(* Title: Pure/General/graph.ML
ID: $Id$
Author: Markus Wenzel and Stefan Berghofer, TU Muenchen
Directed graphs.
*)
signature GRAPH =
sig
type key
type 'a T
exception DUP of key
exception SAME
exception UNDEF of key
val empty: 'a T
val keys: 'a T -> key list
val dest: 'a T -> (key * key list) list
val fold: (key * ('a * (key list * key list)) -> 'b -> 'b) -> 'a T -> 'b -> 'b
val minimals: 'a T -> key list
val maximals: 'a T -> key list
val subgraph: (key -> bool) -> 'a T -> 'a T
val map_nodes: ('a -> 'b) -> 'a T -> 'b T
val fold_map_nodes: (key * 'a -> 'b -> 'c * 'b) -> 'a T -> 'b -> 'c T * 'b
val get_node: 'a T -> key -> 'a (*exception UNDEF*)
val map_node: key -> ('a -> 'a) -> 'a T -> 'a T
val map_node_yield: key -> ('a -> 'b * 'a) -> 'a T -> 'b * 'a T
val imm_preds: 'a T -> key -> key list
val imm_succs: 'a T -> key -> key list
val all_preds: 'a T -> key list -> key list
val all_succs: 'a T -> key list -> key list
val strong_conn: 'a T -> key list list
val new_node: key * 'a -> 'a T -> 'a T (*exception DUP*)
val default_node: key * 'a -> 'a T -> 'a T
val del_nodes: key list -> 'a T -> 'a T (*exception UNDEF*)
val is_edge: 'a T -> key * key -> bool
val add_edge: key * key -> 'a T -> 'a T
val del_edge: key * key -> 'a T -> 'a T
val merge: ('a * 'a -> bool) -> 'a T * 'a T -> 'a T (*exception DUP*)
val join: (key -> 'a * 'a -> 'a) (*exception DUP/SAME*) ->
'a T * 'a T -> 'a T (*exception DUP*)
val irreducible_paths: 'a T -> key * key -> key list list
val all_paths: 'a T -> key * key -> key list list
exception CYCLES of key list list
val add_edge_acyclic: key * key -> 'a T -> 'a T (*exception CYCLES*)
val add_deps_acyclic: key * key list -> 'a T -> 'a T (*exception CYCLES*)
val merge_acyclic: ('a * 'a -> bool) -> 'a T * 'a T -> 'a T (*exception CYCLES*)
val add_edge_trans_acyclic: key * key -> 'a T -> 'a T (*exception CYCLES*)
val merge_trans_acyclic: ('a * 'a -> bool) -> 'a T * 'a T -> 'a T (*exception CYCLES*)
end;
functor GraphFun(Key: KEY): GRAPH =
struct
(* keys *)
type key = Key.key;
val eq_key = is_equal o Key.ord;
val member_key = member eq_key;
val remove_key = remove eq_key;
(* tables and sets of keys *)
structure Table = TableFun(Key);
type keys = unit Table.table;
val empty_keys = Table.empty: keys;
fun member_keys tab = Table.defined (tab: keys);
fun insert_keys x tab = Table.insert (K true) (x, ()) (tab: keys);
(* graphs *)
datatype 'a T = Graph of ('a * (key list * key list)) Table.table;
exception DUP = Table.DUP;
exception UNDEF = Table.UNDEF;
exception SAME = Table.SAME;
val empty = Graph Table.empty;
fun keys (Graph tab) = Table.keys tab;
fun dest (Graph tab) = map (fn (x, (_, (_, succs))) => (x, succs)) (Table.dest tab);
fun fold_graph f (Graph tab) = Table.fold f tab;
fun minimals G = fold_graph (fn (m, (_, ([], _))) => cons m | _ => I) G [];
fun maximals G = fold_graph (fn (m, (_, (_, []))) => cons m | _ => I) G [];
fun subgraph P G =
let
fun subg (k, (i, (preds, succs))) =
if P k then Table.update (k, (i, (filter P preds, filter P succs))) else I;
in Graph (fold_graph subg G Table.empty) end;
fun get_entry (Graph tab) x =
(case Table.lookup tab x of
SOME entry => entry
| NONE => raise UNDEF x);
fun map_entry x f (G as Graph tab) = Graph (Table.update (x, f (get_entry G x)) tab);
fun map_entry_yield x f (G as Graph tab) =
let val (a, node') = f (get_entry G x)
in (a, Graph (Table.update (x, node') tab)) end;
(* nodes *)
fun map_nodes f (Graph tab) = Graph (Table.map (fn (i, ps) => (f i, ps)) tab);
fun fold_map_nodes f (Graph tab) =
apfst Graph o Table.fold_map (fn (k, (i, ps)) => f (k, i) #> apfst (rpair ps)) tab;
fun get_node G = #1 o get_entry G;
fun map_node x f = map_entry x (fn (i, ps) => (f i, ps));
fun map_node_yield x f = map_entry_yield x (fn (i, ps) =>
let val (a, i') = f i in (a, (i', ps)) end);
(* reachability *)
(*nodes reachable from xs -- topologically sorted for acyclic graphs*)
fun reachable next xs =
let
fun reach x (rs, R) =
if member_keys R x then (rs, R)
else apfst (cons x) (fold reach (next x) (rs, insert_keys x R))
in fold_map (fn x => reach x o pair []) xs empty_keys end;
(*immediate*)
fun imm_preds G = #1 o #2 o get_entry G;
fun imm_succs G = #2 o #2 o get_entry G;
(*transitive*)
fun all_preds G = flat o fst o reachable (imm_preds G);
fun all_succs G = flat o fst o reachable (imm_succs G);
(*strongly connected components; see: David King and John Launchbury,
"Structuring Depth First Search Algorithms in Haskell"*)
fun strong_conn G = filter_out null (fst (reachable (imm_preds G)
(flat (rev (fst (reachable (imm_succs G) (keys G)))))));
(* nodes *)
fun new_node (x, info) (Graph tab) =
Graph (Table.update_new (x, (info, ([], []))) tab);
fun default_node (x, info) (Graph tab) =
Graph (Table.default (x, (info, ([], []))) tab);
fun del_nodes xs (Graph tab) =
Graph (tab
|> fold Table.delete xs
|> Table.map (fn (i, (preds, succs)) =>
(i, (fold remove_key xs preds, fold remove_key xs succs))));
(* edges *)
fun is_edge G (x, y) = member_key (imm_succs G x) y handle UNDEF _ => false;
fun add_edge (x, y) G =
if is_edge G (x, y) then G
else
G |> map_entry y (fn (i, (preds, succs)) => (i, (x :: preds, succs)))
|> map_entry x (fn (i, (preds, succs)) => (i, (preds, y :: succs)));
fun del_edge (x, y) G =
if is_edge G (x, y) then
G |> map_entry y (fn (i, (preds, succs)) => (i, (remove_key x preds, succs)))
|> map_entry x (fn (i, (preds, succs)) => (i, (preds, remove_key y succs)))
else G;
fun diff_edges G1 G2 =
flat (dest G1 |> map (fn (x, ys) => ys |> map_filter (fn y =>
if is_edge G2 (x, y) then NONE else SOME (x, y))));
fun edges G = diff_edges G empty;
(* join and merge *)
fun no_edges (i, _) = (i, ([], []));
fun join f (Graph tab1, G2 as Graph tab2) =
let fun join_node key ((i1, edges1), (i2, _)) = (f key (i1, i2), edges1)
in fold add_edge (edges G2) (Graph (Table.join join_node (tab1, Table.map no_edges tab2))) end;
fun gen_merge add eq (Graph tab1, G2 as Graph tab2) =
let fun eq_node ((i1, _), (i2, _)) = eq (i1, i2)
in fold add (edges G2) (Graph (Table.merge eq_node (tab1, Table.map no_edges tab2))) end;
fun merge eq GG = gen_merge add_edge eq GG;
(* irreducible paths -- Hasse diagram *)
fun irreducible_preds G X path z =
let
fun red x x' = is_edge G (x, x') andalso not (eq_key (x', z));
fun irreds [] xs' = xs'
| irreds (x :: xs) xs' =
if not (member_keys X x) orelse eq_key (x, z) orelse member_key path x orelse
exists (red x) xs orelse exists (red x) xs'
then irreds xs xs'
else irreds xs (x :: xs');
in irreds (imm_preds G z) [] end;
fun irreducible_paths G (x, y) =
let
val (_, X) = reachable (imm_succs G) [x];
fun paths path z =
if eq_key (x, z) then cons (z :: path)
else fold (paths (z :: path)) (irreducible_preds G X path z);
in if eq_key (x, y) andalso not (is_edge G (x, x)) then [[]] else paths [] y [] end;
(* all paths *)
fun all_paths G (x, y) =
let
val (_, X) = reachable (imm_succs G) [x];
fun paths path z =
if not (null path) andalso eq_key (x, z) then [z :: path]
else if member_keys X z andalso not (member_key path z)
then maps (paths (z :: path)) (imm_preds G z)
else [];
in paths [] y end;
(* maintain acyclic graphs *)
exception CYCLES of key list list;
fun add_edge_acyclic (x, y) G =
if is_edge G (x, y) then G
else
(case irreducible_paths G (y, x) of
[] => add_edge (x, y) G
| cycles => raise CYCLES (map (cons x) cycles));
fun add_deps_acyclic (y, xs) = fold (fn x => add_edge_acyclic (x, y)) xs;
fun merge_acyclic eq GG = gen_merge add_edge_acyclic eq GG;
(* maintain transitive acyclic graphs *)
fun add_edge_trans_acyclic (x, y) G =
add_edge_acyclic (x, y) G
|> fold add_edge (Library.product (all_preds G [x]) (all_succs G [y]));
fun merge_trans_acyclic eq (G1, G2) =
merge_acyclic eq (G1, G2)
|> fold add_edge_trans_acyclic (diff_edges G1 G2)
|> fold add_edge_trans_acyclic (diff_edges G2 G1);
(*final declarations of this structure!*)
val fold = fold_graph;
end;
structure Graph = GraphFun(type key = string val ord = fast_string_ord);
structure IntGraph = GraphFun(type key = int val ord = int_ord);