Fine-Grain parallel algorithm for LU-decomposition

Question

How would you understand this pseudocode of parallel algorithm for LU-decomposition ? I'm confused mostly with the

min(i; j) - 1,

because I have no idea, what the author wanted to say by it. I know that it means " choose the lesser number of 'i' and 'j' and then substract one", but I don't know what is I and J here (maybe coordinates of the current task? Or is it written from the task's point of view, so the whole pseudocode runs at every task?) etc.

The whole presentation with this code (slide 12) is here.

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Pseudocode and diagram of tasks

Bigger picture here.