Converting trees to dags

The problem with the trees generated in the previous section is that there's a different edge, and therefore a different child, for each possible interval of the field tested, even if those children both execute exactly the same ``original'' arm of the case statement. The code in this section converts the trees to dags, and as part of the process it combines edges pointing to the same node. This can reduce the size of the tree by huge factors.

To make the transformation work, I have to represent a set of intervals on each edge, not just a single interval. Because no two intervals overlap, I can use a wonderful dirty trick, detailed below. I also may convert a node's name string to a namearray mapping field values to strings. The goal is for children of the same parent to share a single name array; that way the edges can be merged and the name operator can be implemented with an array reference. If I don't convert a node's name, the only penalty is that the tree might be bigger. (Code generation will be different for the two cases.)

Now, the dirty representation trick: I can represent a set of numbers S (a union of intervals) as two sets, lo and hi, such that

The procedure addinterval adds a new interval to such a set S, relying on the fact that no two intervals overlap.

<*>= [D->]
procedure addinterval(loset, hiset, lonum, hinum)
    if member(loset, hinum) then delete(loset, hinum) else insert(hiset, hinum)
    if member(hiset, lonum) then delete(hiset, lonum) else insert(loset, lonum)
Defines addinterval (links are to index).

To convert trees to dags I need to be able to compare two nodes for structural identity, and the easiest way is to compute a canonical representation as a string:

node : [fname:patimage(list of edges)] | <NOMATCH> | (image([1].original)) edge : patimage(list of sort(loset ++ hiset)):node

<*>+= [<-D->]
procedure nodetostring(n, depth)
    static cache 
    initial cache := table()
    /depth := 0
    if /cache[n] then
        if *n.children > 0 then {
            result := "[" || || ":"
            every result ||:= edgetostring(!n.children, depth+2)
            cache[n] := result || "]"
        } else if *n.cs.arms = 0 then
            cache[n] := "<NOMATCH>"
            cache[n] := "(" || image( || ":" || image(n.cs.arms[1].original) ||")"
    return \cache[n]

procedure edgetostring(e,depth)
    return left("\n", depth) || 
          "{" || patimage(sort(e.lo ++ e.hi)) || ":" || nodetostring(e.node,depth) || "}"
Defines edgetostring, nodetostring (links are to index).

Conversion to dag is the usual bottom-up hashing; here I compute the string and then use the string to index into a table. The real work of merging edges is done by combinechildren. If edge merging results in a single each, the node is replaced by its child, provided the edge really covers all possible values of the field.

<*>+= [<-D->]
procedure tree2dag(n, nodetable, depth)
    /nodetable := table()
    /depth := 0
    if *n.children > 0 then
        combinechildren(n, nodetable, depth+2)  # converts edges to set form
    if *n.children = 1 then {
        e := n.children[1]
        if covers(n.children[1], n.field.hi - n.field.lo) then
            n := n.children[1].node     # all roads to child: hoist it
            warning("node with one child doesn't match all cases")    
    s := nodetostring(n, depth)
    /nodetable[s] := n
    return nodetable[s]
Defines tree2dag (links are to index).

Here's where I check coverage. Only success or failure of covers is meaningful, not the value returned.

<*>+= [<-D->]
procedure covers(e, width)
    l := sort(e.lo ++ e.hi)
    return *l = 2 & l[1] = 0 & l[2] = 2^width
Defines covers (links are to index).

The complicated stuff here is identifying a name array. At each node, either all edges go in an exiting name array or a new name array is used. If not, I create a new one.

<*>+= [<-D->]
record namearray(field, tbl, hi, codename)
        # field used as index, table[integer] of name, bound on table, name of this array
global natable
Defines namearray, natable (links are to index).

<*>+= [<-D->]
procedure arraycandidates(n)
    initial MAXRANGE := 32
    suspend e := !n.children & type( == "string" & 
            e.hi - e.lo <= MAXRANGE & e

procedure combinechildren(n, nodetable, depth)
    initial natable := table()

    if arraycandidates(n) ~== arraycandidates(n) then {
        <change names of children from strings to namearrays when possible>

    lotable := table()
    hitable := table()
    every e := !n.children & child := tree2dag(e.node, nodetable, depth) do {
        /lotable[child] := set()
        /hitable[child] := set()
        addinterval(lotable[child], hitable[child], e.lo, e.hi)
    n.children := []
    every child := key(lotable) do
        put(n.children, edge(child, lotable[child], hitable[child]))
Defines arraycandidates, combinechildren (links are to index).

<change names of children from strings to namearrays when possible>= (<-U)
mightuse := set()           # name arrays we might use must have right field
every na := !\natable[n.field] do
    insert(mightuse, na)
every e := arraycandidates(n) & na := !mightuse do
    if \na.tbl[e.lo to e.hi - 1] ~== then  # slot used with wrong name
        delete(mightuse, na)
if *mightuse > 0 then
    willuse := ?mightuse
else {
    /natable[n.field] := set()
    insert(natable[n.field], willuse := namearray(n.field, table(), 0))
every e := arraycandidates(n) &
      e.lo - willuse.hi <= MAXRANGE do {
          every willuse.tbl[e.lo to e.hi - 1] :=;
 := willuse
          willuse.hi <:= e.hi
<*>+= [<-D]
procedure namesused(n, result)
    /result := set()
    if type( == "namearray" then insert(result,
    every namesused((!n.children).node, result)
    return result
Defines namesused (links are to index).