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Add Wansdorff's algorithm implementation and benchmarks
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‎source_code/knights_tour.ipynb‎

Lines changed: 151 additions & 49 deletions
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"execution_count": 3,
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"id": "d51322fb-c8f6-4d9e-a378-1b3653b86ae4",
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"metadata": {},
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"outputs": [],
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": 4,
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"id": "31e73e9c-6f07-4033-a1c8-919db0086eaa",
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"metadata": {},
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"outputs": [],
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"execution_count": null,
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"id": "0ccb1546-3cd9-4ef5-9ad7-f4b488508462",
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"metadata": {},
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"outputs": [],
@@ -532,7 +532,7 @@
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"outputs": [],
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"source": [
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"def find_tour(n):\n",
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" '''find a knight's tour on an n-by-n board\n",
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" '''find a knight's tour on an n-by-n board using backtracking\n",
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" \n",
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" Parameters\n",
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" ----------\n",
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "43a3ac96-8353-4828-a0fb-ed5f10421136",
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"execution_count": 88,
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"id": "c7e766e4-ab0d-4bf0-81f9-4bdc5ec1b15e",
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"metadata": {},
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"outputs": [],
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"source": [
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"def compute_initial_accessibility(n):\n",
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" return [len(moves) for moves in compute_all_moves(n)]"
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"def find_wansdorff_tour(n):\n",
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" '''find a knight's tour on an n-by-n board using Wansdorff's algorithm\n",
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" \n",
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" Parameters\n",
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" ----------\n",
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" n: int\n",
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" size of the n-by-n board\n",
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" \n",
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" Returns\n",
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" -------\n",
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" list[int]\n",
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" the sequence of positions on the board that form a valid knight's tour,\n",
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" or an empty list if no tour can be found\n",
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" '''\n",
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" def update_moves(all_moves, pos):\n",
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" for move in all_moves[pos]:\n",
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" try:\n",
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" all_moves[move].remove(pos)\n",
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" except ValueError:\n",
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" pass\n",
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" all_moves = compute_all_moves(n)\n",
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" tour = [0]\n",
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" update_moves(all_moves, tour[-1])\n",
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" for move_nr in range(2, n**2 + 1):\n",
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" min_pos, min_accessibility = None, n**2\n",
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" for move in all_moves[tour[-1]]:\n",
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" accessibility = len(all_moves[move])\n",
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" if move not in tour and 0 <= accessibility < min_accessibility:\n",
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" min_accessibility = accessibility\n",
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" min_pos = move\n",
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" if min_pos is None:\n",
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" return []\n",
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" tour.append(min_pos)\n",
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" update_moves(all_moves, tour[-1])\n",
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" return tour "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "57130a92-ba10-4c43-a7da-b55f32c36d49",
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"execution_count": 90,
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"id": "1cd3254d-c09f-4c57-909f-08fd3f4e9bf5",
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" 1 14 9 20 3\n",
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" 24 19 2 15 10\n",
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" 13 8 25 4 21\n",
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" 18 23 6 11 16\n",
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" 7 12 17 22 5\n"
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]
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}
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],
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"source": [
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"def print_accessibility(accessibility):\n",
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" n = math.isqrt(len(accessibility))\n",
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" for i in range(n):\n",
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" print(' '.join(map(lambda x: f'{x:4d}', accessibility[i*n:(i + 1)*n])))"
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"print_tour(find_wansdorff_tour(5))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "44cdf539-c4c7-4f3c-937b-37a5f4b69ff7",
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"metadata": {},
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"source": [
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"## Performance"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "5e4758f3-fa22-4566-bf89-8e90dd563fa7",
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"execution_count": 91,
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"id": "56eaf07d-91e9-4459-b7a9-8c003402e894",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" 2 3 4 3 2\n",
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" 3 4 6 4 3\n",
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" 4 6 8 6 4\n",
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" 3 4 6 4 3\n",
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" 2 3 4 3 2\n"
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"27.6 µs ± 378 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
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]
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}
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],
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"source": [
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"print_accessibility(compute_initial_accessibility(5))"
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"%timeit find_wansdorff_tour(5)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 23,
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"id": "fabc1605-b986-4d77-9756-5744f122ba30",
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"execution_count": 92,
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"id": "629fd883-54c4-4268-93be-088917c5979f",
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"43.8 µs ± 270 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
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]
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}
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],
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"source": [
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"def wansdorff(n):\n",
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" all_moves = compute_all_moves(n)\n",
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" accessibility =[len(moves) for moves in compute_all_moves(n)]\n",
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" board = [-1]*n\n",
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" pos = 0\n",
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" board[0] = 1\n",
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" for move_nr in range(2, n**2 + 1):\n",
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" pos = min(all_moves[pos], accessibility[k] for k in all_moves[pos] if k not in board)\n",
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" board[pos] = move_nr\n",
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" return board "
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"%timeit find_wansdorff_tour(6)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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"id": "1cd3254d-c09f-4c57-909f-08fd3f4e9bf5",
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"execution_count": 93,
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"id": "ab4f6205-23b7-4962-b277-4efb2a47818f",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"66.2 µs ± 1.58 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
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]
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}
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],
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"source": [
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"%timeit find_wansdorff_tour(7)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 94,
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"id": "b29816be-4099-4a67-b8da-8503878ddd77",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"92.4 µs ± 881 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
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]
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}
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],
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"source": [
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"%timeit find_wansdorff_tour(8)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 96,
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"id": "51b9047e-6986-47e7-9530-b6555c921a4c",
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"metadata": {},
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"outputs": [
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{
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"ename": "IndexError",
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"evalue": "list assignment index out of range",
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"output_type": "error",
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"traceback": [
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"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
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"\u001b[0;31mIndexError\u001b[0m Traceback (most recent call last)",
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"Cell \u001b[0;32mIn[24], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m print_accessibility(wansdorff(\u001b[38;5;241m5\u001b[39m))\n",
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"Cell \u001b[0;32mIn[23], line 9\u001b[0m, in \u001b[0;36mwansdorff\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m move_nr \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(\u001b[38;5;241m2\u001b[39m, n\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m2\u001b[39m \u001b[38;5;241m+\u001b[39m \u001b[38;5;241m1\u001b[39m):\n\u001b[1;32m 8\u001b[0m pos \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mmin\u001b[39m(accessibility[k] \u001b[38;5;28;01mfor\u001b[39;00m k \u001b[38;5;129;01min\u001b[39;00m all_moves[pos] \u001b[38;5;28;01mif\u001b[39;00m k \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;129;01min\u001b[39;00m board)\n\u001b[0;32m----> 9\u001b[0m board[pos] \u001b[38;5;241m=\u001b[39m move_nr\n\u001b[1;32m 10\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m board\n",
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"\u001b[0;31mIndexError\u001b[0m: list assignment index out of range"
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"63.7 ms ± 338 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
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]
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}
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],
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"source": [
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"print_accessibility(wansdorff(5))"
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"%timeit find_wansdorff_tour(50)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 95,
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"id": "e84c70dc-36aa-4103-ae35-f643bf67d4c2",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1.29 s ± 27.8 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
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]
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}
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],
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"source": [
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"%timeit find_wansdorff_tour(100)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f84d6c6e-5cd4-49ce-b8be-903b114366fc",
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"metadata": {},
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"source": [
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"The runtime is very low, and increases slowly with the size of the board, $O(n)$."
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]
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},
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{
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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"version": "3.11.3"
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}
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},
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"nbformat": 4,

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