fix mincut() (JuliaGraphs#325)

* s-t cut weight should be adj_cost. fixes JuliaGraphs#324

* mincut(): simplify cutweight computation

* improve performance by 20x using dequeue_pair!

* _merge_vertex!: use inplace updates

saves another 20% runtime

* set u to new root in heuristic merge

* add comments

---------

Co-authored-by: Etienne dg <etienne.mace_de_gastines@insa-rouen.fr>

src/traversals/maxadjvisit.jl

@@ -26,6 +26,9 @@ assumed to be 1.
     # in which case we'll return immediately.
     (nvg > one(U)) || return (Vector{Int8}([1]), zero(T))
 
+    # to avoid reallocating lists in fadjlist, we have some already merged vertices
+    # still appearing in fadjlist. When iterating neighbors, is_merged makes sure we
+    # don't consider them
     is_merged = falses(nvg)
     merged_vertices = IntDisjointSets(U(nvg))
     graph_size = nvg
@@ -54,7 +57,6 @@ assumed to be 1.
 
     is_processed = falses(nvg)
     @inbounds while graph_size > 1
-        cutweight = zero(T)
         is_processed .= false
         is_processed[u] = true
         # initialize pq
@@ -66,32 +68,23 @@ assumed to be 1.
         for v in fadjlist[u]
             (is_merged[v] || v == u) && continue
             pq[v] = w[u, v]
-            cutweight += w[u, v]
         end
         # Minimum cut phase
+        local cutweight
         while true
             last_vertex = u
-            u, adj_cost = first(pq)
-            dequeue!(pq)
+            u, cutweight = dequeue_pair!(pq)
             isempty(pq) && break
             for v in fadjlist[u]
-                (is_merged[v] || u == v) && continue
-                # if the target of e is already marked then decrease cutweight
-                # otherwise, increase it
-                ew = w[u, v]
-                if is_processed[v]
-                    cutweight -= ew
-                else
-                    cutweight += ew
-                    pq[v] += ew
-                end
+                (is_processed[v] || is_merged[v] || u == v) && continue
+                pq[v] += w[u, v]
             end
             is_processed[u] = true
-            # adj_cost is a lower bound on the cut separating the two last vertices
-            # encountered, so if adj_cost >= bestweight, we can already merge these
+            # cutweight is a lower bound on the cut separating the two last vertices
+            # encountered, so if cutweight >= bestweight, we can already merge these
             # vertices to save one phase.
-            if adj_cost >= bestweight
-                _merge_vertex!(merged_vertices, fadjlist, is_merged, w, u, last_vertex)
+            if cutweight >= bestweight
+                u = _merge_vertex!(merged_vertices, fadjlist, is_merged, w, u, last_vertex)
                 graph_size -= 1
             end
         end
@@ -105,14 +98,14 @@ assumed to be 1.
         end
 
         # merge u and last_vertex
-        root = _merge_vertex!(merged_vertices, fadjlist, is_merged, w, u, last_vertex)
+        u = _merge_vertex!(merged_vertices, fadjlist, is_merged, w, u, last_vertex)
         graph_size -= 1
-        u = root # we are sure this vertex was not merged, so the next phase start from it
     end
     return (convert(Vector{Int8}, parities) .+ one(Int8), bestweight)
 end
 
 function _merge_vertex!(merged_vertices, fadjlist, is_merged, w, u, v)
+    # root is kept, non-root is discarded
     root = union!(merged_vertices, u, v)
     non_root = (root == u) ? v : u
     is_merged[non_root] = true
@@ -122,7 +115,7 @@ function _merge_vertex!(merged_vertices, fadjlist, is_merged, w, u, v)
         w[v2, root] = w[root, v2]
     end
     # update neighbors
-    fadjlist[root] = union(fadjlist[root], fadjlist[non_root])
+    union!(fadjlist[root], fadjlist[non_root])
     for v in fadjlist[non_root]
         if root ∉ fadjlist[v]
             push!(fadjlist[v], root)