Convex hull update type hints

ipynb/Convex Hull.ipynb

@@ -51,7 +51,7 @@
     "- **Set of Points**: a Python set: `{Point(0,0), Point(3,4), ...}` The function `Points(n)` will return a set of `n` random points.\n",
     "- **Polygon**: an ordered list of vertex points.\n",
     "\n",
-    "First, get the necessary imports done, then implement the three types:"
+    "After some necessary imports, here is the implementation:"
    ]
   },
   {
@@ -71,7 +71,9 @@
    },
    "outputs": [],
    "source": [
+    "import csv\n",
     "import collections\n",
+    "from collections.abc import Iterable\n",
     "import math\n",
     "import matplotlib.pyplot as plt\n",
     "import random\n",
@@ -100,7 +102,7 @@
     "Polygon = list[Point]\n",
     "\n",
     "def Points(n: int) -> set[Point]:\n",
-    "    \"\"\"Generate n random points within a unit square.\"\"\"\n",
+    "    \"\"\"Create a set of n random points within a unit square.\"\"\"\n",
     "    return {Point(random.random(), random.random())\n",
     "            for _ in range(n)}"
    ]
@@ -124,9 +126,9 @@
     {
      "data": {
       "text/plain": [
-       "{Point(x=0.019776103178765214, y=0.05751328325941196),\n",
+       "{Point(x=0.17464561621533636, y=0.8295359793654173),\n",
-       " Point(x=0.08201516948907406, y=0.831040189361657),\n",
+       " Point(x=0.21776227621335453, y=0.460000427486846),\n",
-       " Point(x=0.10325343098110484, y=0.36755973349968385)}"
+       " Point(x=0.27479975467800144, y=0.9532283692649101)}"
       ]
      },
      "execution_count": 3,
@@ -224,7 +226,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -273,7 +275,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -426,7 +428,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -475,7 +477,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -520,7 +522,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -565,7 +567,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -610,7 +612,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -655,7 +657,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -700,7 +702,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -745,7 +747,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -790,7 +792,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -835,7 +837,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -853,21 +855,24 @@
    "metadata": {
     "button": false,
     "deletable": true,
+    "editable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
-    }
+    },
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
    },
    "source": [
     "That's all there is to the basic idea of the algorithm, but there are a few edge cases to worry about: \n",
     "\n",
     "* **Degenerate polygons**: A triangle has 3 sides, and no \"true\" polygon has fewer, but it is possible to have a set of 0, 1, or 2 points. Such a set of points is convex because there is no way to draw a line segment that goes outside the points.\n",
     "\n",
-    "* **Colinear points:** if three or more points are colinear, we could keep them all on the hull, or we could keep only the two \"outside\" ones.  I decided to make the convex hull be the minimal possible set of points, and thus only keep the outside points. \n",
+    "* **Colinear points:** if three or more points are colinear and are on the hull, we could keep them all, or we could keep only the two \"outside\" ones.  I decided to make the convex hull be the minimal possible set of points, and thus only keep the outside points. \n",
     "\n",
-    "* **First and last points:** An astute reader might have noticed that our algorithm only rejects the middle point, point B, in the A->B->C turn. That means that the first and last point in sorted order will never be a candidate for rejection, and thus will always end up on the hull.  Is that correct?  Yes it is. The first point is the leftmost point, the one with lowest `x` value (and if there are ties, it is the lowest-leftmost point). That is an extreme corner, so it should always be on the hull. A similar argument holds for the last point in sorted order.\n",
+    "* **First and last points:** An astute reader might have noticed that our algorithm only rejects the middle point, point B, in the A->B->C turn. That means that the first and last point in sorted order will never be a candidate for rejection, and thus will always end up on the hull.  Is that correct?  Yes it is. The first point is the leftmost point, the one with lowest `x` value (and if there are ties, it is the lowest-leftmost point). That is an extreme corner, so it should always be on the hull. A similar argument holds for the last point in sorted order."
-    "\n",
-    "\n"
    ]
   },
   {
@@ -875,10 +880,15 @@
    "metadata": {
     "button": false,
     "deletable": true,
+    "editable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
-    }
+    },
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
    },
    "source": [
     "# Implementation of Convex Hull Algorithm\n",
@@ -893,22 +903,27 @@
     "button": false,
     "collapsed": false,
     "deletable": true,
+    "editable": true,
     "jupyter": {
      "outputs_hidden": false
     },
     "new_sheet": false,
     "run_control": {
      "read_only": false
-    }
+    },
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
    },
    "outputs": [],
    "source": [
     "def convex_hull(points) -> Polygon:\n",
     "    \"\"\"Find the convex hull of a set of points.\"\"\"\n",
-    "    if len(points) <= 3: # Degenerate polygons and triangles\n",
+    "    if len(points) < 3: \n",
-    "        return points\n",
+    "        return Polygon(points) # Degenerate polygons\n",
-    "    # Find the two half-hulls ...\n",
     "    sorted_points = sorted(points)\n",
+    "    # Find the two half-hulls ...\n",
     "    upper = half_hull(sorted_points)\n",
     "    lower = half_hull(reversed(sorted_points))\n",
     "    return upper + lower[1:-1] # ... and append them, but don't repeat first and last points\n",
@@ -928,7 +943,12 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "run_control": {}
+    "editable": true,
+    "run_control": {},
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
    },
    "source": [
     "# Visualization of Results\n",
@@ -987,7 +1007,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1009,12 +1029,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "16 of 1000 points are on the convex hull\n"
+      "17 of 1000 points are on the convex hull\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1053,7 +1073,7 @@
     },
     {
      "data": {
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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1085,7 +1105,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1137,7 +1157,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1147,15 +1167,15 @@
     }
    ],
    "source": [
-    "def parse_cities(text: str, skip=('AK', 'HI'), longitude_scale=-3/4) -> set[Point]:\n",
+    "! [ -e latlong.csv ] || curl -O https://raw.githubusercontent.com/norvig/pytudes/refs/heads/main/data/latlong.csv\n",
-    "    \"\"\"Extract a set of Points from text, skipping the specified states.\"\"\"\n",
+    "    \n",
-    "    # Entries are of the form \"[FLG]  35.13  111.67  Flagstaff,AZ\"\n",
+    "def parse_latlong(rows: Iterable[tuple], skip=('AK', 'HI'), long_scale=-48, lat_scale=69) -> set[Point]:\n",
-    "    entries = re.findall(r'([\\d.]+)\\s+([\\d.]+).+([A-Z][A-Z])', text)\n",
+    "    \"\"\"Make a set of Points from the CSV file of \"long,lat,city,state\", skipping the given states.\"\"\"\n",
-    "    return {Point(longitude_scale * float(long), float(lat))\n",
+    "    return {Point(long_scale * float(long), lat_scale  * float(lat))\n",
-    "                  for (lat, long, state) in entries\n",
+    "                  for (long, lat, city, state) in rows\n",
     "                  if state not in skip}\n",
     "\n",
-    "USA = parse_cities(open('latlong.htm').read())\n",
+    "USA = parse_latlong(csv.reader(open('latlong.csv')))\n",
     "plot_convex_hull(USA)"
    ]
   },
@@ -1193,7 +1213,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1231,12 +1251,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "14 of 100 points are on the convex hull\n"
+      "10 of 100 points are on the convex hull\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1276,12 +1296,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "20 of 720 points are on the convex hull\n"
+      "17 of 720 points are on the convex hull\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1309,12 +1329,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "19 of 360 points are on the convex hull\n"
+      "18 of 360 points are on the convex hull\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 1 Axes>"
       ]
@@ -1345,7 +1365,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 640x480 with 2 Axes>"
       ]
@@ -1355,7 +1375,7 @@
     }
    ],
    "source": [
-    "def hull_stats(n: int, k=30) -> tuple[float, float]:\n",
+    "def hull_stats(n: int, k=40) -> tuple[float, float]:\n",
     "    \"\"\"Run convex_hull(Points(n)) k times, return mean number of points on hull and mean run time in milliseconds.\"\"\"\n",
     "    t0 = time.process_time()\n",
     "    total_hull = sum(len(convex_hull(Points(n))) for _ in range(k))\n",
@@ -1389,7 +1409,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The number of points on the hull of ppoints sampled uniformly from a square is (roughly) a straight line on the log graph, which means it is *O*(log *n*). This agrees with the [literature](https://arxiv.org/abs/1111.5340). The [literature](https://en.wikipedia.org/wiki/Convex_hull_algorithms) also says the run time should be *O*(*n* log *n*), and just from eyeballing the plot that looks right (although it would be more convincing if we did a statistical test to verify it). The rightmost point is for sets with *n* = 2<sup>18</sup> =  262,144 points, and we compute their convex hulls in less than half a second on average (and find an average of about 33 points on the convex hull)."
+    "The number of points on the hull of ppoints sampled uniformly from a square is (roughly) a straight line on the log graph, which means it is *O*(log *n*). This agrees with the [literature](https://arxiv.org/abs/1111.5340). The [literature](https://en.wikipedia.org/wiki/Convex_hull_algorithms) also says the run time should be *O*(*n* log *n*), and just from eyeballing the plot that looks right (although it would be more convincing if we did a statistical test to verify it). The rightmost point is for sets with *n* = 2<sup>18</sup> =  262,144 points, and we compute their convex hulls in less than half a second on average (and find an average of about 33 points on the convex hull).\n",
+    "\n",
+    "# Conclusion\n",
+    "\n",
+    "The convex hull problem is an interesting exercise in algorithm design.\n",
+    "The algorithm covered here is called  [Andrew's Monotone Chain](https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain).\n",
+    "It is a variant of the [Graham Scan](https://en.wikipedia.org/wiki/Graham_scan).\n",
+    "You can read more from  [Wikipedia](https://en.wikipedia.org/wiki/Convex_hull_algorithms)."
    ]
   },
   {
@@ -1406,9 +1433,14 @@
    "execution_count": 31,
    "metadata": {
     "collapsed": false,
+    "editable": true,
     "jupyter": {
      "outputs_hidden": false
-    }
+    },
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
    },
    "outputs": [
     {
@@ -1450,7 +1482,7 @@
     "    assert convex_hull(donut) == convex_hull(convex_hull(donut))\n",
     "    for n in (0, 1, 2, 3):\n",
     "        points = Points(n)\n",
-    "        assert convex_hull(points) == points\n",
+    "        assert set(convex_hull(points)) == points\n",
     "    collinear = {Point(x, 0) for x in range(100)}\n",
     "    assert convex_hull(collinear) == [min(collinear), max(collinear)]\n",
     "    P = Point(5, 5)\n",
@@ -1464,30 +1496,24 @@
    ]
   },
   {
-   "cell_type": "markdown",
+   "cell_type": "code",
+   "execution_count": null,
    "metadata": {
-    "button": false,
+    "editable": true,
-    "deletable": true,
+    "slideshow": {
-    "new_sheet": false,
+     "slide_type": ""
-    "run_control": {
+    },
-     "read_only": false
+    "tags": []
-    }
    },
-   "source": [
+   "outputs": [],
-    "# Concluding Remarks and Further Reading\n",
+   "source": []
-    "\n",
-    "The convex hull problem is an interesting exercise in algorithm design.\n",
-    "The algorithm covered here is called  [Andrew's Monotone Chain](https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain).\n",
-    "It is a variant of the [Graham Scan](https://en.wikipedia.org/wiki/Graham_scan).\n",
-    "You can read more from  [Wikipedia](https://en.wikipedia.org/wiki/Convex_hull_algorithms)."
-   ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
+   "display_name": "Python [conda env:base] *",
    "language": "python",
-   "name": "python3"
+   "name": "conda-base-py"
   },
   "language_info": {
    "codemirror_mode": {
@@ -1499,7 +1525,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.13.3"
+   "version": "3.13.9"
   }
  },
  "nbformat": 4,