From eb519d35d7f11427dd5fc877130b02478f0da80d Mon Sep 17 00:00:00 2001 From: Douglas Rumbaugh Date: Sun, 4 May 2025 16:43:45 -0400 Subject: Began re-writing/updating the sampling extension stuff --- chapters/background.tex | 13 ++++--------- 1 file changed, 4 insertions(+), 9 deletions(-) (limited to 'chapters/background.tex') diff --git a/chapters/background.tex b/chapters/background.tex index 9950b39..69436c8 100644 --- a/chapters/background.tex +++ b/chapters/background.tex @@ -85,6 +85,7 @@ their work on dynamization, and we will adopt their definition, \begin{equation*} F(A \cup B, q) = F(A, q)~ \mergeop ~F(B, q) \end{equation*} + for all $A, B \in \mathcal{PS}(\mathcal{D})$ where $A \cap B = \emptyset$. \end{definition} The requirement for $\mergeop$ to be constant-time was used by Bentley and @@ -101,6 +102,7 @@ problems}, \begin{equation*} F(A \cup B, q) = F(A, q)~ \mergeop ~F(B, q) \end{equation*} + for all $A, B \in \mathcal{PS}(\mathcal{D})$ where $A \cap B = \emptyset$. \end{definition} To demonstrate that a search problem is decomposable, it is necessary to @@ -811,6 +813,7 @@ cost, we could greatly reduce the cost of supporting $C(n)$-decomposable queries. \subsubsection{Independent Range Sampling} +\label{ssec:background-irs} Another problem that is not decomposable is independent sampling. There are a variety of problems falling under this umbrella, including weighted @@ -831,15 +834,7 @@ matching of records in result sets. To work around this, a slight abuse of definition is in order: assume that the equality conditions within the DSP definition can be interpreted to mean ``the contents in the two sets are drawn from the same distribution''. This enables the category -of DSP to apply to this type of problem. More formally, -\begin{definition}[Decomposable Sampling Problem] - A sampling problem $F: (D, Q) \to R$, $F$ is decomposable if and - only if there exists a constant-time computable, associative, and - commutative binary operator $\mergeop$ such that, - \begin{equation*} - F(A \cup B, q) \sim F(A, q)~ \mergeop ~F(B, q) - \end{equation*} -\end{definition} +of DSP to apply to this type of problem. Even with this abuse, however, IRS cannot generally be considered decomposable; it is at best $C(n)$-decomposable. The reason for this is -- cgit v1.2.3