Finish todo in asp notebook

notebooks/asp.ipynb

@@ -31,7 +31,7 @@
    "id": "ade31d26",
    "metadata": {},
    "source": [
-    "## The all pairs of shortest paths (ASP) problem\n",
+    "## The All Pairs of Shortest Paths (ASP) problem\n",
     "\n",
     "Let us start by presenting the all pairs of shortest paths (ASP) problem and its solution with the [Floyd–Warshall algorithm](https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm).\n",
     "\n",
@@ -70,10 +70,21 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 1,
    "id": "4fe447c5",
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "floyd! (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "function floyd!(C)\n",
     "  n = size(C,1)\n",
@@ -99,10 +110,25 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 2,
    "id": "860e537c",
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "4×4 Matrix{Int64}:\n",
+       "  0  9  6  1\n",
+       "  2  0  8  3\n",
+       "  5  3  0  6\n",
+       " 10  8  5  0"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "inf = 1000\n",
     "C = [\n",
@@ -296,7 +322,7 @@
    "id": "db9a3294",
    "metadata": {},
    "source": [
-    "TODO explain this part"
+    "At each iteration, the processor needs the input values `C[i,j]`, `C[i,k]` and `C[k,j]`. Since we split the data row-wise, the process already has values `C[i,j]` and `C[i,k]`. However, `C[k,j]` may be stored in a different process. "
    ]
   },
   {
@@ -314,6 +340,14 @@
     "</div>"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "a01872ef",
+   "metadata": {},
+   "source": [
+    "As we iterate over columns $j$, the process needs input from all values of row $k$. Therefore, at the start of iteration $k$, the whole row $k$ needs to be communicated."
+   ]
+  },
   {
    "attachments": {
     "g12957.png": {
@@ -634,7 +668,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Julia 1.9.0",
+   "display_name": "Julia 1.9.1",
    "language": "julia",
    "name": "julia-1.9"
   },
@@ -642,7 +676,7 @@
    "file_extension": ".jl",
    "mimetype": "application/julia",
    "name": "julia",
-   "version": "1.9.0"
+   "version": "1.9.1"
   }
  },
  "nbformat": 4,