\n",
@@ -273,10 +359,21 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 8,
"id": "75cac17e",
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "floyd2! (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"function floyd2!(C)\n",
" n = size(C,1)\n",
@@ -302,10 +399,19 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 10,
"id": "907bc8c9",
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ " 1.777208 seconds\n",
+ " 2.967238 seconds\n"
+ ]
+ }
+ ],
"source": [
"n = 1000\n",
"C = rand(n,n)\n",
@@ -347,7 +453,36 @@
"```\n",
"\n",
"- The outer loop cannot be parallelized since the operations at iteration $k$ require the result of iteration $k-1$.\n",
- "- The inner loops over $i$ and $j$ can be parallelized at a given iteration $k$."
+ "- The inner loops over $i$ and $j$ can be parallelized at a given iteration $k$.\n",
+ "\n",
+ "\n",
+ "
\n",
+ "Question (hard): Can we really parallelize the loops over `i` and `j` ? To compute `C[i,j]` at iteration `k`, we first need to compute `C[i,k]` and `C[k,j]`. In order words, it seems that the order of the loops over `i` and `j` cannot be arbitrary. The first value of `i` and `j` in the loops should be `k`. However, this is not really necessary, why?\n",
+ "
"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "id": "8d05b686",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "It is not needed to start the loop over `j` with `j=k` as none of the entries of the matrix will be updated\n",
+ "in this particular case. Rememebr: `C[i,j] = min(C[i,j],C[i,k]+C[k,j])` if we substitute `j=k`, we get:\n",
+ "`C[i,k] = min(C[i,k],C[i,k]+C[k,k])`. Rememeber that `C[k,k]=0, thus, `C[i,k] = min(C[i,k],C[i,k])`, or\n",
+ "`C[i,k] = C[i,k]`. In other words, the new value of `C[i,k]` will correspond to the old value.\n",
+ "The same is true for `i=k`.\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "uncover = true\n",
+ "q1_answer(uncover)"
]
},
{
@@ -411,7 +546,7 @@
"id": "db9a3294",
"metadata": {},
"source": [
- "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. "
+ "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. See figure below."
]
},
{
@@ -492,10 +627,18 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 25,
"id": "1bf4de56",
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "It's not correct. Keep trying! 💪\n"
+ ]
+ }
+ ],
"source": [
"answer = \"x\" # replace x with a, b, c or d\n",
"floyd_check(answer)"
@@ -506,7 +649,7 @@
"id": "eba45ea4",
"metadata": {},
"source": [
- "### Efficiency"
+ "### Theoretical performance analysis"
]
},
{
@@ -514,7 +657,9 @@
"id": "a15bc34e",
"metadata": {},
"source": [
- "**Computation cost:** Each process updates $N^2/P$ entries per iteration. "
+ "### Computation complexity\n",
+ "\n",
+ "Each process updates $N^2/P$ entries per iteration. The computation complexity is $O(N^2/P)$."
]
},
{
@@ -537,8 +682,9 @@
"id": "d89d2b45",
"metadata": {},
"source": [
- "**Communication cost:** \n",
- "- One process broadcasts a message of length $N$ to $P-1$ processes per iteration. Thus, the **send cost** is $O(N P)$ per iteration (if we use send/receive instead of broadcast).\n",
+ "### Communication complexity\n",
+ "\n",
+ "- One process broadcasts a message of length $N$ to $P-1$ processes per iteration. Thus, the **send cost** is $O(N P)$ per iteration.\n",
"- $P-1$ processes receive one message of length $N$ per iteration. Hence, the **receive cost** is $O(N)$ per iteration at each process. "
]
},
@@ -562,7 +708,13 @@
"id": "6993b9d0",
"metadata": {},
"source": [
- "In summary, the send/computation ratio is $O(P^2/N)$ and the receive/computation ratio is $O(P/N)$. The algorithm is potentially scalable if $P<
u
+
u_new
+
u_new[i] = u[i-1]+u[i]+u[1+1]
+
i
+
i
+
i+1
+
i-1
+
i
+
i+1
+
i-1
+
i
+
i
+
i+1
+
i-1
+
i
+
u
+
u_new
+
u
+
u_new
+
u
+
u_new
+
u
+
u
+
u_new
+
u
+
u_new
+
i
+
i-1
+
i+1
+
i
+
chnl_next_snd
+
chnl_next_rcv
+
chnl_prev_rcv
+
chnl_prev_snd
+
x
+
u
+
0
+
L
+
-1
+
1
+
0
+
L
+
n+2 points
+
u
+
?
+
-1
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
1
+
u
+
u_new
+
i
+
i-1
+
i+1
+
i
+
i
+
i+1
+
i-1
+
i
+
u
+
u_new
+
k
+
i
+
j
+
u
+
u
+
u
+
u_new
+
i
+
i-1
+
i+1
+
i
+
k
+
i
+
j
+
k
+
i
+
j
+
u
+
?
+
-1
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
1
+
u
+
u_new
+
?
+
-1
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
1
+
-1
+
-1
+
1
+
1
+
s
+
r
+
s
+
u
+
u_new
+
r
+
s
+
r
+
s
+
u
+
u_new
+
r
+
r-1
+
s+1
+
u
+
u
+
u_new
+
communication
+
computation
+
u
+
u_new
+
i-1
+
i+1
+
i
+
i+1
+
-1
+
-1
+
-1
+
1
+
1
+
1
+
u
+
u
+
u_new
+
communication
+
computation
+
-1
+
-1
+
-1
+
1
+
1
+
1
+
(i+1,j)
+
(i,j)
+
(i-1,j)
+
(i,j+1)
+
(i,j-1)
+
+
u
+
u_new
+
local
+
remote
+
remote
+
k
+
i
+
j
+
Graph G
+
...
+
...
+
...
+
...
+
...
+
...
+
Master
+
Workers
+
maxhops
+
u
+
u_new
+
rank 0
+
rank 1
+
rank 2
+
root
+
rand(1:10)
+
u
+
u_new
+
local
+
remote
+
remote
+
local
+
remote
+
Distance matrix C
+
i+1
+
i-1
+
i
+
u
+
u_new
+
i+1
+
i-1
+
i
+
u
+
u_new
+
u
+
u_new
+
-1
+
-1
+
1
+
1
+
-1
+
1
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
?
+
-1
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_08.png"
+ y="303.1282"
+ x="218.77849"
+ height="5.1131492"
+ width="5.1131492"
+ id="rect847-5-7-49"
+ style="fill:none;fill-opacity:0.21383099;stroke:#999999;stroke-width:0.41446099;stroke-linecap:square;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none" />-1
+
1
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_08.png"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect8341);display:inline;stroke-width:1"
+ id="text8339-6"
+ xml:space="preserve"
+ x="62.376186"
+ y="190.4624">1
+
i-1
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_09.png"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect1044-2-1-7);display:inline;stroke-width:1"
+ id="text1042-1-7-4-6"
+ xml:space="preserve"
+ x="108.95108"
+ y="250.83264">i-1
+
i+1
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_09.png"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect961-6-9-1-1);display:inline;stroke-width:1"
+ id="text959-2-1-1-4-3"
+ xml:space="preserve"
+ x="118.81889"
+ y="250.66995">i+1
+
i
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_09.png"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect8986);display:inline;stroke-width:1"
+ id="text8984-3"
+ xml:space="preserve"
+ x="115.70671"
+ y="264.6026">i
+
-1
+ style="fill:none;fill-opacity:0.21383099;stroke:#999999;stroke-width:0.41446099;stroke-linecap:square;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none"
+ id="rect23749"
+ width="5.1131492"
+ height="5.1131492"
+ x="218.77849"
+ y="316.91107"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_08.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200" />-1
+
1
+ y="204.2453"
+ x="62.376186"
+ xml:space="preserve"
+ id="text23779"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect8341);display:inline;stroke-width:1"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_08.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200">1
+
-1
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/francesc/fig09.png"
+ style="fill:none;stroke:#000000;stroke-width:0.26499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#marker26176)"
+ d="m 175.04236,305.96308 4.22907,13.17794"
+ id="path6407-1-9-5"
+ sodipodi:nodetypes="cc"
+ inkscape:connector-curvature="0" />-1
+
1
+ y="228.58638"
+ x="62.376186"
+ xml:space="preserve"
+ id="text26120"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect8341);display:inline;stroke-width:1"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_08.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200">1
+
i-1
+ y="288.9566"
+ x="114.24284"
+ xml:space="preserve"
+ id="text26126"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect1044-2-1-7);display:inline;stroke-width:1"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_09.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200">i-1
+
i+1
+ y="288.79391"
+ x="124.1106"
+ xml:space="preserve"
+ id="text26130"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect961-6-9-1-1);display:inline;stroke-width:1"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_09.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200">i+1
+
i
+ y="302.72656"
+ x="120.99844"
+ xml:space="preserve"
+ id="text26134"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect8986);display:inline;stroke-width:1"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_09.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200">i
+
-1
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_08.png"
+ y="355.03506"
+ x="218.77849"
+ height="5.1131492"
+ width="5.1131492"
+ id="rect26136"
+ style="fill:none;fill-opacity:0.21383099;stroke:#999999;stroke-width:0.41446099;stroke-linecap:square;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none" />-1
+
1
+ inkscape:export-ydpi="200"
+ inkscape:export-xdpi="200"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_08.png"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.17499995px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect8341);display:inline;stroke-width:1"
+ id="text26166"
+ xml:space="preserve"
+ x="62.376186"
+ y="242.36928">1
+
"red" phase
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="cc"
+ id="path26170"
+ d="m 180.33401,344.08706 4.22907,13.17794"
+ style="fill:none;stroke:#000000;stroke-width:0.26499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#marker6411-7-1-9)"
+ inkscape:export-filename="/home/francesc/fig09.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200" />"red" phase
+
"black" phase
+ y="264.14084"
+ x="169.92833"
+ xml:space="preserve"
+ id="text28138"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.69665527px;line-height:1.25;font-family:Hack;-inkscape-font-specification:Hack;letter-spacing:0px;word-spacing:0px;white-space:pre;shape-inside:url(#rect7541);display:inline;stroke-width:1.16430092"
+ inkscape:export-filename="/home/fverdugo/fig_jacobi_06.png"
+ inkscape:export-xdpi="200"
+ inkscape:export-ydpi="200">"black" phase
+
1234921531234123412935000012341234123350000Input
\ No newline at end of file
+ inkscape:export-filename="/home/francesc/Downloads/fig18.png"
+ xml:space="preserve"
+ id="flowRoot14827-5"
+ style="font-style:normal;font-weight:normal;font-size:40px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none"
+ transform="matrix(0.12547462,0,0,0.12547462,988.0062,316.39023)">Output968561081234921531234123412935000012341234123350000InputOutput968561081234921531234123412935000012341234129350000Inputk=01234321531234123412935000012341234123350000Inputk=45656108810968968infinfinfinfinfinfinf1234921531234123412935000012341234123350000InputOutput96856108infinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinf1234321531234123412935000012341234123350000Inputk=1infinfinfinfinfinfinfinfinfinfinfinfinfinf1234321531234123412935000012341234123350000Inputk=25656infinfinfinfinfinfinfinfinfinfinfinf1234321531234123412935000012341234123350000Inputk=35656108810infinfinfinfinfinfinfinfinfinfinfinfinfinfinfinfinf
\ No newline at end of file
diff --git a/notebooks/matrix_matrix.ipynb b/notebooks/matrix_matrix.ipynb
index 5f1fd22..9aad890 100644
--- a/notebooks/matrix_matrix.ipynb
+++ b/notebooks/matrix_matrix.ipynb
@@ -255,6 +255,19 @@
"Question: Which is the complexity (number of operations) of the serial algorithm? Assume that all matrices are $N$-by-$N$ matrices. \n",
"![](\"attachment:g37385.png\")
\n",
+ "![](\"attachment:g32490.png\")
\n",
"![](\"attachment:g38106.png\")
\n",
+ "![](\"attachment:g33734.png\")
\n",
"![](\"attachment:g38354.png\")
\n",
+ "![](\"attachment:g32779.png\")
\n",
"![](\"attachment:g38620.png\")
\n",
+ "![](\"attachment:g33077.png\")
\n",
+ "![](\"attachment:g33443.png\")
\n",
"