hi

docs/ALGORITHM.md

@@ -120,11 +120,31 @@ and `ys`. Then we can define a general
 applyMerge :: (Ord c) => (a -> b -> c) -> [a] -> [b] -> [c]
 ```
 where `applyMerge f xs ys` is an ordered list of all `f x y`, where `x ∈ xs` and `y ∈ ys`. In particular, `smooth3 = applyMerge (*) powersOf2 powersOf3`.
+We can also define a more general
+```haskell
+applyMergeBy :: (c -> c -> Ordering) -> (a -> b -> c) -> [a] -> [b] -> [c]
+```
 
 ## Implementation and complexity
 We can use a priority queue to maintain the frontier of elements. To determine in step 3 whether a down- or right-neighbor is unblocked, we maintain two `IntSet`s, for the `x`- and `y`- indices of all elements in the frontier. Then in step 3, if the `x`- and `y`-index of a neighbor are not in the respective `IntSet`s, the neighbor is unblocked and can be added to the frontier.
 
-After producing $n$ elements, the size of the frontier is at most $O(\sqrt{n})$ elements. <i>(TODO: prove this)</i> Manipulating the priority queue and the `IntSet`s to yield one element takes $O(\text{log } \sqrt{n}) = O(\text{log } n)$ time. Therefore, producing $n$ elements of `applyMerge f xs ys` takes $O(n \log n)$ time and $O(\sqrt{n})$ space.
+After producing $n$ elements, the size of the frontier is at most $O(\sqrt{n})$ elements. <i>(TODO: prove this)</i> Manipulating the priority queue and the `IntSet`s to produce the next element takes $O(\text{log } \sqrt{n}) = O(\text{log } n)$ time. Therefore, producing $n$ elements of `applyMerge f xs ys` takes $O(n \log n)$ time and $O(\sqrt{n})$ auxiliary space.
+
+## Other applications
+
+`applyMerge` is versatile!
+```haskell
+-- Sieve of Erastosthenes
+primes :: [Int]
+primes = 2 : ([3..] `minus` composites)    -- `minus` from data-ordlist
+
+composites :: [Int]
+composites = applyMerge (*) primes [2..]
+
+-- Gaussian integers in the first quadrant, ordered by norm
+gaussianInts :: [Complex Int]
+gaussianInts = applyMergeBy (comparing (\a b -> a*a + b*b)) (:+) [1..] [1..]
+```
 
 # Prior work
 In <code>[data-ordlist](https://www.stackage.org/lts/package/data-ordlist)</code>,