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Select row based upon the MAX of one column and the related MIN of another

I have a table set up like this:

CREATE TABLE dbo.IntervalCounts (
    item_id int NOT NULL,
    interval_time time(0) NOT NULL,
    interval_count int DEFAULT 0 NOT NULL
)

Each item_id has 96 interval_time s, from 00:00 to 23:45 in 15 minute increments. each interval_time has an interval_count >= 0. This table has approx. 200 million rows.

I need to select values from a table where the count is the highest, then, if there are multiple qualifying rows with the same count, pick the one with the lowest interval time.

So, if I have a item_id 1, whose max count is 100:

item_id   interval_time interval_count
1         00:00         100
1         13:15         100
1         07:45         100
1         19:30         100

I'd like to get just one row:

item_id   interval_time interval_count
1         00:00         100

Getting the first selection is easy enough, I've got:

SELECT a.item_id, a.interval_time, a.interval_count
    FROM dbo.IntervalCounts a
    LEFT JOIN dbo.IntervalCounts b
        ON a.item_id = b.item_id
        AND a.interval_count < b.interval_count
    WHERE 1=1
    AND b.interval_count IS NULL

However, getting it down to just one row has proven tricky for me.

This triple self-join ran for an hour and a half before I killed it (I'll be running it regularly, ideally it would run no more than 15 minutes max).

SELECT a.item_id, a.interval_time, a.interval_count
    FROM dbo.IntervalCounts a
    LEFT JOIN dbo.IntervalCounts b
        ON a.item_id = b.item_id
        AND a.interval_count < b.interval_count
    LEFT JOIN dbo.IntervalCounts c
        ON a.item_id = c.item_id
        -- if I remove this line, it will ALWAYS give me the 00:00 interval
        -- if I keep it, it runs way too long
        AND a.interval_count = c.interval_count
        AND a.interval_time > c.interval_time
    WHERE 1=1
    AND b.interval_count IS NULL
    AND c.interval_time IS NULL

Doing something like this just seems ungainly, and I was also forced to kill the execution after about an hour and a half:

DECLARE @tempTable TABLE
    (
    item_id int,
    interval_time time(0),
    interval_count int
    )

INSERT INTO @tempTable
SELECT a.item_id, a.interval_time, a.interval_count
FROM dbo.IntervalCount a
LEFT JOIN dbo.IntervalCount b
    ON a.item_id = b.item_id
    AND a.interval_count < b.interval_count
WHERE 1=1
AND b.interval_count IS NULL

SELECT a.item_id, a.interval_time, a.interval_count
FROM @tempTable a
LEFT JOIN @tempTable b
    ON a.item_id = b.item_id
    AND a.interval_time > b.interval_time
WHERE 1=1
AND b.interval_time IS NULL

There must be a better way, but I'm stumped. How can I do this in a manner that won't take forever to run?

You are overthinking it, you can use ROW_NUMBER :

WITH CTE AS
(
    SELECT  *,
            RN = ROW_NUMBER() OVER(PARTITION BY item_id 
                                   ORDER BY interval_count DESC, interval_time)
    FROM dbo.IntervalCounts
)
SELECT *
FROM CTE
WHERE RN = 1;

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