How to use a window function to determine when to

I am new to SQL and need to be able to solve the following problem in both Hive and Postgres.

Data

I have a some data showing the start day and end day for different pre-prioritised tasks per person:

   person      task_key start_day end_day
1    Kate             A         1       5
2    Kate             B         1       5
3    Adam             A         1       5
4    Adam             B         2       5
5     Eve             A         2       5
6     Eve             B         1       5
7   Jason             A         1       5
8   Jason             B         4       5
9   Jason             C         3       5
10  Jason             D         5       5
11  Jason             E         4       5

NOTE: Task key is ordered so that higher letters have higher priorities.

Question

I need to work out which task each person should be working on each day, with the condition that:

Higher lettered tasks take priority over lower lettered tasks.
If a higher lettered task overlaps any part of a lower lettered task, then the lower lettered task gets set to NA (to represent that the person should not work on it ever).

Simplification In the real data the end_day is always 5 in the original table i.e. only the start_day varies but the end_day is constant. This means my desired output will have the same number of rows as my original table :)

Output

This is the sort of output I need (Jason is more representative of the data I have which can be over 100 tasks covering a period of 90 days):

   person    task_key start_day end_day valid_from valid_to
1    Kate           A         1       5         NA       NA
2    Kate           B         1       5          1        5
3    Adam           A         1       5          1        2
4    Adam           B         2       5          2        5
5     Eve           A         2       5         NA       NA
6     Eve           B         1       5          1        5
7   Jason           A         1       5          1        3
8   Jason           B         4       5         NA       NA
9   Jason           C         3       5          3        4
10  Jason           D         5       5         NA       NA
11  Jason           E         4       5          4        5

Thank you for your time in advance.

P.S. Similar question I have asked but in R: How to use a window function to determine when to perform different tasks?

The solution in Postgres is fairly easy, because it supports generate_series(). First, explode the data for one row per day for each row in your table:

select d.*, gs.dy
from data d, lateral
     generate_series(start_day, end_day) gs(dy);

Then, aggregate to get the task for each day:

select d.person, d.dy, max(d.task_key) as task_key
from (select d.*, gs.dy
      from data d, lateral
           generate_series(start_day, end_day) gs(dy)
     ) d
group by d.person, d.dy;

You can then re-aggregate, but this is tricky because you might have "split" the original rows (see my comment). This answers your question about which task to perform on which day.

You can do all of this without a lateral join or generate_series() by using a number/tally table.