In SQL Server, there are often scenarios where we need to solve gaps and islands problems. These problems involve finding patterns or sequences in data that have gaps or islands of missing values. One common example is scheduling constraints, where we need to ensure that certain features are present over fixed time intervals.

For example, let’s consider the problem of rostering flight attendants. One of the scheduling constraints might be that rosters of one month’s duration must have at least 24 consecutive hours free of duty among any 7 consecutive days. To represent this constraint, we can use a sequence of binary digits, where 0 represents rest and 1 represents work.

To solve this problem, we can use a SQL Server function called Sliding(W, N, M, L), where W is the work period, N is the length of the work period, M is the length of the subsequence, and L is the minimum number of consecutive 0s required. This function checks if the work period satisfies the constraint by using a select query and some algebraic calculations.

The Sliding function works by first identifying the rest periods in the work period. A rest period is a subsequence of 0s, and a maximal rest period is a rest period that is not contained in a larger rest period. The function then checks if the work period has at least L consecutive 0s among any consecutive M binary digits. If the constraint fails, the function returns the position where the failure occurred.

To compute the rest periods and check the constraint, the Sliding function uses a few key observations and the Sliding Theorem. The Sliding Theorem states that the constraint fails if any of the following conditions hold:

1. There are no basic periods (maximal rest periods)
2. The starting position of the first basic period is greater than or equal to M – L + 2
3. The ending position of the last basic period is less than or equal to N – M + L – 1
4. The gap between adjacent basic periods is greater than or equal to M – 2L + 3

By checking these conditions, the Sliding function can determine if the constraint is satisfied or not.

In addition to the Sliding function, there is also a function called fnBasic that removes non-basic rest periods by converting their 0s to 1s. This is done because non-basic rest periods do not affect the constraint and can be safely ignored. Removing these non-basic rest periods simplifies the computation and ensures that adjacent rest periods are always basic.

Overall, the Sliding function and the fnBasic function provide a simple and efficient way to solve gaps and islands problems in SQL Server. By using algebraic calculations and select queries, we can check if a work period satisfies a given constraint without the need for loops or cursors.

For more information and code examples, you can refer to the Resource section below.

Resource

For more information and code examples, you can refer to the following resources:

• Sliding function: [link to code]
• fnBasic function: [link to code]
• Sliding Theorem: [link to theorem]
• spString2Table stored procedure: [link to code]