Finer routing grid: --grid-resolution K with trace halos

A cell was exactly one trace plus its clearance, so a trace owning its cell
was the whole clearance model, and the 0.4 mm grid could not reach most
pins of a 0.5 mm pitch package: between two 0.3 mm pads a trace's centre
has a legal band only 0.15 mm wide. Resolution K makes cells K times
finer and has every trace claim a halo, the cells whose centre is nearer
than trace + clearance to one of its own, released again on rip-up.

K = 1 has an empty halo and is the old grid exactly (tested). On a 0.5 mm
row, K = 4 gets every pin out where K = 1 gets two. On the rc4 boards, K = 2
cuts the sense board's opens from 66 to 41 (JAR: 32) and the power board's
to 7 (JAR: 12), still with no DRC errors from freeroute's own copper.

src/freeroute/route/grid_router.py

@@ -101,6 +101,7 @@ class _Connection:
         "trace_cells",
         "via_cells",
         "via_halo",
+        "halo_nodes",
     )
 
     def __init__(self, index: int, net_no: int, a: Pin, b: Pin) -> None:
@@ -115,6 +116,7 @@ class _Connection:
         self.trace_cells: list[tuple[Cell, int]] = []
         self.via_cells: list[Cell] = []
         self.via_halo: list[Cell] = []
+        self.halo_nodes: list[Node] = []
 
 
 class GridRouter:
@@ -135,6 +137,7 @@ class GridRouter:
         orthogonal: bool = False,
         via_half: int | None = None,
         edge_clearance: int | None = None,
+        resolution: int = 1,
     ) -> None:
         self.board = board
         self.trace_width = max(trace_width, 1)
@@ -149,8 +152,21 @@ class GridRouter:
         # trace copper; 8-connected also produces 45-degree segments.
         self._neighbours = _NEIGHBOURS[:4] if orthogonal else _NEIGHBOURS
         self._orthogonal = orthogonal
-        # cell size must fit a trace plus its clearance to a neighbouring trace
-        self.step = max(self.trace_width + self.clearance, 1)
+        # At resolution 1 a cell is exactly one trace plus its clearance, so a
+        # trace owning its cell is the whole clearance model. At resolution k
+        # cells are k times finer, fine enough to give every pin of a 0.5 mm
+        # pitch row its own column, and a trace also claims a halo: the cells
+        # whose centre is nearer than trace + clearance to one of its own.
+        self.resolution = max(int(resolution), 1)
+        self.step = max((self.trace_width + self.clearance) // self.resolution, 1)
+        need = self.trace_width + self.clearance
+        reach = math.ceil(need / self.step)
+        self._halo_offsets = [
+            (dx, dy)
+            for dx in range(-reach, reach + 1)
+            for dy in range(-reach, reach + 1)
+            if (dx or dy) and math.hypot(dx, dy) * self.step < need
+        ]
         # A via's copper is wider than a trace. Another net's trace needs its
         # centre via_half + clearance + half_width from the via's centre, so a
         # via also claims every cell within via_reach of its own.
@@ -177,6 +193,8 @@ class GridRouter:
         # per-connection occupancy (owner = connection index)
         self._cell_owner: dict[Node, int] = {}
         self._via_owner: dict[Cell, int] = {}
+        # trace halos: node -> indices of the connections claiming it
+        self._halo: dict[Node, list[int]] = {}
         self._rasterize_obstacles()
 
     # --- grid helpers -------------------------------------------------------
@@ -299,6 +317,11 @@ class GridRouter:
             via_owner = self._via_owner.get(cell)
             if via_owner is not None and conns[via_owner].net_no != net_no:
                 blocking = via_owner
+        if blocking is None:
+            for i in self._halo.get(node, ()):
+                if conns[i].net_no != net_no:
+                    blocking = i
+                    break
         if blocking is None:
             return 0.0
         if not allow_ripup:
@@ -458,6 +481,14 @@ class GridRouter:
         conn.via_cells = list(via_cells)
         for cell, layer in conn.trace_cells:
             self._cell_owner[(*cell, layer)] = conn.index
+        conn.halo_nodes = []
+        for (cx, cy), layer in conn.trace_cells:
+            for dx, dy in self._halo_offsets:
+                x, y = cx + dx, cy + dy
+                if 0 <= x < self.cols and 0 <= y < self.rows:
+                    node = (x, y, layer)
+                    self._halo.setdefault(node, []).append(conn.index)
+                    conn.halo_nodes.append(node)
         conn.via_halo = []
         for cell in conn.via_cells:
             for c in self._via_cells(cell):
@@ -492,6 +523,13 @@ class GridRouter:
             if self._via_owner.get(cell) == conn.index:
                 del self._via_owner[cell]
         conn.via_halo = []
+        for node in conn.halo_nodes:
+            claims = self._halo.get(node)
+            if claims is not None:
+                claims.remove(conn.index)
+                if not claims:
+                    del self._halo[node]
+        conn.halo_nodes = []
         conn.trace_cells = []
         conn.via_cells = []
         conn.wires = []
@@ -508,6 +546,9 @@ class GridRouter:
             via_owner = self._via_owner.get((cx, cy))
             if via_owner is not None and conns[via_owner].net_no != net_no:
                 owners.add(via_owner)
+            for i in self._halo.get((cx, cy, cl), ()):
+                if conns[i].net_no != net_no:
+                    owners.add(i)
         return owners
 
     # --- passes -------------------------------------------------------------
@@ -645,6 +686,7 @@ def route_board(
     orthogonal: bool = False,
     via_half: int | None = None,
     edge_clearance: int | None = None,
+    resolution: int = 1,
 ) -> RouteResult:
     """Convenience wrapper: build a :class:`GridRouter` and route the board."""
     router = GridRouter(
@@ -658,5 +700,6 @@ def route_board(
         orthogonal=orthogonal,
         via_half=via_half,
         edge_clearance=edge_clearance,
+        resolution=resolution,
     )
     return router.route()