Optimizing ORDER BY LIMIT queries in MySQL - php

Question

In my web app I made an internal messaging system. I want to place a 'previous' and a 'next' link on each page (where the user viewing the message).
In order to get the next and previous id I execute two queries:
For the previous one:
SELECT id FROM pages WHERE (id<$requestedPageId) ORDER BY id DESC LIMIT 1
And for the next one:
SELECT id FROM pages WHERE (id>$requestedPageId) ORDER BY id LIMIT 1
EXPLAIN says the the query type is "range" and the rows column says it would examine all rows that has smaller or bigger id than the page's id (a big number). The Extra row says "Using where".
It seems MySQL ignores that I want only one row. Isn't MySQL smart enough to optimize this kind of query so it would find the row for the page and search for the first matching row back/forth?
Is there a better approach to get the next and previous page's id?
Additional notes:
This problem seems to exist on every ORDER BY LIMIT type queries (eg.: when I split a long list to multiple pages.).
Where clause is not this simple (I want to let the user access the next/previous page he has permission to access. No joins though.)
All columns appear in WHERE are indexed (id is the primary key)
variables are protected against injection.
EDIT1:
So the query I'm using currently:
SELECT id
FROM reports
WHERE (id<$requestedPageId) AND ((isPublic=1) OR (recipientId=$recipient))
ORDER BY id DESC
LIMIT 1
Or when I re-factor it as the answer said:
SELECT MAX(id)
FROM reports
WHERE (id<$requestedPageId) AND ((isPublic=1) OR (recipientId=$recipient))

Answer 1

For the previous
SELECT MAX(id) FROM pages WHERE id<$requestPageId
And for the next
SELECT MIN(id) FROM pages WHERE id>$requestedPageId

Answer 2

The database is behaving as expected. Your query is a range query because of the less-than symbol (id < $requestedPageId). The OR statement makes it harder to use a single index to find the results. And, sorting the results means it has to get all matching rows to perform the sort, even though you only want 1 row.
You're not going to be able to make this a "const" type query, but you may be able to optimize it using indexes, sub-queries, and/or union statements.
Here is one query to rule them all. I'm not saying this is the best solution, but just one way of approaching the problem. To start, this query will work better if you create two indexes, one on recipientId and another on isPublic.
SELECT
GREATEST(
( SELECT MAX( id ) FROM reports
WHERE id < $requestedPageId AND recipientId = $recipient ),
( SELECT MAX( id ) FROM reports
WHERE id < $requestedPageId AND isPublic = 1 )
) AS prev_id
LEAST(
( SELECT MIN( id ) FROM reports
WHERE id > $requestedPageId AND recipientId = $recipient ),
( SELECT MIN( id ) FROM reports
WHERE id > $requestedPageId AND isPublic = 1 )
) AS next_id