solver: break board dihedral symmetry

Constrain the unique unit square to a closed D4 fundamental region once its placement is known. This preserves one representative of every board-orientation orbit without assigning identities to repeated squares.

Keep a symmetry-disabled benchmark path, document the proof and measurements, and cover generic, diagonal, midline, corner, and centre orbits.

Tests: Release, Debug, ASan, and UBSan CTest (11 passed each)

Refs: #3

benchmarks/benchmark.cc

@@ -28,6 +28,7 @@ namespace {
 
   auto solve(std::uint64_t order, bool instrument,
              SearchPolicy policy, bool direct_search,
+             SymmetryBreaking symmetry,
              SearchCounters &counters)
       -> TimedSolution {
     auto const predecessor_order =
@@ -36,8 +37,8 @@ namespace {
     auto predecessor =
         instrument
             ? search_solution_instrumented(predecessor_order, counters,
-                                           policy)
-            : search_solution(predecessor_order, policy);
+                                           policy, symmetry)
+            : search_solution(predecessor_order, policy, symmetry);
     auto const search_end = Clock::now();
 
     auto const construction_begin = Clock::now();
@@ -93,9 +94,10 @@ namespace {
 }
 
 int main(int argc, char **argv) {
-  if (argc < 3 || argc > 5) {
+  if (argc < 3 || argc > 6) {
     std::cerr << "usage: partridge_benchmark ORDER counters|plain "
-                 "[ascending|descending|best-fit] [public|direct]\n";
+                 "[ascending|descending|best-fit] [public|direct] "
+                 "[d4|none]\n";
     return 2;
   }
   auto const order = static_cast<std::uint64_t>(std::strtoull(argv[1], nullptr, 10));
@@ -117,15 +119,25 @@ int main(int argc, char **argv) {
     return 2;
   }
   auto const direct_search =
-      argc == 5 && std::string_view(argv[4]) == "direct";
-  if (argc == 5 && std::string_view(argv[4]) != "public" &&
+      argc >= 5 && std::string_view(argv[4]) == "direct";
+  if (argc >= 5 && std::string_view(argv[4]) != "public" &&
       std::string_view(argv[4]) != "direct") {
     std::cerr << "search route must be public or direct\n";
     return 2;
   }
+  auto const symmetry =
+      argc < 6 || std::string_view(argv[5]) == "d4"
+          ? SymmetryBreaking::d4_unit_square
+          : SymmetryBreaking::disabled;
+  if (argc == 6 && std::string_view(argv[5]) != "d4" &&
+      std::string_view(argv[5]) != "none") {
+    std::cerr << "symmetry mode must be d4 or none\n";
+    return 2;
+  }
 
   SearchCounters counters;
-  auto timed = solve(order, instrument, policy, direct_search, counters);
+  auto timed =
+      solve(order, instrument, policy, direct_search, symmetry, counters);
 
   auto const validation_begin = Clock::now();
   auto const validation_ok = valid(order, timed.result);
@@ -150,6 +162,9 @@ int main(int argc, char **argv) {
       << "\""
       << ",\"search_route\":\""
       << (direct_search ? "direct" : "public") << "\""
+      << ",\"symmetry_breaking\":\""
+      << (symmetry == SymmetryBreaking::d4_unit_square ? "d4" : "none")
+      << "\""
       << ",\"solved\":" << boolean(!timed.result.squares().empty())
       << ",\"valid\":" << boolean(validation_ok)
       << ",\"timing_seconds\":{\"solve\":" << timed.search_seconds

benchmarks/run.py

@@ -79,11 +79,20 @@ def environment(binary):
     }
 
 
-def run_once(binary, order, mode, search_policy, search_route, timeout):
+def run_once(
+    binary, order, mode, search_policy, search_route, symmetry, timeout
+):
     started = time.monotonic()
     try:
         process = subprocess.run(
-            [str(binary), str(order), mode, search_policy, search_route],
+            [
+                str(binary),
+                str(order),
+                mode,
+                search_policy,
+                search_route,
+                symmetry,
+            ],
             check=False,
             capture_output=True,
             text=True,
@@ -223,6 +232,11 @@ def main():
         default="ascending",
     )
     parser.add_argument("--direct-search", action="store_true")
+    parser.add_argument(
+        "--no-symmetry",
+        action="store_true",
+        help="disable unique-unit-square D4 symmetry breaking",
+    )
     parser.add_argument(
         "--measure-overhead",
         action="store_true",
@@ -254,6 +268,7 @@ def main():
                     mode,
                     args.search_policy,
                     "direct" if args.direct_search else "public",
+                    "none" if args.no_symmetry else "d4",
                     args.timeout,
                 )
         for trial in range(args.repetitions):
@@ -265,6 +280,7 @@ def main():
                         mode,
                         args.search_policy,
                         "direct" if args.direct_search else "public",
+                        "none" if args.no_symmetry else "d4",
                         args.timeout,
                     )
                 )
@@ -293,6 +309,7 @@ def main():
             "per_run_timeout_seconds": args.timeout,
             "candidate_order": args.search_policy,
             "search_route": "direct" if args.direct_search else "public",
+            "symmetry_breaking": "none" if args.no_symmetry else "d4",
             "stdout": "captured; rendered grid suppressed by probe",
         },
         "cases": cases,