Stop promoting garbage. If you precompute certain values, you might lose the effect of the birthday paradox in the sense that you are reducing randomness and possible combinations. Precomputation can make the problem more deterministic and less dependent on the random coincidences that make the birthday paradox so surprising. You are leading people into a void.
Excellent!
I think now it's time for some magic circles, don't hesitate!
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The birthday paradox is based on probability and randomness, which allows for finding collisions more efficiently than with brute force. By modifying the kangaroo algorithm in a more deterministic way, this probabilistic advantage is lost, and it becomes an approach closer to brute force.
The essence of the birthday paradox is that, with enough randomness, it is more likely to find collisions in a large set of data. By eliminating or reducing randomness, this property is lost, and the algorithm become less efficient.