Caterpillars and Leaves. Can we do better than O(n*c)?

Question

Found this question while preparing for interviews.

Suppose that some caterpillars start from the bottom and jump to the next leaf. They eat the leaf before jumping to n

Answer 1

This is in reference to O(n*c) algo that you mentioned. It's O(n logc) if you look closely.

A caterpillar eats all multiple of its 'jump step' j, thus if it were alone, each caterpillar would eat floor(n/j) leaves.

This complexity is bounded by: n+n/2+n/3+...+n/c <= n log(c)

It won't make a difference as c is small but just pointing out :)

Check this link for implementation of Inclusion-exclusion by Cimbali: figure out Uneaten Leaves algorithm bug

EDIT: Here is the proof that the harmonic series is bounded by log(c). We use the inequality for lower limit of integration.