Point-Cover-Interval Problem

Author: rasbt

ipython_nbs/essentials/greedy-algorithm-intro.ipynb

@@ -205,7 +205,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## 0-1 Knapsack"
+    "### 0-1 Knapsack"
    ]
   },
   {
@@ -336,7 +336,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Fractional Knapsack"
+    "### Fractional Knapsack"
    ]
   },
   {
@@ -457,6 +457,100 @@
     "knappsack_fract(capacity=10, weights=weights, values=values)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example 3: Point-Cover-Interval Problem"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The classic Point-Cover-Interval problem is another example that is well suited for demonstrating greedy algorithms. Here, we are given a set of Intervals *L*, and we want to find the minimum set of points so that each interval is covered at least once by a given point as illustrated in the example below:\n",
+    "\n",
+    "![](./images/point-cover-interval-ex.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Our greedy strategy, which finds the optimal solution for this problem, can be as follows:\n",
+    "\n",
+    "- sort intervals in increasing order by the value of their endpoints\n",
+    "- for interval in interval-set:\n",
+    "    - if interval is not yet covered:\n",
+    "        - add interval-endpoint to the set of points"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "def min_points(intervals):\n",
+    "    s_ints = sorted(intervals, key=lambda x: x[1])\n",
+    "\n",
+    "    points = [s_ints[0][-1]]\n",
+    "\n",
+    "    for interv in s_ints:\n",
+    "        if not(points[-1] >= interv[0] and points[-1] <= interv[-1]):\n",
+    "            points.append(interv[-1])\n",
+    "        \n",
+    "    return points"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[3]"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pts = [[2, 5], [1, 3], [3, 6]] \n",
+    "min_points(pts)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[3, 6]"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pts = [[4, 7], [1, 3], [2, 5], [5, 6]]\n",
+    "min_points(pts)"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},