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Diffstat (limited to 'chapters/beyond-dsp.tex')
| -rw-r--r-- | chapters/beyond-dsp.tex | 24 |
1 files changed, 12 insertions, 12 deletions
diff --git a/chapters/beyond-dsp.tex b/chapters/beyond-dsp.tex index 9e6adf0..7632261 100644 --- a/chapters/beyond-dsp.tex +++ b/chapters/beyond-dsp.tex @@ -3,7 +3,7 @@ \begin{center} \emph{The following chapter is an adaptation of work completed in collaboration with Dr. Dong Xie and Dr. Zhuoyue Zhao and published - in PVLDB Volume 17, Issue 11 (July 2024) under the title "Towards Systematic Index Dynamization". + in PVLDB Volume 17, Issue 11 (July 2024) under the title ``Towards Systematic Index Dynamization''. } \hrule \end{center} @@ -248,8 +248,8 @@ using the following interface, a final query result, $R$. \end{itemize} -Let $P(n)$ be the cost of $\mathbftt{local\_preproc}$, $D(n)$ be -the cost of $\mathbftt{distribute\_query}$, $\mathscr{Q}_\ell(n)$ +Let $P(n)$ be the cost of $\mathbftt{local\_preproc}$, $\mathscr{D}(n)$ be +the cost of $\mathbftt{distribute\_query}$, $\mathscr{Q}_S(n)$ be the cost of $\mathbftt{local\_query}$, and $C_e(n)$ be the cost $\mathbftt{combine}$. To solve a search problem with this interface requires calling $\mathbftt{local\_preproc}$ and $\mathbftt{local\_query}$ @@ -258,12 +258,12 @@ $\mathbftt{combine}$ once. For a Bentley-Saxe dynamization then, with $O(\log_2 n)$ blocks, the worst-case cost of answering an eDSP is, \begin{equation} \label{eqn:edsp-cost} -O \left( \log_2 n \cdot P(n) + D(n) + \log_2 n \cdot \mathscr{Q}_\ell(n) + C_e(n) \right) +O \left( \log_2 n \cdot P(n) + \mathscr{D}(n) + \log_2 n \cdot \mathscr{Q}_S(n) + C_e(n) \right) \end{equation} As an example, we'll express IRS using the above interface and analyze its complexity to show that the resulting solution is the -same $\Theta(log^2 n + k)$ cost as the specialized solution from +same $\Theta(\log^2 n + k)$ cost as the specialized solution from Chapter~\ref{chap:sampling}. We use $\mathbftt{local\_preproc}$ to determine the number of records on each block falling on the interval $[l, u]$ and return this, as well as $i_l$ and $i_u$ as the @@ -332,8 +332,8 @@ each of these operations in pseudo-code. } \end{algorithm} -These operations result in $P(n) \in \Theta(\log n)$, $D(n) \in -\Theta(\log n)$, $\mathscr{Q}(n,k) \in \Theta(k)$, and $C_e(n) \in +These operations result in $P(n) \in \Theta(\log n)$, $\mathscr{D}(n) \in +\Theta(\log n)$, $\mathscr{Q}_S(n,k) \in \Theta(k)$, and $C_e(n) \in \Theta(1)$. At first glance, it would appear that we arrived at a solution with a query cost of $O\left(\log_2^2 n + k\log_2 n\right)$, and thus fallen short of our goal. However, Equation~\ref{eqn:edsp-cost} @@ -455,7 +455,7 @@ following provides an upper bound on the worst-case query complexity of an IDSP, \begin{equation*} - O\left(\log_2 n \cdot P(n) + D(n) + R(n) \left(\log_2 n \cdot Q_s(n) + + O\left(\log_2 n \cdot P(n) + \mathscr{D}(n) + R(n) \left(\log_2 n \cdot Q_S(n) + C_e(n)\right)\right) \end{equation*} @@ -1147,7 +1147,7 @@ a constant number of times per level (at most once for Bentley-Saxe, exactly once for tiering, and at most $s$ times for leveling), and there are at most $\log_s n$ levels. Thus, the amortized insertion cost is, \begin{equation*} - I_a(n) \in \Theta\left(\frac{B(n)}{n} \cdot \log_s n\right) + I_A(n) \in \Theta\left(\frac{B(n)}{n} \cdot \log_s n\right) \end{equation*} for standard search problems. Slightly more efficient solutions are possible for merge decomposable search problems based on the cost of @@ -1319,7 +1319,7 @@ in Section~\ref{ssec:dyn-idsp}, with slight modifications to account for the different cost function of buffer querying and preprocessing. The cost is, \begin{equation*} -\mathscr{Q}(n) \in O \left(P_B(N_B) + \log_s n \cdot P(n) + D(n) + R(n)\left( +\mathscr{Q}(n) \in O \left(P_B(N_B) + \log_s n \cdot P(n) + \mathscr{D}(n) + R(n)\left( Q_B(n) + \log_s n \cdot Q_s(n) + C_e(n)\right)\right) \end{equation*} where $P_B(n)$ is the cost of pre-processing the buffer, and $Q_B(n)$ is @@ -1747,7 +1747,7 @@ standard DDSP, we compare with the Bentley-Saxe Method (\textbf{BSM})\footnote{ point-lookup directly on the VPTree. } and a dynamic data structure for the same search problem called an M-tree~\cite{mtree,mtree-impl} (\textbf{MTree}), which is an example of a so-called -"ball tree" structure that partitions high dimensional space using nodes +``ball tree'' structure that partitions high dimensional space using nodes representing spheres, which are merged and split to maintain balance in a manner not unlike a B+tree. We also consider a static instance of a VPTree built over the same set of records (\textbf{VPTree}). We used @@ -1875,7 +1875,7 @@ same massive degradation in query performance that PGM's native update support does in its own update-optimized configuration.\footnote{ It's also worth noting that PGM implements tombstone deletes by inserting a record with a matching key to the record to be deleted, - and a particular "tombstone" value, rather than using a header. This + and a particular ``tombstone'' value, rather than using a header. This means that it can not support duplicate keys when deletes are used, unlike our approach. It also means that the records are smaller, which should improve query performance, but we're able to beat it even |