Arrange numbers in order - variables

I've some variables, Lets say a, b, c, d. All belongs to a fixed interval [0, e]
Now i've some relations between them like
a > b
a > c
b > d
Something like this; I want to make a function which print all the possible cases for this.
Example:
a b c d
a c b d
a b d c
a c b d


In essence, what you have is a directed acyclic graph.
A relatively simple approach is to store, for each variable, a set of the variables that must precede them. (In your example, this storage would map b to {a}, c to {a}, and d to {b}.) You can then write a recursive function that generates all valid tails consisting of a subset of these variables (in your case, for example, the subset {c,d} produces two valid tails: [c,d] and [d,c]). This recursive function examines each variable in the subset and determines whether its prerequisites are already met. (For example, since b maps to {a}, any subset including both a and b cannot produce a tail that begins with b.) If so, then it can recursively call itself on the subset excluding that variable.
There are some optimizations you can then perform, if desired. For example, you can use dynamic programming to avoid repeatedly re-computing the set of valid tails for the same subset.

Related

SQL: Finding all members of a group when they are only listed in pairs, and not all pairs are listed

I have some data that looks like this, and identifies pairs that are related:
From_ID To_ID
A C
B C
D E
E D (note this is the same pair as above, in a different order)
E F
A F
G H
Using the logic of 'if x is paired with y, and y is paired with z, then x is paired with z', how can I run an SQL query to return all members of a group?
So for the table above I would like a set of results that identifies or returns two groups: 'A, B, C, D, E, F' and 'G, H', not fussy about how this is done.
It feels like some kind of iterative query but I really have no idea where to start with this so any pointers would be appreciated.
edit: could be run in SQL Developer or HiveQL.

What is impossible?

Hi recently i appeared in an aptitude,there was a problem that i realy cant understand please provide some idea, how to solve it.(and sorry to for poor English.)
(Question)-> Three candidates, Amar, Birendra and Chanchal stand for the local election. Opinion polls are
conducted and show that fraction a of the voters prefer Amar to Birendra, fraction b prefer Birendra to
Chanchal and fraction c prefer Chanchal to Amar. Which of the following is impossible?
(a) (a, b, c) = (0.51, 0.51, 0.51);
(b) (a, b, c) = (0.61, 0.71, 0.67);
(c) (a, b, c) = (0.68, 0.68, 0.68);
(d) (a, b, c) = (0.49, 0.49, 0.49);
(e) None of the above.

If you tried to list of possible preferences people can have
are either
ABC (means you prefer A to B, prefer B to C and therefore also prefer A to C)
ACB
BAC
BCA
CAB
CBA
in this case you'll find that each fraction of the population represents:
a=ABC+ACB+CAB
b=ABC+BAC+BCA
c=BCA+CAB+CBA
therefore a+b+c = 2(ABC+BCA+CAB)+ACB+BAC+CBA
as you notice not all groups within the population are repeated. we can therefore assume than (a+b+c) can never be more than twice the population since each member of the population is represented twice at the most.
out of the options C is the one where the sum is more than 2. and is therefore the impossible value.

Which languages have a primitive operation for swapping variables?

In most languages, if you want to swap two variables, it's something like:
var c = b
b = a
a = c
Yes, you can do fancy hacks with XOR if you like but it's generally 3 lines of code for a single operation. Are there any languages that have swapping variables as a primitive in the language?

Lua, Python, Ruby and more support this notation:
a, b = b, a
And javascript sure needs no temporary variable either ;)
a = -(b = (a += b) - b) + a;
For more examples on how to swap variables (in 86 languages), see: http://rosettacode.org/wiki/Generic_swap

In most dynamic languages you can do something like this to swap:
a, b = b, a
Now a have the value of b, and b has the value of a. I am not sure if this is what you meant or not.

Approaches to converting a table of possibilities into logical statements

I'm not sure how to express this problem, so my apologies if it's already been addressed.
I have business rules summarized as a table of outputs given two inputs. For each of five possible value on one axis, and each of five values on another axis, there is a single output. There are ten distinct possibilities in these 25 cells, so it's not the case that each input pair has a unique output.
I have encoded these rules in TSQL with nested CASE statements, but it's hard to debug and modify. In C# I might use an array literal. I'm wondering if there's an academic topic which relates to converting logical rules to matrices and vice versa.
As an example, one could translate this trivial matrix:
A B C
-- -- -- --
X 1 1 0
Y 0 1 0
...into rules like so:
if B OR (A and X) then 1 else 0
...or, in verbose SQL:
CASE WHEN FieldABC = 'B' THEN 1
WHEN FieldABX = 'A' AND FieldXY = 'X' THEN 1
ELSE 0
I'm looking for a good approach for larger matrices, especially one I can use in SQL (MS SQL 2K8, if it matters). Any suggestions? Is there a term for this type of translation, with which I should search?

Sounds like a lookup into a 5x5 grid of data. The inputs on axis and the output in each cell:
Y=1 Y=2 Y=3 Y=4 Y=5
x=1 A A D B A
x=2 B A A B B
x=3 C B B B B
x=4 C C C D D
x=5 C C C C C
You can store this in a table of x,y,outvalue triplets and then just do a look up on that table.
SELECT OUTVALUE FROM BUSINESS_RULES WHERE X = #X and Y = #Y;

database index: why pairing

I have a table with multiple indexes, several of which duplicate the same columns:
Index 1 columns: X, B, C, D
Index 2 columns: Y, B, C, D
Index 3 columns: Z, B, C, D
I'm not very knowledgeable on indexing in practice, so I'm wondering if somebody can explain why X, Y and Z were paired with these same columns. B is an effective date. C is a semi-unique key ID for this table for a specific effective date B. D is a sequence that identifies the priority of this record for the identifier C.
Why not just create 6 indexes, one for each X, Y, Z, B, C, D?
I want to add an index to another column T, but in some contexts I'll only be querying on T alone while in others I will also be specifying the B, C and D columns... so should I create just one index like above or should I create one for T and one for (T, B, C, D)?
I've not had as much luck as expected when googling for comprehensive coverage of indexing. Any resources where I can get a through explanation and lots of examples of B-tree indexing?

The rule with indexing is that an index can be used to filter on any list of columns that constitute a prefix of the columns used for that index.
In other words, we can use Index 1 when we filter on X and B, or X, B and C, or just X, or all four.
However, we cannot use the index to filter "in the middle". This is because indexes work not entirely unlike concatenating the values of those columns for each row, and sorting the result. If we know what the thing we're looking for begins with, we can figure out where in the index to look - just like when doing binary search.
That's why a single index is no good: if we need to filter on B, C, D, and one of X, Y and Z, we need three indexes; X, Y is no good as an index for just filtering on Y, because the prefix of the values we're looking for - the X - is not known.
As Daniel mentioned, a covering index is a possible explanation for repeating B, C, and D: even if D is never filtered on, it may be the case that we need exactly the columns which you see in your indexes, and we can then just read the columns from the index instead of just using the index to locate the row.

One reason for having B, C and D in those indexes might be to have a covering index for frequently used queries. You will have a covering index when the index itself contains all the required data fields for a particular query.
A covering index can dramatically speed up data retrieval, since only the index pages, not the data pages, will be used to retrieve the data.
Below is an example query where index 1 would be a covering index:
SELECT B, C, D FROM table WHERE X = '10'

You should create it in (T, B, C, D).
Let's say you have two fields with an index in a table: A and B. When you create a separate index on each one of the columns, and have a query such as:
SELECT * FROM table WHERE A = 10 AND B = 20
What happens is either:
1) The DB creates two intermediate result-sets, one with rows where A = 10, and another one with rows where B = 20. It then has to merge these two result-sets into one (and also check for duplicate rows).
2) The DB creates one result-set with rows where A = 10. It then has to go manually through all of the rows in this intermediate result-set and check in each one where B = 10.
However when you know that index B depends on index A, and your query uses A before B, you can create one index for both of the columns: (A, B)
What this means that now the DB will first find all rows where A = 10, but because B is part of the same index, it can use the same index information to filter the result-set into rows where B is also 20. It doesn't have to make two intermediate result-sets + merge them, or only use one of the indexes and do manual scan for the other.
There might be other ways that the DB deals with these situations as well, it largely depends on an implementation.

The indexes in the form (X, B, C, D) can be used to optimize queries like:
... WHERE X rel sthg (possibly ORDER BY B, C, D)
... WHERE X = sthg AND B rel sthg (possibly ORDER BY C, D)
... WHERE X = sthf AND B = sthg AND C rel sthg (possibly ORDER BY D)
etc. where rel are arbitrary relation operators (<, >, =, <=, >=) and sthg are values or expressions. Especially the second two, and the sorting variants wouldn't be optimized by the "single column indexes variant".
OTOH, it cannot optimize a query
... WHERE B = sthg
because it starts in the middle of the index; here, the single column index would work.

For a resource where you can get a through explanation and lots of examples regarding indexes on Oracle (and any other Oracle-related issue), you should visit and bookmark askTom.