# Generalized Search Algorithm

Author

Quanzheng Long

Last Updated

6 years ago

License

Creative Commons CC BY 4.0

Abstract

A new algorithm can speed up searching anything as a generalization of B/R-Trees

Author

Quanzheng Long

Last Updated

6 years ago

License

Creative Commons CC BY 4.0

Abstract

A new algorithm can speed up searching anything as a generalization of B/R-Trees

Tags

```
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\input{preamble.tex}
\newtheorem{definition}{DEFINITION}
\newtheorem{proposition}{Proposition}
\newtheorem{example}{\textbf{Example}}
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\lstset{
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var, val, until%
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\begin{document}
\title{Generalized Search Algorithm}
\numberofauthors{5}
\author{%
% author names are typeset in 11pt, which is the default size in the author block
{A new algorithm to speed up searching anything as a generalization of B/R-Trees}\\
{Quanzheng Long}
%% add some space between author names and affils
\vspace{1.6mm} \\
\fontsize{10}{10}\selectfont\itshape
% separate superscript on following line from affiliation using narrow space
%$^{\dagger}$
%Purdue University\\%,$^{\ast}$\\
\href{mailto:prclqz@gmail.com}{prclqz@gmail.com}}
\maketitle
\begin{abstract}
Many search algorithms have been proposed in the literature that differ from one another by the distance function used. Typically, the objective is to minimize this distance function. Based on the nature of the distance function, the search algorithm varies. The question is: Can one develop a generic search algorithm independent of the distance function?
%Multi-dimensional algebraic functions are often used to make abstract models for real-world data.
%One common problem is to search for data that has the smallest function value.
Another problem is to search for data whose function values are in a given range. As the dataset is larger, linear search methods
become increasingly more expensive. Existing techniques are only applicable to specific scenarios, or have some limitations. In this paper,
we present an efficient search algorithm and data structure that are applicable to any algebraic distance function. We demonstrate the applicability of our approach in several search problems, e.g., k-nearest-neighbor, aggregate-nearest-neighbors, and
%WGA: need to complete here. Also, need to comment on the performance results, specifically the search time in contrast to tailored search algorithms.
\end{abstract}
\category{H.3.4} {Search}\ {Index}\ {Query processing}
\section{Introduction}
One of the common problems is to search for the data element with the smallest function value.
Consider the following motivating applications.
\noindent
{\bf Example 1 - Determining a server location for clients}: The server location is chosen from a limited set of candidate locations. The optimal server location has the minimum total
%of summary
distance from the server to all the clients. When a company has only one client, the solution is the location nearest to that client.
This problem is solved by a classic kNN algorithm, e.g.,~\cite{kNN}.
%WGA: List other references above.
%Qlong: How can I get more other references?
%WGA: I will add some references later.
However, if a company has more than one client, the problem is out of the scope of the classic kNN algorithm, and the problem becomes an aggregate nearest-neighbor search~\cite{CANN2013}. In this case,
%WGA: Please add this reference: H.G. Elmongui, M.F. Mokbel, W.G. Aref. Continuous aggregate nearest neighbor queries. GeoInformatica 17 (1), 63-95.
the search objective is to minimize the aggregate distance function (sum or maximum) from the server locations to all the client locations, i.e.,
$\sqrt{(x-c_{11})^2+(y-c_{12})^2}+\sqrt{(x-c_{21})^2+(y-c_{22})^2}$,
where
$<c_{11},\ c_{12}> and <c_{21},\ c_{22}>$ are the locations of the two clients, $<x,y>$ is the location of a candidate server.
Tailored Group Nearest Neighbor (GNN) algorithms (e.g., see~\cite{gnn,CANN2013}) exist to specifically address this scenario.
%WGA: Need to comment about the class of kNN algorithms whose distance function is measured over a road network and hence does not use Euclidean distance but rather distance over the network. There are many such algorithms.
%Qlong: Addressed
The question is: Instead of a tailored search algorithm per distance function, can we develop a generalized search algorithm that works regardless of which distance function used?
{\bf Example 2 -
Searching for speeding vehicles in a given time interval and a given
region}: In this scenario, the input contains the locations and corresponding times of
all vehicles. The objective speed function is:
$\frac{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}{t_2-t_1}$,
where $<x_1,\ y_1>$ is the location at Time $t_1$, and
$<x_2,\ y_2>$ is the location at Time
$t_2$. The problem is to search for any vehicle whose objective function value is
greater than the maximum speed limit.
%WGA: State what is it that you are trying to find. For example, find ... that maximizes ... or the search function is > ...
%Qlong: Addressed
Some indexing structures and algorithms e.g., the TB-Tree and STB-Tree\cite{tbtree},
are already designed specifically for this scenario. Again, the question is: Can we formulate the above query as a search problem with an objective function that needs to fall in a given range and apply the same generalized search algorithm instead of building tailored indexing structures and search algorithms when only the distance function is what is different?
{\bf Example 3 -
Customized recommendations}:
Internet companies, e.g., Airbnb or Yelp provide search capabilities for the items most ``relevant" to their customers. The relevant value of an item is often calculated by certain algebraic functions that take care of many factors, e.g., the distance to the customer and the number of starred reviews. The weight value for each factor can be customized. Therefore, the objective function can be expressed as follows:
$W_1*\sqrt{(x-c_1)^2+(y-c_2)^2}+ W_2 * z $,
where $<x,\ y>$ is the location of a candidate, $z$ is its number of starred reviews, $<c_1,\ c_2>$ is the location of a customer, and $W_1\ and\ W_2$ are the customized weight values for that customer.
%Qlong: is there any paper designed for this problem?
%WGA: I am not sure. But we can refer to any of the papers on recommender systems. You can pick one from Mokbel's papers on the subject of recommender systems.
The three example applications above demonstrate the need for search using different algebraic functions.
%WGA: State what...
%Linearly searching through a data set can be very expensive.
%Qlong: addressed
Linearly searching through the dataset and applying the objective function for each data item is expensive.
Indexing techniques, e.g., the M-tree~\cite{m-tree}, speeds up the search by avoiding this linear search.
However, one limitation of the M-tree is that it applies only to the metric space and depends heavily on the triangular inequality.
This paper investigates
the problem of searching datasets given arbitrary algebraic distance functions,
and proposes a unified algorithm that has
overcome
limitations of traditional search techniques.
%Specifically,
%WGA: I rewrote and rephrased the contribution paragraphs below. Please verify.
More specifically, we focus on efficient ways for answering the top-k and range
query problems when using generic algebraic functions. Distance functions in metric and non-metric spaces are both supported.
%searching for tuples with the k-smallest (similarly,
%K-largest and k-nearest) function values (i.e., ) . and tuples with
%values in a given range (i.e., the range-query problem).
%problems are also investigated.
Given the various objective functions, one important target is to avoid as much as possible
accessing data that does not contribute to the query results.
%WGA: Can we claim that the solution works for both metric and non-metric spaces?
%QL: Addressed
%WGA: Write a sentence here that reflects a summary of the findings of the experiments.
%Qlong:Addressed
The results of the conducted extensive experiment demonstrate that the I/O requests are much less than linear searching technique.
The rest of this paper proceeds as follows. Section~\ref{sec-prelim} formally defines the generalized algebraic function search problem and provides some
background material.
Section~\ref{sec-alg} presents the generalized search algorithm and analyzes its time complexity.
Section~\ref{sec-rstar} introduces an improved R*-tree that enhances the performance of the proposed generalized search algorithm.
Section~\ref{sec-experiment} presents experiment results.
Section~\ref{sec-summary} concludes the paper.
\section{Preliminaries}
\label{sec-prelim}
In this section, we define the algebraic search problem formally.
We overview some necessary background material including
spatial indexing and partial derivatives that will facilitate the explanation of the proposed algorithm in Section~\ref{sec-alg}.
\subsection{Generalized Search using Algebraic Distance Functions}
Consider an n-dimensional dataset $S$ that contains $N$ points (vectors), i.e., each vector data point,
say $x \in S$,
has $n$ dimensions, $x=<x_1,\ ...,\ x_i, ...,\ x_n>,$
where $i=1, \cdots, n$.
An algebraic ranking function $f(x)$ is defined on $x$ for all the three motivating examples in the introduction after representing the variables in vector format.
\begin{flushleft}
\begin{small}
\label{fomula-serverlocation}
$f(x)=\sqrt{(x_1-c_{11})^2+(x_2-c_{12})^2}+\sqrt{(x_1-c_{21})^2+(x_2-c_{22})^2}$
\end{small}
$f(x)\frac{\sqrt{(x_2-x_1)^2+(
%4_2
x_4-x_3)^2}}{x_6-x_5}$
$f(x)=W_1*\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}+ W_2 * x_3 $
\end{flushleft}
The search problem is formalized in an algebraic form. In this section, for simplicity, we only define the k-smallest and range searches. Other search types, e.g., k-nearest search, are discussed further in Section~\ref{sec-alg}.
\begin{sloppypar}
\begin{definition}
(\textbf{k-smallest search}) Given an integer number k and a dataset $S$, find the data points (vectors) in $S$ that have the k smallest function values of f(x). Formally, the search result denoted as KS(S,f(x),k), is a set of vectors that $\forall o \in KS(S,f(x),k), \forall s \in S-KS(S,f(x),k), f(o)\leq f(s)$
\end{definition}
\end{sloppypar}
\begin{sloppypar}
\begin{definition}
(\textbf{range search}) Given two values {\em LEFT} and {\em RIGHT}, where {\em LEFT} $\leq$ {\em RIGHT}, and Dataset $S$, find all data points (vectors) in $S$ whose function values are between {\em LEFT} and {\em RIGHT}. Formally, the search result, denoted as $RT( S, f(x), LEFT, RIGHT)$, is a set of vectors that $\forall o \in RT( S, f(x), LEFT, RIGHT ), LEFT \leq f(o)\leq RIGHT$.
\end{definition}
\end{sloppypar}
\subsection{Partial Derivative}
To study the properties of generic algebraic functions, their partial derivatives are used. Intuitively, a partial derivative represents how fast the function value increases (when the derivative is positive) or decreases (when the derivative is negative) on a given dimension (the derivative dimension).
%Formally,
The derivative to dimension $x_i$ is denoted by $f'_{x_i}$.
\begin{center}
$f'_{x_i} = \lim_{h->0} \frac{f(x_1,...,x_i+h,...,x_n)-f(x_1,...,x_i,...,x_n)}{h}, $
$\forall i, \ 1\leq i \leq n $
\end{center}
\textbf{COROLLARY 1} When $f'_{x_i}\geq 0$, f(x) is not decreasing as $x_i$ increases; when $f'_{x_i}\leq 0$, f(x) is not increasing as $x_i$ decreases.
%Formally, given
When $f'_{x}\geq 0$, $\forall x_i, x_j, \in [MIN,\ MAX]$ such that $MIN\leq x_i\leq x_j\leq MAX,\ then\ f(x_i)\leq f(x_j),$ And vise versa.
\bigskip
For example, consider the partial derivatives of the objective function in Motivating Example 1.
\begin{flushleft}
\begin{small}
$f(x)=\sqrt{(x_1-c_{11})^2+(x_2-c_{12})^2}+\sqrt{(x_1-c_{21})^2+(x_2-c_{22})^2}$
\end{small}
\end{flushleft}
$f'_{x_1} = \frac{x_1-c_{11}}{\sqrt{(x_1-c_{11})^2+(x_2-c_{12})^2}}+\frac{x_1-c_{21}}{\sqrt{(x_1-c_{21})^2+(x_2-c_{22})^2}}$, and
$ f'_{x_2} = \frac{x_2-c_{12}}{\sqrt{(x_1-c_{11})^2+(x_2-c_{12})^2}}+\frac{x_2-c_{22}}{\sqrt{(x_1-c_{21})^2+(x_2-c_{22})^2}}$.
%Matured mathematical algorithms and systems, such as Maxima\cite{maxima}, is able to compute the partial derivative formulas of a given function.
Using the partial derivative formula, we calculate the \textbf{value ranges} of derivatives in postfix notation (Reverse Polish notation)\cite{reverse_polish_notation}. These notations and the procedures to calculate them are presented in the next section.
%WGA: Why is this needed ? and why is the postfix notation needed here? You need to have a justification.
\subsection{Value Range}
\begin{sloppypar}
\begin{definition}
(\textbf{Value Range:}) The value range is a set of pairs of values $[vmin, vmax]$, where $vmin \leq vmax$. A value range of a derivative means that the value of the derivative is inside the range designated by one of the pairs.
More specifically, let $VR$ be a value range. Then,
$VR=\{[vmin_1,vmax_1],\ ...,\ [vmin_t,vmax_t]\}$,
where $vmin_i\leq vmax_i\ for\ 1 \leq i \leq t$. The value range of $f'_{x_i}$ means that $\exists 1 \leq j \leq t, vmin_j\leq f'_{x_i} \leq vmax_j $.
\end{definition}
\end{sloppypar}
We devise a procedure to calculate the value range of a derivative. Given a mathematical formula for a partial derivative, and the value ranges of its variable,
%WGA: You have not defined what the value ranges of a variable (not a function) mean.
the procedure is to use the postfix notation\cite{reverse_polish_notation}.
The procedure is similar to calculating a exact value of a mathematical expression given exact variable values. However, the mathematical operators take different behaviors when processing value range calculation.
Therefore, we redefine the mathematical operators for value range calculation. Only the procedures of addition and multiplication operators (+ and *) are presented for simplicity, as shown in Algorithm \ref{alg-vr}. Other operators, such as division or sine, are not difficult to design.
\begin{algorithm}
\caption{Operations for value range }
\label{alg-vr}
\begin{algorithmic}[1]
\Procedure{plus}{VR1,VR2}
\State initialize VR
\For {pair( vmini, vmaxi) in VR1}
\For {pair( vminj, vmaxj ) in VR2}
\State VR.add( vmini+ vminj, vmaxi+ vmaxj)
\EndFor
\EndFor
\State \textbf{return} VR
\EndProcedure
\Procedure{multiply}{VR1,VR2}
\State initialize VR
\For {pair( vmini, vmaxi) in VR1}
\For {pair( vminj, vmaxj ) in VR2}
\State vmin = min ( vmini * vminj, vmini * vmaxj, vmaxi * vminj, vmaxi * vmaxj )
\State vmax = max ( vmini * vminj, vmini * vmaxj, vmaxi * vminj, vmaxi * vmaxj )
\State VR.add( (vmin, vmax) )
\EndFor
\EndFor
\State \textbf{return} VR
\EndProcedure
\end{algorithmic}
\end{algorithm}
\noindent
{\bf Example}: \\
Calculating the derivative of $fx_1$, \\
$f'x_1 = x_1 * (x_1 + x_2)$.
Initially, $VR(x_1) = \{(-1, 2)\}$, $VR(x_2) = \{(-1, 2)\}$.
Then, $VR(x_1 + x_2)=\{(-2, 4)\}$.
Finally, $VR(f'x_1) = \{(-4, 8)\}$.
%WGA: This example is not clear. Please explain further. How is VR(F') calculated? Do we need to have the second derivative?
\bigskip
Notice that
the value range
may get amplified, i.e., the value range calculated may become enflated and wider than the actual value of the expression. This is expected and does not affect the proposed algorithm.
To avoid this amplification, we can try to simplify the formula. The proposed algorithm only cares whether or not the value range is non-negative or non-positive. For example, consider the Euclidean distance function $f(x)=\sqrt{x_1^{2}+x_2^{2}}$. The derivative with respect to
%WGA: with respect to what?$x_1$? $x_2$? please advise.
i.e., $f'_{x_1} =\frac{x}{\sqrt{x_1^{2}+x_2^{2}}}$. However, we use $g'_{x_1}=x$ instead as an equivalent function because $\sqrt{x_1^{2}+x_2^{2}}>=0$.
\subsection{Multi-dimensional Index And MBR}
To avoid accessing the entire dataset upon search, we use a multi-dimensional index. Any spatial (multi-dimensinoal) indexes, e.g., the R-Tree and its variants~\cite{r-tree}, or Quad-Tree and its variants~\cite{quad-tree}, exist in the literature and in systems. The proposed search algorithm can use any multi-dimensional index as long as the index has the following basic features:
\begin{enumerate}
\item The index groups the data points (vectors) using some \textbf{MBRs} when the vectors are close to each other.
\item An MBR is stored in an index node along with links to other child index nodes.
\item Build a top-down tree architecture where a parent node points to its child nodes.
\end{enumerate}
\bigskip
%WGA: I am not sure we need to state this as it may not necessarily be true.
%Most spatial indexes are developed on top of MBR (Minimum Bounding Rectangle). We are using the same definition as in \cite{knn}.
\begin{sloppypar}
\begin{definition}
\label{def-mbr}
(\textbf{MBR}) An MBR is defined by two end points, $S$ and $T$, of the rectangle's major diagonal.
$MBR\ =\ [S,T], where$
$S = <s_1,...,s_i,...,s_n>,$
$T = <t_1,...,t_i,...,t_n>,$
$\forall x=<x_1,...,x_i,...,x_n>\ under\ the\ node, $
$s_i\leq x_i \leq t_i\ for\ 1\leq i\leq n.$
An MBR has a minimum property that:
%Wen: Should be s_i-\epsilon for x>T' and s_i+\epsilon for x<S'
$\forall \epsilon>0, \forall 1\leq i \leq n, MBR'=[S,T'],T = <t_1,...,t_i-\epsilon, \cdots, t_n>, \exists x=<x_1,...,x_i,...,x_n>\ under\ the\ node\ that\
%x>T'. $
%WGA: What does it mean that x>T'? I am not sure what this means. Probably what you mean is:
x_i>t'_i. $
Similarly, $\forall \epsilon>0, \forall 1\leq i \leq n, MBR'=[S',T], S' = <s_1,...,s_i+\epsilon,...,s_n>, \exists x=<x_1,...,x_i,...,x_n>\ under\ the\ node\ that\
%x<S'. $
x_i<s'_i.$
\end{definition}
\end{sloppypar}
There are four types of index nodes in a tree structured index:
1) An \textbf{Entry node} stores the original data of a row (including vectors and unindexed data). An entry node also has an MBR even there is only one vector in it. Both the two endpoints of the MBR are the exactly the vector stored in the node.
2) A \textbf{Leaf node} stores its MBR and the links to the entry nodes that are inside the MBR.
3) An \textbf{Internal node} stores its MBR and the links to its children nodes (internal or leaf nodes).
4) A \textbf{Root node} is a special internal node that represents the entire space and that has no parents. The root node serves as the starting node to begin the search process at.
Figure~\ref{fig:tree} gives an example multi-dimensional tree structure index.
\begin{figure}
\centering
\includegraphics[width=0.4 \textwidth]{pics/tree2.png}
\caption{\label{fig:tree}A multi-dimensional tree structured index}
\end{figure}
\begin{sloppypar}
\begin{definition}
\label{def-monotones}
%First explain a dimension is monotonic iff VR(f'(x_i)\leq 0) for all x in [min, max] , or (f'(x_i)\geq 0) for all x in [min, max].
%Then explain a index node is monotonic iff it's monotonic on every dimension.
\textbf{(Monotonicity)} An index node is \textbf{monotonic} w.r.t. $f(x)$ if and only if the node's MBR is monotonic w.r.t. $f(x)$ in all the dimensions. For each dimension $1\leq i \leq n$ of vector $x$, $MBR=[S,\ T]$ is \textbf{monotonic} w.r.t. $i$ if and only if $any\ S\leq x\leq T$, $f(x)$ is derivable w.r.t. $x_i$, $and\ VR(f'_{x_i})\leq 0 or\geq 0 $.
\end{definition}
\end{sloppypar}
Definition~\ref{def-monotones} is a principal component of this paper. When an index node, say $P$ is monotonic, the algorithm can calculate distance metrics and function measurements about the entire node $P$ without havinig to access any of the vectors (data points) in $P$ in case $P$ is a leaf node, and without having to access the child nodes of $P$ in case $P$ is a non-leaf node. We discuss the monotonicity propery further in the next section.
\section{Search Algorithm}
\label{sec-alg}
In this section, we present details of k smallest search algorithm. The k-nearest and range search algorithms are also discussed briefly at the end of the section.
First, we present the metrics used by the search algorithm to prune unnecessary branches during the search. Then, we outline and discuss the pseudo code of the algorithm.
%The metrics are based on the preliminaries given above.
\subsection{Metrics}
As discussed in the previous section, an MBR represents the domain of all data vectors under the corresponding node of the MBR. To avoid accessing the original data vectors (the entry nodes), two metrics of the node, MINVAL and MINMAXVAL, explained below, are computed.
\begin{sloppypar}
\begin{definition}
(\textbf{MINVAL}) Let $nd$ be an index node, p be a data vector in an entry node, $q_i$ be the corners of an \textbf{MBR}, $1\le i\le 4$, and $ch_j$ be the children nodes of $nd$, $1\le j \le c$. The MINVAL (minimum value) of nd, denoted by $MINVAL(nd,f(x))$, is computed as follows:
\begin{eqnarray}
\begin{cases}
= f(p)& if\ p\ is\ an\ entry\ node; \cr
= min_{i=1,4}(f(q_i)) & if\ p\ is\ monotonic; \cr
= min_{1 \le j \le c}(MINVAL(ch_j,f(x))) & otherwise.
\end{cases}
\end{eqnarray}
\end{definition}
\end{sloppypar}
\begin{sloppypar}
\begin{definition}
\label{def-mbr-corner}
(\textbf{MBR corners}) The corners of an MBR=(S,T) is a set of vectors, $\{x|<x_1,...,x_i,...,x_n>\},\ where\ 1\leq i \leq n,\ x_i=s_i\ or\ t_i$
\end{definition}
\end{sloppypar}
%WGA: I am not sure that taking the two corners only is the correct thing. The min or max can be in any of the other two corners. Please let me know what you think.
The above definition is explained as follows. When $nd$ is an entry node, $MINVAL$ is the value of the function $f(x)$. When the MBR of $nd$ is monotonic w.r.t. $f(x)$, $MINVAL$ is the minimum of all the \textbf{corners of the MBR}. If neither of the two cases apply, then $MINVAL$ is calculated as the minimum of all the $MINVAL$s of the child nodes.
\bigskip
\textbf{THEOREM 1}
The function value of any vector under Node $nd$ is greater than or equal to $MINVAL(nd,\ f(x))$.
\bigskip
\textbf{PROOF}:
According to Definition~\ref{def-monotones}, there are the following three cases to consider.
\noindent
Case 1) If $nd$ is an entry node, there is only one vector $p$ under $nd$. Hence, the function value under $nd$ is equal to $f(p)$.
\noindent
Case 2) When $nd$ is monotonic w.r.t. $f(x)$, this means that when $x_i$ increases for $\forall 1\leq i\leq n$, $f(x)$ consistently increases or decreases over the area covered by the MBR of node $nd$. Therefore, the minimum value for $f(x)$ occurs at one of the corners of the MBR and is smaller than or equal to the function value of any point inside the MBR. Thus, $MINVAL$ is actually the minimum over all function values of all the corners of $nd$'s MBR.
\noindent
Case 3) When Node $nd$ is not monotonic, we need to apply $MINVAL$'s definition recursively into $nd$ child nodes. In this case, $MINVAL$ will be the minimum of all $MINVAL$s of the child nodes. Hence, the function value of any data vector under $nd$ is still greater than or equal to $MINVAL$.
Thus, $MINVAL$ of a node $nd$ is the lower-bound of the all the function evaluations of function $f$ over all the data vectors under the subtree rooted at $nd$.
%MINVAL guarantees that every vector is greater or equal to it.
%WGA: the above sentence is repetitive.
Similarly, \textbf{MINMAXVAL} serves as an upper-bound that guarantees that there exists at least one data vector in the underlying subtree of which the function value is smaller than or equal to MINMAXVAL.
%WGA: Need to fix the next statement. One suggestion is:
In order to compute MINVAL and MINMAXVAL for all the nodes in the entire tree, we make use of one of the properties of MBRs, namely that for each face of an MBR, there is at least one data vector that touches that face, i.e., is inside the face
%WGA: BTW, this statement is not true for quadtrees because they are space partitioning trees, so it can be that an MBR for the points in the space does not touch any of the partitions. Talk to me if you want to discuss further.
%original sentence: The basic idea of construction is to use the property of MBR. Imagine the faces of an MBR, and for each face, there exists at least one vector inside the face
-- otherwise we can reduce the size of the MBR. Therefore, when a non-entry node is monotonic w.r.t. $f(x)$, the proposed search algorithm calculates the maximum values of all the faces, and the minimum of these maximum values is $MINMAXVAL$.
\begin{sloppypar}
\begin{definition}
\label{def-minmaxval}
(\textbf{MINMAXVAL}) Let $nd$ be an index node, $p$ be the data vector in an entry node, and $ch_j$ be the children nodes of $nd$, $1\le j \le c$.
The MINMAXVAL (mini-maximum value) of $nd$, denoted by $MINMAXVAL(nd,f(x))$, can be calculated as follows:
\begin{eqnarray}
\begin{cases}
= f(p),& if\ p\ is\ an\ entry\ node; \cr
= min( max( f(corners\ of\ \textbf{MBR faces}) ), & if\ MBR\ is\ monotonic; \cr
= min_{1 \le j \le c}(MINMAXVAL(ch_j,f(x))) &otherwise;
\end{cases}
\end{eqnarray}
\end{definition}
\end{sloppypar}
\begin{sloppypar}
\begin{definition}
(\textbf{MBR faces}) An MBR face is an (n-1) dimensional MBR. There are $2n$ faces. Each face has $2^{n-1}$ corners, because there are $2^{n-1}$ combinations of choosing S or T.
Formally, for the $j^{th}$ corner of $i^{th}$ face, corner $q=<q_1,...,q_k,...q_n>,$ where
$q_k$=
\begin{eqnarray}
\begin{cases}
s_k, &k=(i+1)/2\ AND \ i\ mod\ 2 = 1;
\cr t_k, &k=(i+1)/2\ AND\ i\ mod\ 2 = 0;
\cr s_k\ or\ t_k, &depends\ the\ \textbf{MBR\ corners};
\end{cases}
\end{eqnarray}
where $1\leq i \leq 2*n,1\leq j \leq 2^{n-1},1\leq k \leq n$.
\end{definition}
\end{sloppypar}
\bigskip
\textbf{THEOREM 2}
There exists at least one data vector under node $nd$, whose function value is less than or equal to $MINMAXVAL(nd,f(x))$.
\bigskip
\textbf{PROOF}:
According to Definition~\ref{def-minmaxval}, there are the following three cases to consider.
\noindent
{\bf Case 1)} If $nd$ is an entry node, there is only one data tuple $p$ under $nd$. Hence, the function value of any data tuple under $nd$ is equal to $f(p)$.
\noindent
{\bf Case 2)} According to the minimum property of MBR in Definition~\ref{def-mbr}, there exists at least one vector on each face
%WGA: that satisfies what? Please state.
%.
Also, according to Definition \ref{def-monotones}, when $nd$ is monotonic w.r.t. $f(x)$, then $f(x)$ is consistently increasing or decreasing over all the MBR's faces. Therefore, for any face, there exists a vector whose function value is less than or equal to the maximum of MBR face corners' function values. MINMAXVAL is chosen from one of the MBR faces hence there is always existing a vector whose the function value is less than or equal to MINMAXVAL.
\noindent
{\bf Case 3)} When the node is not monotonic, the MINMAXVAL is a minimum of all MINMAXVALs of the child nodes. Hence, there exists at least one vector under nd with the function value less than or equal to the MINMAXVAL.
\bigskip
Note that there is a more efficient method for calculating MINMAXVAL than the one implied by the definition. This method is more efficient because it avoids iterating over all possible corners of each face. There are $2^{n}$ corners in an MBR. However, each face has half of the MBR's corners ($2^{n-1}$). To obtain the maximum value of the corners of each face, sort all the corners based on their function values in increasing order. Then, for each face, iterate in that order and choose the first corner that belongs to the face. The function value of that first corner is the maximum value of corners for the face.
\bigskip
\textbf{THEOREM 3}
Given two index nodes $nd_1$ and $nd_2$, if MINVAL($nd_1$,f(x))>MINMAXVAL($nd_2$,f(x)), then the vector with the smallest function value must be under node $nd_2$.
\bigskip
\subsection{Pseudo Code}
In this section, we present the pseudo code of the k-smallest search algorithm. We assume that the dataset has at least k vectors so that edge checking is ignored.
As Algorithm \ref{alg-kss}, it has two main steps to get the top-k tuples: to search for smallest value, then search for the remaining smallest values.
\begin{algorithm}
\caption{k-smallest-search}
\label{alg-kss}
\begin{algorithmic}[1]
\Procedure{KSmallestSearch}{$k,root,f(x), T$}
\State $init lstStk,minStk,actLst$
\State $actLst.put(root)$
\State $SmallestSearch(k,f(x),T,actLst,minStk,lstStk)$
\State $RemainingSearch(k,f(x),T,actLst,minStk,lstStk)$
\State \textbf{return} $actLst$
\EndProcedure
\end{algorithmic}
\end{algorithm}
Initially, an active list is initialized for R-Tree nodes. The root node is put into the active list. Then recursively execute the following process until reaching entry nodes.
1) Retrieve all child nodes of nodes in the active list;
2) Visit the child nodes and compute their MINVALs and MINMAXVALs;
3) Based on the metrices and Theorem 3, prune some nodes to avoid further access.
4) Since the pruned nodes may be used later, they are pushed into a stack.
The details are shown in Algorithm \ref{alg-ss}.
\begin{algorithm}
\caption{smallest-search}
\label{alg-ss}
\begin{algorithmic}[1]
\Procedure{SmallestSearch}{
$k,f(x),T,actLst,minStk,lstStk$}
\While{$NonEntry(actLst.first())$}
\State $init tmpLst$
\For{$ node\;n\;in\; actLst$}
\For{$ cn\; in\; node.children$}
\State{$tmpLst.add(cn)$}
\EndFor
\EndFor
\State{$actLst.clear()$}
\State{min\_MINMAX = min ( MINMAXVAL ( tmpLst.nodes, f(x), T ) )}
\For{$ n\; in\; tmpLst$}
\If{min\_MINMAX >= MINVAL ( n, f(x), T )}
\State{actLst.add(n)}
\State{tmpLst.del(n)}
\EndIf
\EndFor
\If{tmpLst.size()>0)}
\State{min\_MIN = min ( MINVAL ( tmpLst.nodes, f(x), T ) )}
\State{minStk.push(min\_MIN)}
\State{lstStk.push(tmpLst)}
\EndIf
\EndWhile
\EndProcedure
\end{algorithmic}
\end{algorithm}
At the end of the first step, there might be more than one vector that have the same smallest function value. Assume m vectors, where $1\leq m\leq k$.
In the second step, the algorithm looks for the remaining k-m vector with the smallest function values. The pruned nodes are visited one by one, in order of popping from the stack, until obtaining the remaining k-m vectors. We skip the pseudo code for simplicity. It is similar to Algorithm \ref{alg-ss}, and not difficult to implement.
\bigskip
This algorithm exploit the usage of index. The key idea is to calculate MINVALs and MINMAXVALs for all index nodes visited, only using their MBRs.
\subsection{Other Search Query Types}
Other types of algebraic function searches, such as range and k-nearest searches are also supported.
Range search is much more straightforward than k-smallest search. In addition to MINVAL, MAXVAL is defined similarly. It guarantees that every function value of vectors in the nodes is less than or equal to MAXVAL. The result to range search query is also retrieved by visiting nodes in active list from top to bottom. In the meantime of visiting, prune nodes that are out of the queried range until all nodes are in the range.
\begin{sloppypar}
\begin{definition}
(\textbf{k-nearest search}) Given a target value T and an integer number k, search for the k tuples out of a dataset DS, with the function values of f(x) nearest to T. Formally, the search result denoted as KN(DS,f(x),T,k), is a set of tuples that $\forall o \in KN(DS,f(x),T,k), \forall s \in DS-KN(DS,f(x),T,k), |f(o)-T|\leq |f(s)-T|$
\end{definition}
\end{sloppypar}
\textbf{THEOREM 4}
A k-nearest search problem can be converted into a k-smallest search problem.
\bigskip
\textbf{PROOF}
Construct another function g(x)=|f(x)-T|, then the k smallest search for g(x) is equivalent to the k nearest search for f(x) with the target value T.
\bigskip
Theorem 4 shows that we can use the k smallest search algorithm to solve the k nearest search problem. However, a function expression in a form of $g(x)=|f(x)-T|$ is not easy and efficient to apply the solution. Because when computing the metrics for an index node, the algorithm requires the formulas of partial derivatives. Since $g(x)=|f(x)-T|=\sqrt{(f(x)-T)^{2}}$, the partial derivatives obtained through this formula might be very complex.
To overcome this difficulty, two different metrics for the k nearest search, MINDIST and MINMAXDIST can be designed to replace MINVAL and MINMAXVAL. MINDIST is the lower bound to replace MINVAL, and MINMAXDIST is the upper bound in place of MINMAXVAL. They are more efficient to compute than MINVAL and MINMAXVAL. The definitions are trivial and can be written similarly.
\subsection{Algorithm Performance}
The algorithm complexity depends on the quality of the index structure and the nature of the algebraic function applied.
In best case, the algorithm would exploit the usage of index, visiting very few internal nodes to locate the target entry nodes. However in worst case, the algorithm would iterate every nodes as linear scanning.
When a function is not derivable, or the MBRs are not monotonic, the algorithm performs closed to the worst case. For any Weierstrass function\cite{Weierstrass}, the algorithm always performs at the worst case because this type of function is not derivative at all.
Most algebraic functions in real life are derivable. However, some algebraic function, such as application example 2, performs badly in some dataset even the function is derivable. Because whether the MBRs are monotonic or not heavily depends on the dataset used, and the multi-dimensional index applied.
One of the well known data structure for multi-dimensional data is R*-tree. It provides balanced structure and splitting rectangles with very little overlap. It performs well for lots of common algebraic functions. However there are still some potential improvements.
In next section, we investigate the bad performace issue of the R*-Tree and propose improvements on the R*-Tree to make it works better for some difficult case like speed function.
\section{Improved R*-Tree}
\label{sec-rstar}
R*-Tree\cite{rstar} is one of most popular data structure for multi-dimensional data. It is developed from R-Tree\cite{r-tree} as a member of the R-Tree family.
The original R-Tree is an natural extension of B-Tree. Similarly, it provides insert/delete/search operations. When inserting a new item to a R-Tree, choosing a node to insert and how to split a full node are two critical problems. The two problems lead researchers to develop different members of R-Tree family. Among those members, R*-Tree is proved to be one of the best members providing good quality of choosing and splitting nodes.
\subsection{Root Causes of Bad Performance}
\label{sec-root-cause}
When applying R*-Tree to our algorithm, most of the algebraic functions perform well. The algorithm can save at least 80\% of disk page accessing in most cases. However, some kind of algebraic functions are not able to avoid too many disk accessing as expected. Such as the speed function in motivating application example 2:
$f(x)=\frac{\sqrt{(x_1-x_2)^2+(x_3-x_4)^2}}{x_5-x_6}$.
One of the root causes is $\frac{1}{x_5-x_6}$. That leads the algorithms of choosing inserting nodes and splitting nodes perform badly.
Assume that
$t_{11}<=x_6<=t_{12}$,
$t_{21}<=x_5<=t_{22}$,
$t_{01}<=x_5-x_6<=t_{02}$,
What if $t_{01}<0$ and $t_{02}>0$? Then $\frac{1}{x_5-x_6}$ reaches $+\inf$ and $-\inf$. Hence the MBR with this value range is not monotonic.
To avoid this from happening, we conduct the following.
Let $t_{01}<=0, t_{02}<=0$ or $t_{01}>=0,t_{02}>=0$, then
$t_{12}>=t_{11}>=t_{22}>=t_{21}$ or
$t_{22}>=t_{21}>=t_{12}>=t_{11}$.
Intuitively, an MBR is better not across line $x_5=x_6$ as shown in Figure \ref{fig:across}.
\begin{figure}
\centering
\includegraphics[width=0.4 \textwidth]{pics/across.png}
\caption{\label{fig:across}an MBR is better not across line $t_2=t_1$}
\end{figure}
The R*-Tree and other implementation of R-Trees, do not take care the factor that whether or not a MBR should across a line. Consequently, lots of MBRs are not monotonic.
Another root cause is that, every time splitting a R*-Tree node, it only affect one dimension. As more dimensions are used, it is less possible that the algorithm would split one specific critical axis. However, some axis's might be more promising to make a monotonic MBR, than other axis's.
To address those issues, we propose two improvements on R*-Tree. One improvement is on choosing subtree to insert new items, and another is on choosing split axis and index.
The final root cause we discover is that some algebraic functions have stronger locality than some others. A function with weak locality may require some distribution in the dataset to have very good performance. A strong locality means that the vectors closed to each other always have some closed function values, and the vectors far way may have very different function values.
For example, we consider $f(x)=x_1*x_2$ has stronger locality than $f(x)=sin(x_1*x_2)$. Therefore, algebraic function $f(x)=sin(x_1*x_2)$ only has good performance when the dataset is very dense. Instead $f(x)=sin(x_1*x_2)$ doesn't have that request.
\subsection{Choosing Subtree}
When choosing a subtree for inserting a new item, the original R*-Tree\cite{rstar} takes into account factors including overlap enlargement, area enlargement and rectangle area. Those factors are beneficial for choosing a subtree that leads to little overlap and area enlargement. However, they might also lead a index node changing from monotonic to non-monotonic for certain functions.
Therefore we propose that choosing subtree also needs to consider monotones change as an extra factor.
After inserting an item to a subtree node, the subtree node's MBR might have three cases of changing monotones:
1) Change from non-monotonic to monotonic;
2) Keep being monotonic;
3) Keep being non-monotonic;
4) Change from monotonic to non-monotonic;
Above is the preferred order for choosing a subtee. 1) > 2) > 3) > 4). The improving algorithm would choose the subtree in that above order. Then resolve the tier using the original R*-Tree's algorithm.
\subsection{Choosing Split Axis and Index}
When splitting a index node, a splitting axis and a splitting index need to be chosen. The original R*-Tree algorithm has a issue of causing some MBR changing from monotonic to non-monotonic.
Similar to the improvement on choosing subtree for inserting, we propose that the monotones is also considered. For each splitting axis and index, the number of child nodes that are monotonic is recorded. Then choosing the axis and index having the maximum number. Also, we resolve the tier using the original R*-Tree algorithm.
\subsection{Visualized Comparison}
\begin{figure}
\centering
\includegraphics[width=0.25 \textwidth]{pics/orgin-6.png}
\caption{\label{fig:origin-6} The original R*-Tree in $x_5$(horizontal) and $x_6$(vertical) dimensions. Many rectangles cross the dash red line $x_5=x_6$}.
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.25 \textwidth]{pics/improved-6.png}
\caption{\label{fig:improved-6}The improved R*-Tree has few rectangles across line $x_5=x_6$, which leaves an empty line. }
\end{figure}
We conduct an experiment to visually compare the original R*-Tree to the improved R*-Tree. The experiment uses the speed monitoring function, which has six variables (dimensions). 3000 randomly generated elements are inserted into both trees with the same order.
The results are shown in Figure \ref{fig:origin-6} and Figure \ref{fig:improved-6}. We can intuitively observe the rectangles in dimension $x_5$ and $x_6$. The original R*-Tree put lots of rectangles across the line $x_5=x_6$, which let them be non-monotonic. On the other hand, the improved R*-Tree avoid doing that as much as possible, hence Figure \ref{fig:improved-6} has an empty line along $x_5=x_6$.
The visual results give us confidence that the improved R*-Tree can work better than the original one. Consequently, we conduct benchmark tests for comparing their performance.
\section{Performance Experiments}
\label{sec-experiment}
In this section, we study the performance of the proposed data structure and algorithm. Different algebraic functions are applied in different dataset. The number of disk page access is used as the performance metric.
\subsection{Customized Recommendation}
\label{exp:customized-recommendation}
\textbf{Algebraic function}:
$f(x)=W_1*\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}+ W_2 * x_3 $
\textbf{Dataset}:
In this case we use the yelp academic data\ref{yelp-data} as business locations. Each business location contains a pair of longitude and latitude, and also the number of reviews serving as $x_3$. We manually set up different $<c_1,c_2>$ as customer locations, and the weight values for distance and factor $x_3$. The customer location should not be very far away from all candidates. As long as the location is reasonable, the algorithm is always efficient.
\begin{figure}
\centering
\includegraphics[width=0.3 \textwidth]{pics/Reinforcement.png}
\caption{\label{fig:customized-recommendation}Page access for customized recommendation problem via different algorithms}
\end{figure}
\textbf{Result}:
In Figure \ref{fig:customized-recommendation}, the page access grows much slower as the dataset size increases, when using the new algorithm(yellow curve), compared to linear scanning technique( blue curve).
\subsection{Server Location}
\textbf{Algebraic function}:
\begin{small}
$f(x)=\sqrt{(x_1-c_{11})^2+(x_2-c_{12})^2}+\sqrt{(x_1-c_{21})^2+(x_2-c_{22})^2}$
\end{small}
\textbf{Dataset}:
In this case we still use Yelp academic data\cite{yelp-data} as candidate locations for server. Since it is the exact same dataset, we can use the exact same index built for Section \ref{exp:customized-recommendation}.
Also, we manually set up two locations as clients. Those constant can also affect the performance. The two client locations should not be very far away from all candidates. As long as the locations are reasonable, the algorithm is always efficient. Also the number of clients can be vary.
\begin{figure}
\centering
\includegraphics[width=0.3 \textwidth]{pics/Serverlocation.png}
\caption{\label{fig:serverlocation}Page access for server location problem via different algorithms}
\end{figure}
\textbf{Result}:
In Figure \ref{fig:serverlocation}, the page access grows significantly slower using the proposed technique(yellow curve), compared to linear scanning technique(blue curve), as the dataset size increases.
Note that we are using the exact same dataset and index as we use for server location problem. The index is built with more dimensions ($x_1,x_2,x_3$) than the dimensions that are queried($x_1\ and\ x_2$). This suggests that one index can be reused in different algebraic functions. It exploits the usage of index and enables more functionality of database system.
\subsection{Speed Monitoring}
\textbf{Algebraic function}:
$f(x)=\frac{\sqrt{(x_3-x_1)^2+(x_4-x_2)^2}}{x_6-x_5}$
\textbf{Dataset 1}:
In this case we use BerlinMOD data\cite{berlin-mod} for vehicles trips. The dataset contains the start and end of locations and time.
\begin{figure}
\centering
\includegraphics[width=0.3 \textwidth]{pics/Speed-bad.png}
\caption{\label{fig:speed-bad}Page access for speed monitoring problem via different algorithms}
\end{figure}
\textbf{Result}:
In Figure \ref{fig:speed-bad}, the page access grows mostly the same via different techniques. However the improved R*-Tree(gray curve) works better than original R*-Tree(yellow curve) and linear scanning(blue curve).
Section \ref{sec-root-cause} discussed the root causes of the bad performance. One of the cause the nature of the algebraic function. This function has very week locality.
Another cause is the dataset distribution. The speeds are mostly in a small range and in a particular pattern. The start and end time are in a sequential order, which would make the rectangles too large. We verify this cause using a different dataset.
\textbf{Dataset 2}:
In this dataset, we still use BerlinMon, however we change the start and end time. Instead, we use the values of $x_1$ and $x_4$ and start and end times. This would break the $x_5$ and $x_6$ into more random data.
\begin{figure}
\centering
\includegraphics[width=0.3 \textwidth]{pics/Speed-nice.png}
\caption{\label{fig:speed-nice}Page access for speed monitoring problem via different algorithms using random start and end time}
\end{figure}
\textbf{Result}:
In Figure \ref{fig:speed-nice}, our proposed algorithm with improved R*-Tree performs better than with original R*-Tree. And they both work better than linear scanning. This verify that the data distribution can affect the performance.
\section{Conclusion}
\label{sec-summary}
In this paper, we presented the new algorithm for generic algebraic function search. We analyzed the performance and complexity of the algorithm and proposed improvement on data structure. Our extensive experimental results shows the efficiency and availability of the techniques.
Essentially, our new technique is a natural extension of spatial index and classic kNN algorithm. This extension is enabling the database indexing to be more powerful in real-world problems.
\bibliographystyle{abbrv}
\bibliography{sigproc}
\end{document}
```

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