Query Processing

To find $f^k(x)$, write $k$ in binary. For each bit that is $1$, make the corresponding jump. This decomposes $k$ into a sum of powers of $2$. Start at $x$. For each bit position $i$ from high to low, if bit $i$ is set in $k$, update $x = \text{jump}[i][x]$.

Process all set bits. After processing all bits, you are at $f^k(x)$. This takes $O(\log k)$ time because $k$ has at most $\log k$ bits. Each bit is a constant-time lookup.