  The task of finding an entry in an unsorted list of $N$ elements famously
takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$
queries for a quantum computer using Grover's algorithm. Reformulated as a
spatial search problem, this corresponds to searching the complete graph, or
all-to-all network, for a marked vertex by querying an oracle. In this
tutorial, we derive how discrete- and continuous-time (classical) random walks
and quantum walks solve this problem in a thorough and pedagogical manner,
providing an accessible introduction to how random and quantum walks can be
used to search spatial regions. Some of the results are already known, but many
are new. For large $N$, the random walks converge to the same evolution, both
taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$.
In contrast, the discrete-time quantum walk asymptotically takes
$\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$,
while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a
success probability of $1$.
