"how to avoid “using temporary” in many-to-many queries?" Code Answer
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Here's a simplified example I did for a similar performance related question sometime ago that takes advantage of innodb clustered primary key indexes (obviously only available with innodb !!)
You have 3 tables: category, product and product_category as follows:
The most import thing is the order of the product_catgeory clustered composite primary key as typical queries for this scenario always lead by cat_id = x or cat_id in (x,y,z...).
We have 500K categories, 1 million products and 125 million product categories.
So let's see how this schema performs for a query similar to yours. All queries are run cold (after mysql restart) with empty buffers and no query caching.
So that's 0.70 seconds cold - ouch.
Hope this helps :)
EDIT
Having just read your reply to my comment above it seems you have one of two choices to make:
or.
depends on your typical queries as to how you define your clustered PK.