{D})$) such that $cost(C)$ is minimized:

[TABLE]

5. qwLSH

In this section, we describe our proposed index structure, qwLSH. Given a query workload, our goal is to intelligently divide the cache based on the cardinality and dimensionality of the given dataset. In order to do this, we leverage the detailed cost model presented in Section 4. In this section, we will present the two strategies for dividing the cache, along with unique cost models.

Naive Solution: In LSH (and its variants), the cache is not intelligently divided into two parts: $C_{I}$ and $C_{D}$. For a given query workload, the cache (using the default cache replacement policy, such as MRU) will treat the index files and the data object files equal, regardless of the cardinality and the dimensionality of the dataset. As shown in Figures 1 and 2, the cardinality and the dimensionality of the dataset affects the total size of index files that are read into the cache and the total size of data objects respectively. We leverage this observation that naive solutions do not. In Section 6, we show the benefit of leveraging this observation into designing the cache intelligently.

5.1. Design of qwLSH

For each query, LSH index structures access each of the $m$ hash functions in order to find the candidate data object IDs. Once the candidate data object IDs are found for the query, the data objects are brought from the external storage in order to remove the false positives. Given the total cache size, our goal is to find how much of the total cache size to allocate to the index cache, $C_{I}$, and the data cache, $C_{D}$. Once we determine the appropriate split between $C_{I}$ and $C_{D}$, we use the popular cache replacement policy, MRU, to decide which index files need to be evicted from $C_{I}$ and which data files need to be evicted from $C_{D}$. With the intuition that queries in query workloads are often times near each other (Aly:2015:AAQ:2831360.2831361, ; DBLP:journals/pvldb/NagarkarCB15, ), we choose to use MRU. Before we present the cost models that we used to determine the appropriate split between $C_{I}$ and $C_{D}$, we first present two strategies based on how the index files are accessed and cached.

Strategy 1: In this strategy, the index cache, $C_{I}$, is further divided uniformly between the $m$ hash functions, as shown in Figure 3(a). In each of the sub-index cache, we have an MRU replacement policy specific to the sub-index cache.

Strategy 2: In this strategy, the index cache, $C_{I}$, is not further divided uniformly between the $m$ hash functions, as shown in Figure 3(b). We have a single MRU replacement policy for the index cache that stores (and if needed, evicts) index files from different hash functions.

In both strategies, there is a separate MRU replacement policy for the data cache, $C_{D}$. The intuition behind Strategy 1 is that each projection will use the cache uniformly. On the other hand, the intuition behind Strategy 2 is that some projections might require bringing more hash buckets into the cache than others (which would be the case when the data distribution is skewed or the usage of individual projections is different).

5.2. Cost Models of qwLSH

We have seen in Figures 1 and 2 that the cardinality and dimensionality of the dataset affects the size of index files and data files that need to be read into the cache respectively. In order to determine the appropriate split between the index cache, $C_{I}$, and the data cache, $C_{D}$, we train a model based on the size of index files and data files that are read from the external storage for different settings (cardinality and dimensionality) on the DeepSat dataset. Figure 4 shows the optimal setting of the index cache size for different dataset characteristics that returns the least amount of total IO. This model validates our observation that different data characteristics utilize the cache differently. For instance, datasets with low dimensionality (=50) require more index cache utilization ($\geq$80%) since the data points are small in size. On the contrary, datasets with low cardinality ($\leq$100K) and high dimensionality ($\geq$500) require less index cache utilization since the data points are large in size and the number of index files needed to be brought in memory are low. Hence, it is beneficial to cache more data points than index files to reduce the total amount of IO.

Figure 5 shows the effect of different Index Cache size setting on the IndexIO, DataIO, and the TotalIO for P53 dataset. As the Index Cache size setting increases (i.e. more % of the cache is allocated to the index cache), as expected, the DataIO cost increases while the IndexIO cost decreases. When Index Cache size setting is at 40%, the system incurs the least amount of IO while processing the entire query workload. The model presented in Figure 4 also estimates the Index Cache size setting for P53 dataset to be at 40%. Our model is dependent on the cache size and the number of queries in the query workload. We show in Section 6 that our novel index structure can still adapt for different cache sizes and number of queries.

5.3. Workflow of qwLSH

In order to find the optimal index cache size for efficient utilization of the cache, we have to first generate a training model that shows the behavior of different cache sizes for different dataset settings (Figure 4). Note that, this process is done offline and does not require the knowledge of the query workloads beforehand. During query processing, once we know the characteristics of the dataset (i.e. the cardinality and dimensionality), we refer to our model to determine the index cache size and the data cache size. Note that, the underlying LSH index (that is stored on the disk) is not changed. In qwLSH, we use a linked list (to decide which objects to evict or store in the cache) and an unordered hash map (for fast retrieval of the objects). In Section 6, we show that this overhead is minimal and the gains achieved from our novel models outweigh the cost of the overhead.

6. Experimental Evaluation

In this section, we evaluate the effectiveness of our proposed index structure, qwLSH. For these evaluations, we use several real data sets with different cardinality and dimensionality, under different system parameters. All experiments were run on machines with the following specifications: Intel Core i7-6700, 16GB RAM, 2TB HDD, and Ubuntu 16.04 operating system. The reported results are an average of 5 runs. We used the state-of-the-art QALSH (Gao:2015:SHC:2783258.2783284, ) as our base implementation333qwLSH can be implemented over any state-of-the-art LSH variant.. All codes were written in C++-11. We implement a cache in the existing QALSH code. For all following alternatives, we use the default settings of QALSH ($c=2$, $w=2.719$, $\delta=1/e$) that are mentioned in (Gao:2015:SHC:2783258.2783284, ).

Since there is no work that directly aims at solving our problem, we compare our work with the following alternatives:

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QALSH_Naive: We compare against the baseline QALSH algorithm.

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QALSH_Ci: In this alternative, we allocate the maximum size of 99% (of the total cache size) to $C_{I}$. Hence, $C_{D}$ will have 1% of the total cache size allocation.

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QALSH_Cd: Here, we allocate the maximum size of 99% (of the total cache size) to $C_{D}$. Hence, $C_{I}$ will have 1% of the total cache size allocation.

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QALSH_CiCd: In this alternative, we equally divide the cache into $C_{I}$ and $C_{D}$, i.e. we allocate 50% (of the total cache size) to $C_{I}$ and 50% to $C_{D}$.

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QALSH_Opt: We try 11 different settings for $C_{I}$ ($C_{I}=1,10,20,...,90,99$) and report the most efficient setting.

6.1. Datasets

In order to train our models, we needed a large dataset with high-cardinality and high-dimensionality. Due to the lack of such datasets, we followed the same technique used to generate the Mnist444http://yann.lecun.com/exdb/mnist/ dataset by downloading 324,000 images from the Deepsat dataset (that consists of airborne images of different land surfaces)555https://www.kaggle.com/crawford/deepsat-sat6/home. Each image has 2828 pixels. For each pixel, we store the RGB and Near-Infrared value, and hence for each image, we get a 2828*4=3136-dimensional point. From this dataset, we generated 20 different datasets (with different cardinalities and dimensionalities) to create our model (Figure 4). In order to test our model and the effectiveness of qwLSH, for our experiments, we used the following four commonly used real high-dimensional datasets:

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Audio666http://www.cs.princeton.edu/cass/audio.tar.gz This dataset consists of 54387 192-dimensional points. It consists of human-labled sound clips.

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LabelMe777http://labelme.csail.mit.edu/Release3.0/browserTools/php/dataset.php This dataset consists of 181093 512-dim. points.

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P53888https://archive.ics.uci.edu/ml/datasets/p53+Mutants This dataset consists of 31008 5409-dimensional points. Since our test dataset consisted of only 3136 dimensions, we reduce the dimensionality of each point to 3000.

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Sift999http://corpus-texmex.irisa.fr/ This dataset consists of 250,000 128-dim. points.

6.2. Evaluation Criteria and Parameters

Note that, our goal is not to improve the accuracy of the queries in the query workload, but to improve the total runtime of the query workload. Since we do not change the logic of the LSH algorithm, the accuracy of each query executed in qwLSH is the same as the accuracy of the underlying LSH algorithm. Hence, due to space limitations, we do not report the accuracy of the individual queries. For that, we ask the reader to refer to the QALSH paper (Gao:2015:SHC:2783258.2783284, ). We evaluate the effectiveness of qwLSH by comparing the size of the data (i.e. the index files and the data objects) that is needed to be brought into the cache. We do not report the index size or the index construction cost, since they would be the same as the underlying LSH implementation that we use, which in our case is QALSH (Gao:2015:SHC:2783258.2783284, ). We focus on one specific criterion: size of the data that is needed to be brought into the cache.

On our test datasets, we found that Strategy 1 and Strategy 2 (Section 5.1) gave similar results (mostly because of the fact that QALSH treats each projection as the same, i.e., the number of hash buckets brought into the main memory is the same for each projection). Due to space constraints, we only evaluate Strategy 1 and compare with the alternatives presented earlier in this section.

Table 1 shows our default parameter settings and the different ranges that we consider. We generate query workloads by choosing queries from dense regions of the dataset (that reflects our motivation and prior works (Aly:2015:AAQ:2831360.2831361, )). We chose the maximum cache size setting based on (Sundaram:2013:SSS:2556549.2556574, ).

6.3. Discussion of the Performance Results

We first show the effect of different values of top-k for the Audio and the P53 dataset in Figures 6 and 7 respectively. From these figures, it can be seen that k does not have a noticeable effect on the performance of the algorithms. Hence, in further charts, we only consider k=50 due to space limitations and for simplicity purposes. In our experiments, we also observed that the alternative, QALSH_Cd, always gave the worst result (because of insufficient space to cache the index files). Hence we also omit QALSH_Cd from the charts.

In Figure 8, we compare qwLSH against all alternatives on the 4 real datasets for the default settings [16MB cache, —Q—=250]. While for Audio, LabelMe, and Mnist, our proposed model always matches with the optimal setting (QA_Opt), QA_Opt for P53 returns a slightly lower IO than qwLSH (154.66 vs. 157.22). This is because our model returns an Index Cache size setting of 45.41%. Since we only compare the setting sizes with increments of 10 (i.e. for 40% and 50%), our model does not return the most optimal answer. Even with this small drawback, the difference between the IOs is small. For LabelMe and Sift, the Index Cache size setting of 99% is always the most optimal because the IndexIO cost is always very dominant. Due to this reason (and space limitations), we only present the results for Audio and P53 datasets in our next experiments.

**Effect of Varying Total Cache Size: **Figure 9 shows the effect of different cache sizes on qwLSH and its proposed models. It can be seen that our proposed model can adapt to different cache sizes. While qwLSH does not always return the most optimal setting, it is still always very close to the optimal setting. For some scenarios (e.g. Audio, 8MB), the optimal answer is close to QA_Ci, whereas for some scenarios (e.g. P53, 20MB), the optimal answer is closer to QA_CiCd. In both these scenarios, qwLSH is able to adapt and return results closer to the optimal setting.

**Effect of Varying Number of Queries in the Query Workload: **Figure 10 shows the effect of different number of queries in the query workload. It can be seen from Figure 10 that the optimal answer for Audio is closer to QA_Ci, whereas the optimal answer for P53 is closer to QA_CiCd. qwLSH can always adapt and return results closer to the optimal setting.

**Overhead of qwLSH’s Cache Implementation: **Figure 11 shows the negligible overhead of qwLSH when compared with QALSH_Naive for Audio and P53 datasets. Even with the caching overhead, qwLSH is still faster than QALSH_Naive because of the efficient cache utilization and the resultant savings in the total IO.

7. Conclusion

In this paper, we presented a novel cache-conscious index structure, qwLSH, for efficient execution of query workloads in high-dimensional spaces. Traditional LSH-based index structures are not designed to efficiently query workloads in high-dimensional spaces. Based on important observations about the effect of cardinality and dimensionality of a dataset on cache utilization, we intelligently divided a given cache during processing of a query workload by using novel cost models. Experimental analysis over different real datasets under different settings showed the effectiveness of qwLSH.