refined Graph implementation: more abstract/scalable Graph.Keys instead of plain lists -- order of adjacency is now standardized wrt. Key.ord;
(* Title: Pure/General/graph.ML
Author: Markus Wenzel and Stefan Berghofer, TU Muenchen
Directed graphs.
*)
signature GRAPH =
sig
type key
structure Keys:
sig
type T
val is_empty: T -> bool
val fold: (key -> 'a -> 'a) -> T -> 'a -> 'a
val fold_rev: (key -> 'a -> 'a) -> T -> 'a -> 'a
val dest: T -> key list
end
type 'a T
exception DUP of key
exception SAME
exception UNDEF of key
val empty: 'a T
val is_empty: 'a T -> bool
val keys: 'a T -> key list
val dest: 'a T -> (key * key list) list
val get_first: (key * ('a * (Keys.T * Keys.T)) -> 'b option) -> 'a T -> 'b option
val fold: (key * ('a * (Keys.T * Keys.T)) -> 'b -> 'b) -> 'a T -> 'b -> 'b
val subgraph: (key -> bool) -> 'a T -> 'a T
val get_entry: 'a T -> key -> key * ('a * (Keys.T * Keys.T)) (*exception UNDEF*)
val map: (key -> 'a -> 'b) -> 'a T -> 'b T
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 -> Keys.T
val imm_succs: 'a T -> key -> Keys.T
val immediate_preds: 'a T -> key -> key list
val immediate_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 minimals: 'a T -> key list
val maximals: 'a T -> key list
val is_minimal: 'a T -> key -> bool
val is_maximal: 'a T -> key -> bool
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 del_node: key -> 'a T -> 'a T (*exception UNDEF*)
val is_edge: 'a T -> key * key -> bool
val add_edge: key * key -> 'a T -> 'a T (*exception UNDEF*)
val del_edge: key * key -> 'a T -> 'a T (*exception UNDEF*)
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 UNDEF | CYCLES*)
val add_deps_acyclic: key * key list -> 'a T -> 'a T (*exception UNDEF | CYCLES*)
val merge_acyclic: ('a * 'a -> bool) -> 'a T * 'a T -> 'a T (*exception CYCLES*)
val topological_order: 'a T -> key list
val add_edge_trans_acyclic: key * key -> 'a T -> 'a T (*exception UNDEF | CYCLES*)
val merge_trans_acyclic: ('a * 'a -> bool) -> 'a T * 'a T -> 'a T (*exception CYCLES*)
exception DEP of key * key
val schedule: ((key * 'b) list -> key * 'a -> 'b) -> 'a T -> 'b list (*exception DEP*)
end;
functor Graph(Key: KEY): GRAPH =
struct
(* keys *)
type key = Key.key;
val eq_key = is_equal o Key.ord;
structure Table = Table(Key);
structure Keys =
struct
abstype T = Keys of unit Table.table
with
val empty = Keys Table.empty;
fun is_empty (Keys tab) = Table.is_empty tab;
fun member (Keys tab) = Table.defined tab;
fun insert x (Keys tab) = Keys (Table.insert (K true) (x, ()) tab);
fun remove x (Keys tab) = Keys (Table.delete_safe x tab);
fun fold f (Keys tab) = Table.fold (f o #1) tab;
fun fold_rev f (Keys tab) = Table.fold_rev (f o #1) tab;
fun make xs = Basics.fold insert xs empty;
fun dest keys = fold_rev cons keys [];
fun filter P keys = fold (fn x => P x ? insert x) keys empty;
end;
end;
(* graphs *)
datatype 'a T = Graph of ('a * (Keys.T * Keys.T)) Table.table;
exception DUP = Table.DUP;
exception UNDEF = Table.UNDEF;
exception SAME = Table.SAME;
val empty = Graph Table.empty;
fun is_empty (Graph tab) = Table.is_empty tab;
fun keys (Graph tab) = Table.keys tab;
fun dest (Graph tab) = map (fn (x, (_, (_, succs))) => (x, Keys.dest succs)) (Table.dest tab);
fun get_first f (Graph tab) = Table.get_first f tab;
fun fold_graph f (Graph tab) = Table.fold f tab;
fun subgraph P G =
let
fun subg (k, (i, (preds, succs))) =
if P k then Table.update (k, (i, (Keys.filter P preds, Keys.filter P succs)))
else I;
in Graph (fold_graph subg G Table.empty) end;
fun get_entry (Graph tab) x =
(case Table.lookup_key tab x of
SOME entry => entry
| NONE => raise UNDEF x);
fun map_entry x f (G as Graph tab) = Graph (Table.update (x, f (#2 (get_entry G x))) tab);
fun map_entry_yield x f (G as Graph tab) =
let val (a, node') = f (#2 (get_entry G x))
in (a, Graph (Table.update (x, node') tab)) end;
(* nodes *)
fun map_nodes f (Graph tab) = Graph (Table.map (apfst o f) tab);
fun get_node G = #1 o #2 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 Keys.member R x then (rs, R)
else Keys.fold reach (next x) (rs, Keys.insert x R) |>> cons x;
fun reachs x (rss, R) =
reach x ([], R) |>> (fn rs => rs :: rss);
in fold reachs xs ([], Keys.empty) end;
(*immediate*)
fun imm_preds G = #1 o #2 o #2 o get_entry G;
fun imm_succs G = #2 o #2 o #2 o get_entry G;
fun immediate_preds G = Keys.dest o imm_preds G;
fun immediate_succs G = Keys.dest o imm_succs G;
(*transitive*)
fun all_preds G = flat o #1 o reachable (imm_preds G);
fun all_succs G = flat o #1 o reachable (imm_succs G);
(*minimal and maximal elements*)
fun minimals G = fold_graph (fn (m, (_, (preds, _))) => Keys.is_empty preds ? cons m) G [];
fun maximals G = fold_graph (fn (m, (_, (_, succs))) => Keys.is_empty succs ? cons m) G [];
fun is_minimal G x = Keys.is_empty (imm_preds G x);
fun is_maximal G x = Keys.is_empty (imm_succs G x);
(*strongly connected components; see: David King and John Launchbury,
"Structuring Depth First Search Algorithms in Haskell"*)
fun strong_conn G =
rev (filter_out null (#1 (reachable (imm_preds G) (all_succs G (keys G)))));
(* nodes *)
fun new_node (x, info) (Graph tab) =
Graph (Table.update_new (x, (info, (Keys.empty, Keys.empty))) tab);
fun default_node (x, info) (Graph tab) =
Graph (Table.default (x, (info, (Keys.empty, Keys.empty))) tab);
fun del_nodes xs (Graph tab) =
Graph (tab
|> fold Table.delete xs
|> Table.map (fn _ => fn (i, (preds, succs)) =>
(i, (fold Keys.remove xs preds, fold Keys.remove xs succs))));
fun del_node x (G as Graph tab) =
let
fun del_adjacent which y =
Table.map_entry y (fn (i, ps) => (i, (which (Keys.remove x) ps)));
val (preds, succs) = #2 (#2 (get_entry G x));
in
Graph (tab
|> Table.delete x
|> Keys.fold (del_adjacent apsnd) preds
|> Keys.fold (del_adjacent apfst) succs)
end;
(* edges *)
fun is_edge G (x, y) = Keys.member (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, (Keys.insert x preds, succs)))
|> map_entry x (fn (i, (preds, succs)) => (i, (preds, Keys.insert y succs)));
fun del_edge (x, y) G =
if is_edge G (x, y) then
G |> map_entry y (fn (i, (preds, succs)) => (i, (Keys.remove x preds, succs)))
|> map_entry x (fn (i, (preds, succs)) => (i, (preds, Keys.remove 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, (Keys.empty, Keys.empty));
fun join f (G1 as Graph tab1, G2 as Graph tab2) =
let fun join_node key ((i1, edges1), (i2, _)) = (f key (i1, i2), edges1) in
if pointer_eq (G1, G2) then G1
else fold add_edge (edges G2) (Graph (Table.join join_node (tab1, Table.map (K no_edges) tab2)))
end;
fun gen_merge add eq (G1 as Graph tab1, G2 as Graph tab2) =
let fun eq_node ((i1, _), (i2, _)) = eq (i1, i2) in
if pointer_eq (G1, G2) then G1
else fold add (edges G2) (Graph (Table.merge eq_node (tab1, Table.map (K 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 (Keys.member X x) orelse eq_key (x, z) orelse member eq_key path x orelse
exists (red x) xs orelse exists (red x) xs'
then irreds xs xs'
else irreds xs (x :: xs');
in irreds (immediate_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 Keys.member X z andalso not (member eq_key path z)
then maps (paths (z :: path)) (immediate_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;
fun topological_order G = minimals G |> all_succs G;
(* maintain transitive acyclic graphs *)
fun add_edge_trans_acyclic (x, y) G =
add_edge_acyclic (x, y) G
|> fold_product (curry add_edge) (all_preds G [x]) (all_succs G [y]);
fun merge_trans_acyclic eq (G1, G2) =
if pointer_eq (G1, G2) then G1
else
merge_acyclic eq (G1, G2)
|> fold add_edge_trans_acyclic (diff_edges G1 G2)
|> fold add_edge_trans_acyclic (diff_edges G2 G1);
(* schedule acyclic graph *)
exception DEP of key * key;
fun schedule f G =
let
val xs = topological_order G;
val results = (xs, Table.empty) |-> fold (fn x => fn tab =>
let
val a = get_node G x;
val deps = immediate_preds G x |> map (fn y =>
(case Table.lookup tab y of
SOME b => (y, b)
| NONE => raise DEP (x, y)));
in Table.update (x, f deps (x, a)) tab end);
in map (the o Table.lookup results) xs end;
(*final declarations of this structure!*)
val map = map_nodes;
val fold = fold_graph;
end;
structure Graph = Graph(type key = string val ord = fast_string_ord);
structure Int_Graph = Graph(type key = int val ord = int_ord);