[DMDB] Fix issues, improve query processing with notes on time complexities

semester4/dmdb/parts/03_systems/02_indexing/06_art-trees.tex

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+Adaptive Radix Trees (ARTs) are mainly used to ensure primary key constraints and to speed up point and highly selective queries (selectivity <0.1\%).
+It can also be manually created using \texttt{CREATE INDEX} and are automatically created for columns with a \texttt{UNIQUE} or \texttt{PRIMARY KEY} constraint.
+
+The above applies to DuckDB, which is often similar to Postgres.
+ARTs are not balanced and the internal node size is modified to accommodate the data distribution, thus not requiring nodes to have fixed sizes, which would contain empty pointers.
+
+The internal node size however typically isn't \textit{entirely} variable, but often comes in four different sizes, facilitating traversal.
+Each node stores the keys and pointers to the next node to follow for a given key.
+
+Common optimizations includes path compression, where particularly with long keys, some nodes could potentially only contain a single pointer.
+These nodes can be collapsed and removed to make the tree smaller (i.e. the pointer in the parent is updated to directly point to the data instead).
+If required by insertions, nodes can be added as needed (akin to B+ trees).
+
+For speed, ART require indexed values to be binary-comparable.
+This can become an issue with data types such as signed numbers (extra bit for sign), strings (termination character) and nulls (must be mapped to a non-value sequence).
+Thus, the data needs to be transformed to be able to index it and search the tree.

semester4/dmdb/parts/03_systems/02_indexing/07_r-trees.tex

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+R-Trees are typically used for geospatial datasets, where searching for spatial relationship is expensive (requiring a full table scan).
+
+R-Trees create indexes on bounding boxes as opposed to points in 2- or 3D space.
+We can locate the smallest bounding box where a tuple may lay by traversing the index and looking up the leaf to see if the point is there (or to look for a near neighbour).
+
+For more information, see \hlhref{https://en.wikipedia.org/wiki/R-tree}{Wikipedia}
+
+{\scriptsize (yes, they actually just linked to Wikipedia, so it probably isn't very exam-relevant)}

semester4/dmdb/parts/03_systems/02_indexing/08_other-optimizations.tex

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+For groups of data that don't change often, we can compute statistics on insert, such as \texttt{sum}, \texttt{avg}, \texttt{min}, \texttt{max}, etc.
+These aggregates can be used to check whether data needed is within that group.
+This also enable computing (some) aggregates without reading the entire table and is commonly used in data mining and data cubes.
+
+A concept that is not used in DB engines, is count min sketch. It can be very useful in data streaming engines and large scale data processing.
+What this does is it computes statistics on how many times a given value was seen in a list or stream of data and is a variation of the bloom filter,
+but with several arrays and counting. The overestimation in count is bounded by the size in the matrix.