How can I speed up a query that orders by a calculated field?
If you don't really need zero-second actuality, you could just run your query time to time and cache the results.

If you still need to have real-time data on this (sacrificing insert performance), I would do this:

Since self-joins are not allowed in indexed views, you need to create two copies of each table:

CREATE TABLE personBrother
        (
        personId INT NOT NULL,
        brotherName INT NOT NULL
        )

CREATE TABLE personBrother2
        (
        personId INT NOT NULL,
        brotherName INT NOT NULL
        )


Create an indexed view on their join:

CREATE VIEW
        commonBrothers
WITH SCHEMABINDING
AS
        SELECT  p1.personId AS p1,
                p2.personId AS p2,
                COUNT_BIG(*) AS cnt
        FROM    dbo.personBrother p1
        JOIN    dbo.personBrother2 p2
        ON      p1.brotherName = p2.brotherName
        WHERE   p1.personId &lt; p2.personId
        GROUP BY
                p1.personId, p2.personId

CREATE UNIQUE CLUSTERED INDEX
        ux_commonBrothers_p1_p2
ON      commonBrothers (p1, p2)

CREATE INDEX
        ix_commonBrothers_cnt
ON      commonBrothers (cnt)


Same for sisters.

You should manually maintain these tables to have same data (write a trigger, insert/update/delete both etc).

Now we can easily get pairs with the most brothers and most sisters:

SELECT  TOP 1 WITH TIES
        *
FROM    commonBrothers
ORDER BY
        cnt DESC


All we need now is to fetch a greatest sum. Unfortunately, we cannot index a join of these views (it's a pure implementation flaw, there's no theoretical limitation for this).

So we need to do the following: the top pair cannot have less brothers than the top sis pair. Same holds for the sisters. So we have this query:

SELECT  TOP 1 WITH TIES
        cb.p1, cb.p2, cb.cnt + cs.cnt AS totalCnt
FROM    commonBrothers cb
JOIN    commonSisters cs
ON      cs.p1 = cb.p1
        AND cs.p2 = cb.p2
WHERE   cs.cnt &gt;=
        (
        SELECT  MAX(cst.cnt)
        FROM    (
                SELECT  TOP 1 WITH TIES
                        p1, p2
                FROM    commonBrothers 
                ORDER BY
                        cnt DESC
                ) cbt
        JOIN    commonSisters cst
        ON      cst.p1 = cbt.p1
                AND cst.p2 = cbt.p2
        )
        AND cb.cnt &gt;=
        (
        SELECT  MAX(cbt.cnt)
        FROM    (
                SELECT  TOP 1 WITH TIES
                        p1, p2
                FROM    commonSisters
                ORDER BY
                        cnt DESC
                ) cst
        JOIN    commonBrothers cbt
        ON      cbt.p1 = cst.p1
                AND cbt.p2 = cst.p2
        )
ORDER BY
        totalCnt DESC


If the numbers of common brothers and sisters are correlated, this query will be very fast.

This solution has two drawbacks:


DML performance: if you insert or delete a record for a name shared by million brothers, the indexed view will get 2M inserts or delete. This is the price you pay for real-time query: the kind of data you are asking for cannot be easily indexed.
Persons with 0 brothers or 0 sisters will not be indexed. If there's a chance that top pair will not have brothers or sisters, you should amend the last query a little.