The execution cost of user-defined method
The selectivity of user-defined method
How to use the user-defined method to test the predicates in the queries
The compabilities of user-defined methods to different join methods such as
sort merge, next loop

Run the user-defined method in a seperated process, and return the result using by passing messages to the kernel process.
Run the user-defined function on a virtual-machine. The user function is first compile in into intermediate binary code. And the run in the virtual machine. 
Provide a new programming language to the user which can avoid dangerous operation to the DBMS

In the parallel DBMS we horizontally partition the data into different sites. The this scenario, the tuples and the objects it points to might be in differnet machines. So if the attribute in question is not processed as the first operator in the query tree. The decision need to be make about whether the objects will be retrieved from the remote site. One proposal is that once

2
a(i)Page level nested join : 101
 (ii) Sort Merge : 333
 (iii) Grace Hash Join : 12
 (iv) Hybrid hash join : 12

b In the case the data do not distribute uniformly among the partitions. The memory needed for the join is equal to the size of the largest partition. So the memory is not utilized efficiently. 

c. Use the virtual memory to put the hash partition. In order to guarantee the performance of the hash probe, a portion of physical memory is reserved for the join. 


3.  simlarity : Both index are  dynamic, they can adjust graceful with
the repeated     insert and  delete.   They   both  have balance  tree
structure. The data are  stored in the  leaf node and the search begin
at the  root  node. They  both use node   split and merge  to keep the
balance for the tree.

difference : The B+  tree index singel  value while  the R tree  index
one-dimension  segments.  In order to  look up a  record in the index,
B-Tree only need to check one path while for the R-Tree, we might need
to check multiple paths from the root to the leaves.

B+ tree is suitable to index value data. And it can be used to execute
the exact match  query and range queries. R-Tree  is suitable to index
segment data. It can be  used to  execute the  exact match, range  and
overlap queries.

(b)  Similarity : Dynamic  sturecutre, Both  can guanrantee the  space
utility. both use merge and split to optimze the structure of the index.

Difference :  R-Tree is a mutillevel index.  Grid file is  a two level
index.  R-TRee partition the data set. Grid File partition the embeded
space.  For the  exact match query,  Grid File  only need  2 IO access
while R-Tree  might need to check multiple  paths from the root to the
leaves hence might need  more than 2 IO. Generaly,  R-Tree can be used
for  both 2-dimension objects and 2-dimensional  point, while the Grid
file generaly only can be used for 2-dimensional points. 

When the data  we want to index  is points, then Grid  File is  a good
choice.