\section{Sparse Persistent}
We want to use (nearly) the whole field
as our index space, without cost.
\begin{outline}
  \1 What if you want $\vec a$ and $\vec b$ to be index-value pairs, where the
  indices are from a large space (rather than $[N]$)?
\1 params
  \2 Now, $N$ becomes the capacity
  \2 $C$ is the number of creations
\1 Three operations:
  \2 read
  \2 conditional write
  \2 create
    \3 these happen *before* the reads/writes
\1 approach
  \2 use a dummy key value to indicate dead spots
     \3 zero is a natural choice
  \2 essentially, you inject creates into the initial store,
     and filter out some dummies afterwards
  \2 you just add a conditional uniqueness check
    \3 uniqueness check: \csPerN{4}
    \3 with a condition: \csPerN{2}
    \3 computing the condition (field eq): \csPerN{2}
  \2 base cost of the permutation: \csPerN{4}
    \3 one from initial coefficient hash
    \3 one from root hash
    \3 one from root hash again
    \3 one from coefficient hash again
\1 accounting
  \2 \csPerN{12}
  \2 very similar dependence on $A$
  \2 some dependence on $C$ too (like 10--30)
\end{outline}