mirror of
https://github.com/fverdugo/XM_40017.git
synced 2026-11-15 02:57:17 +01:00
3 files changed
+482
-468
No files matched your search
+2
-2
@@ -121,8 +121,8 @@ makedocs(;
|
||||
"Distributed computing in Julia" => "julia_distributed.md",
|
||||
"Distributed computing with MPI" => "mpi_tutorial.md",
|
||||
"Matrix-matrix multiplication"=>"matrix_matrix.md",
|
||||
# "Jacobi" => "jacobi_method.md",
|
||||
# "ASP" => "asp.md",
|
||||
"Jacobi method" => "jacobi_method.md",
|
||||
"All pairs of shortest paths" => "asp.md",
|
||||
# "Solutions" => "solutions.md",
|
||||
],
|
||||
],
|
||||
|
||||
+138
-93
@@ -65,33 +65,22 @@
|
||||
"source": [
|
||||
"### Floyd's sequential algoritm\n",
|
||||
"\n",
|
||||
"The ASP problem can be solved with the [Floyd–Warshall algorithm](https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm). A sequential implementation of this algorithm is given in this function."
|
||||
"The ASP problem can be solved with the [Floyd–Warshall algorithm](https://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm). A sequential implementation of this algorithm is given in the following function:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"execution_count": null,
|
||||
"id": "4fe447c5",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"floyd! (generic function with 1 method)"
|
||||
]
|
||||
},
|
||||
"execution_count": 1,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"function floyd!(C)\n",
|
||||
" n = size(C,1)\n",
|
||||
" @assert size(C,2) == n\n",
|
||||
" for k in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" @inbounds C[i,j] = min(C[i,j],C[i,k]+C[k,j])\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
@@ -110,25 +99,10 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"execution_count": null,
|
||||
"id": "860e537c",
|
||||
"metadata": {},
|
||||
"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"
|
||||
}
|
||||
],
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"inf = 1000\n",
|
||||
"C = [\n",
|
||||
@@ -154,8 +128,8 @@
|
||||
"```julia\n",
|
||||
"n = size(C,1)\n",
|
||||
"for k in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" C[i,j] = min(C[i,j],C[i,k]+C[k,j])\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
@@ -248,6 +222,69 @@
|
||||
"</div>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c7027ac3",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Serial performance\n",
|
||||
"\n",
|
||||
"This algorithm is memory bound, meaning that the main cost is in getting and setting data from the input matrix `C`. In this situations, the order in which we traverse the entries of matrix `C` has a significant performance impact.\n",
|
||||
"\n",
|
||||
"The following function computes the same result as for the previous function `floyd!`, but the nesting of loops over i and j is changed.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "75cac17e",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"function floyd2!(C)\n",
|
||||
" n = size(C,1)\n",
|
||||
" @assert size(C,2) == n\n",
|
||||
" for k in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" @inbounds C[i,j] = min(C[i,j],C[i,k]+C[k,j])\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
" C\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "399385e8",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
" Compare the performance of both implementations (run the cell several times)."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "907bc8c9",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"n = 1000\n",
|
||||
"C = rand(n,n)\n",
|
||||
"@time floyd!(C)\n",
|
||||
"C = rand(n,n)\n",
|
||||
"@time floyd2!(C);"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ad811b10",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The performance difference is significant. Matrices in Julia are stored in memory in column-major order (like in Fortran, unlike in C). It means that it is more efficient to access the data also in column-major order (like in function `floyd!`). See this section of [Julia's performance tips](https://docs.julialang.org/en/v1/manual/performance-tips/#man-performance-column-major) if you are interested in further details."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0c95ea88",
|
||||
@@ -264,8 +301,8 @@
|
||||
"```julia\n",
|
||||
"n = size(C,1)\n",
|
||||
"for k in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" C[i,j] = min(C[i,j],C[i,k]+C[k,j])\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
@@ -401,8 +438,12 @@
|
||||
"source": [
|
||||
"- Each process updates $N^2/P$ entries per iteration\n",
|
||||
"- 1 process broadcasts a message of length $N$ to $P-1$ processes per iteration\n",
|
||||
"- The send cost in this process is $O(N P)$ per iteration (if we use send/receive instead of broadcast)\n",
|
||||
"- $P-1$ processes receive one message of length $N$ per iteration\n",
|
||||
"- The receive/computation ration is $O(P/N)$ which would be small if $P<<N$"
|
||||
"- The receive cost is $O(N)$ per iteration at each process\n",
|
||||
"- The send/computation ratio is $O(P^2/N)$\n",
|
||||
"- The receive/computation ratio is $O(P/N)$\n",
|
||||
"- The algorithm is potentially scalable if $P<<N$"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -425,50 +466,7 @@
|
||||
"id": "494aa965",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Parallel Implementation"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c624722a",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Generating test data"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "09937668",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"function rand_distance_table(n)\n",
|
||||
" threshold = 0.4\n",
|
||||
" mincost = 3\n",
|
||||
" maxcost = 10\n",
|
||||
" infinity = 10000*maxcost\n",
|
||||
" C = fill(infinity,n,n)\n",
|
||||
" for j in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" if rand() > threshold\n",
|
||||
" C[i,j] = rand(mincost:maxcost)\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
" C[j,j] = 0\n",
|
||||
" end\n",
|
||||
" C\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "3116096c",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"rand_distance_table(10)"
|
||||
"## Parallel Implementation\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -511,7 +509,9 @@
|
||||
"id": "680e56cf",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Code"
|
||||
"### Code\n",
|
||||
"\n",
|
||||
"We split the code in two functions. The first function is called on the main process (the process running this notebook). It splits the input matrix into blocks of rows. Then, we call `floyd_worker!` (see below) remotely on each worker using the corresponding block of rows.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -534,6 +534,14 @@
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "9fc3be11",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The second function is the one run on the workers. Note that we considered MPI for communication in this case."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
@@ -560,8 +568,8 @@
|
||||
" else\n",
|
||||
" MPI.Recv!(C_k,comm,source=MPI.ANY_SOURCE,tag=0)\n",
|
||||
" end\n",
|
||||
" for i in 1:m\n",
|
||||
" for j in 1:n\n",
|
||||
" for j in 1:n\n",
|
||||
" for i in 1:m\n",
|
||||
" @inbounds Cw[i,j] = min(Cw[i,j],Cw[i,k]+C_k[j])\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
@@ -570,6 +578,39 @@
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c624722a",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Testing the implementation"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "09937668",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"function rand_distance_table(n)\n",
|
||||
" threshold = 0.4\n",
|
||||
" mincost = 3\n",
|
||||
" maxcost = 10\n",
|
||||
" infinity = 10000*maxcost\n",
|
||||
" C = fill(infinity,n,n)\n",
|
||||
" for j in 1:n\n",
|
||||
" for i in 1:n\n",
|
||||
" if rand() > threshold\n",
|
||||
" C[i,j] = rand(mincost:maxcost)\n",
|
||||
" end\n",
|
||||
" end\n",
|
||||
" C[j,j] = 0\n",
|
||||
" end\n",
|
||||
" C\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
@@ -654,21 +695,25 @@
|
||||
"- Use synchronous send MPI_SSEND (less efficient). Note that the blocking send MPI_SEND used above does not guarantee that the message was received.\n",
|
||||
"- Barrier at the end of each iteration over $k$ (simple solution, but synchronization overhead)\n",
|
||||
"- Order incoming messages (buffering and extra user code needed)\n",
|
||||
"- Use a specific rank id instead of `MPI.ANY_SOURCE` (one needs to know which are the rows owned by the other ranks)"
|
||||
"- Use a specific rank id instead of `MPI.ANY_SOURCE` or use `MPI.Bcast!` (one needs to know which are the rows owned by the other ranks)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "db2b586f",
|
||||
"cell_type": "markdown",
|
||||
"id": "c789dc7a",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
"source": [
|
||||
"# License\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"This notebook is part of the course [Programming Large Scale Parallel Systems](https://www.francescverdugo.com/XM_40017) at Vrije Universiteit Amsterdam and may be used under a [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/) license."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Julia 1.9.1",
|
||||
"display_name": "Julia 1.9.0",
|
||||
"language": "julia",
|
||||
"name": "julia-1.9"
|
||||
},
|
||||
@@ -676,7 +721,7 @@
|
||||
"file_extension": ".jl",
|
||||
"mimetype": "application/julia",
|
||||
"name": "julia",
|
||||
"version": "1.9.1"
|
||||
"version": "1.9.0"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
+342
-373
@@ -62,8 +62,7 @@
|
||||
"gauss_seidel_1_check(answer) = answer_checker(answer,\"c\")\n",
|
||||
"jacobi_1_check(answer) = answer_checker(answer, \"d\")\n",
|
||||
"jacobi_2_check(answer) = answer_checker(answer, \"b\")\n",
|
||||
"jacobi_3_check(answer) = answer_checker(answer, \"c\")\n",
|
||||
"jacobi_4_check(anwswer) = answer_checker(answer, \"d\")"
|
||||
"jacobi_3_check(answer) = answer_checker(answer, \"c\")"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -332,13 +331,333 @@
|
||||
"id": "513f1f7e",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Efficiency\n",
|
||||
"### Communication overhead\n",
|
||||
"- We update $N/P$ entries in each process at each iteration, where $N$ is the total length of the vector and $P$ the number of processes\n",
|
||||
"- We need to get remote entries from 2 neighbors (2 messages per iteration)\n",
|
||||
"- We need to communicate 1 entry per message\n",
|
||||
"- Communication/computation ration is $O(P/N)$ (potentially scalable if $P<<N$)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "f6b54b7b",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## 1D Implementation\n",
|
||||
"\n",
|
||||
"We consider the implementation using MPI. The programming model of MPI is generally better suited for data-parallel algorithms like this one than the task-based model provided by Distributed.jl. In any case, one can also implement it using Distributed, but it requires some extra effort to setup the remote channels right for the communication between neighbor processes."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "1b3c8c05",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Ghost (aka halo) cells\n",
|
||||
"\n",
|
||||
"A usual way of implementing the Jacobi method and related algorithms is using so-called ghost cells. Ghost cells represent the missing data dependencies in the data owned by each process. After importing the appropriate values from the neighbor processes one can perform the usual sequential Jacobi update locally in the processes."
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {
|
||||
"fig13.png": {
|
||||
"image/png": "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 truncated
|
||||
}
|
||||
},
|
||||
"cell_type": "markdown",
|
||||
"id": "98e0eb71",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div>\n",
|
||||
"<img src=\"attachment:fig13.png\" align=\"left\" width=\"450\"/>\n",
|
||||
"</div>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0148f9b3",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Thus, the algorithm is usually implemented following two main phases at each iteration Jacobi:\n",
|
||||
"\n",
|
||||
"1. Fill the ghost entries with communications\n",
|
||||
"2. Do the Jacobi update sequentially at each process"
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {
|
||||
"fig15.png": {
|
||||
"image/png": "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 truncated
|
||||
}
|
||||
},
|
||||
"cell_type": "markdown",
|
||||
"id": "baccd833",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div>\n",
|
||||
"<img src=\"attachment:fig15.png\" align=\"left\" width=\"450\"/>\n",
|
||||
"</div>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "8ed4129c",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Code\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Take a look at the implementation below and try to understand it. Note that we have used MPIClustermanagers and Distributed just to run the MPI code on the notebook. When running it on a cluster, MPIClustermanagers and Distributed are not needed.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e15082fb",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"] add MPI MPIClusterManagers"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "a66cbf9a",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"using MPIClusterManagers \n",
|
||||
"using Distributed"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e0d63c6b",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"if procs() == workers()\n",
|
||||
" nw = 3\n",
|
||||
" manager = MPIWorkerManager(nw)\n",
|
||||
" addprocs(manager)\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d7fb9177",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"First, we implement the function to be called on the MPI ranks."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "d39c7bb2",
|
||||
"metadata": {
|
||||
"code_folding": []
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"@everywhere workers() begin\n",
|
||||
" using MPI\n",
|
||||
" comm = MPI.Comm_dup(MPI.COMM_WORLD)\n",
|
||||
" function jacobi_mpi(n,niters)\n",
|
||||
" nranks = MPI.Comm_size(comm)\n",
|
||||
" rank = MPI.Comm_rank(comm)\n",
|
||||
" if mod(n,nranks) != 0\n",
|
||||
" println(\"n must be a multiple of nranks\")\n",
|
||||
" MPI.Abort(comm,1)\n",
|
||||
" end\n",
|
||||
" n_own = div(n,nranks)\n",
|
||||
" u = zeros(n_own+2)\n",
|
||||
" u[1] = -1\n",
|
||||
" u[end] = 1\n",
|
||||
" u_new = copy(u)\n",
|
||||
" for t in 1:niters\n",
|
||||
" reqs = MPI.Request[]\n",
|
||||
" if rank != 0\n",
|
||||
" neig_rank = rank-1\n",
|
||||
" req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" req = MPI.Irecv!(view(u,1:1),comm,source=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" end\n",
|
||||
" if rank != (nranks-1)\n",
|
||||
" neig_rank = rank+1\n",
|
||||
" s = n_own+1\n",
|
||||
" r = n_own+2\n",
|
||||
" req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" end\n",
|
||||
" MPI.Waitall(reqs)\n",
|
||||
" for i in 2:(n_own+1)\n",
|
||||
" u_new[i] = 0.5*(u[i-1]+u[i+1])\n",
|
||||
" end\n",
|
||||
" u, u_new = u_new, u\n",
|
||||
" end\n",
|
||||
" return u\n",
|
||||
" end\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "6eab32d0",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"In order to check the result, we will compare it against the serial implementation. To this end, we need to define the serial implementation also in the workers."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "f1a8db8f",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"@everywhere workers() function jacobi(n,niters)\n",
|
||||
" u = zeros(n+2)\n",
|
||||
" u[1] = -1\n",
|
||||
" u[end] = 1\n",
|
||||
" u_new = copy(u)\n",
|
||||
" for t in 1:niters\n",
|
||||
" for i in 2:(n+1)\n",
|
||||
" u_new[i] = 0.5*(u[i-1]+u[i+1])\n",
|
||||
" end\n",
|
||||
" u, u_new = u_new, u\n",
|
||||
" end\n",
|
||||
" u\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d2b04c67",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Finally, we call the parallel function on the workers, gather the results on the root rank, and compare against the sequential solution."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "68851107",
|
||||
"metadata": {
|
||||
"code_folding": []
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"@everywhere workers() begin\n",
|
||||
" # Call jacobi in parallel\n",
|
||||
" niters = 10\n",
|
||||
" load = 4\n",
|
||||
" nranks = MPI.Comm_size(comm)\n",
|
||||
" n = load*nranks\n",
|
||||
" u = jacobi_mpi(n,niters)\n",
|
||||
" # Gather results in root process and check\n",
|
||||
" rank = MPI.Comm_rank(comm)\n",
|
||||
" n_own = div(n,nranks)\n",
|
||||
" if rank == 0\n",
|
||||
" results = zeros(n+2)\n",
|
||||
" results[1] = -1\n",
|
||||
" results[n+2] = 1\n",
|
||||
" rcv = view(results, 2:n+1)\n",
|
||||
" else\n",
|
||||
" rcv = nothing\n",
|
||||
" end\n",
|
||||
" MPI.Gather!(view(u,2:n_own+1),rcv,comm;root=0)\n",
|
||||
" if rank == 0\n",
|
||||
" @show results ≈ jacobi(n,niters)\n",
|
||||
" end \n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "eff25246",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div class=\"alert alert-block alert-success\">\n",
|
||||
"<b>Question:</b> In function jacobi_mpi, how many messages per iteration are sent from a process away from the boundary?\n",
|
||||
"</div>\n",
|
||||
"\n",
|
||||
" a) 1\n",
|
||||
" b) 2\n",
|
||||
" c) 3\n",
|
||||
" d) 4\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "98bd9b5e",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"answer = \"x\" # replace x with a, b, c or d\n",
|
||||
"jacobi_2_check(answer)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "075dd6d8",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div class=\"alert alert-block alert-success\">\n",
|
||||
"<b>Question:</b> At the end of function jacobi_mpi ...\n",
|
||||
"</div>\n",
|
||||
"\n",
|
||||
" a) each rank holds the complete solution.\n",
|
||||
" b) only the root process holds the solution. \n",
|
||||
" c) the values of the ghost cells of u are not consistent with the neighbors\n",
|
||||
" d) the ghost cells of u contain the initial values -1 and 1 in all ranks"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "c3b58002",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"answer = \"x\" # replace x with a, b, c or d\n",
|
||||
"jacobi_3_check(answer)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c9aa2901",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Latency hiding\n",
|
||||
"\n",
|
||||
"Can our implementation above be improved? Note that we only need communications to update the values at the boundary of the portion owned by each process. The other values (the one in green in the figure below) can be updated without communications. This provides the opportunity of overlapping the computation of the interior values (green cells in the figure) with the communication of the ghost values. This technique is called latency hiding, since we are hiding communication latency by overlapping it with communication that we need to do anyway.\n",
|
||||
"\n",
|
||||
"The modification of the implementation above to include latency hiding is leaved as an exercise (see below).\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {
|
||||
"fig16.png": {
|
||||
"image/png": "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 truncated
|
||||
}
|
||||
},
|
||||
"cell_type": "markdown",
|
||||
"id": "7d66b1a2",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div>\n",
|
||||
"<img src=\"attachment:fig16.png\" align=\"left\" width=\"450\"/>\n",
|
||||
"</div>\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "9d4de5a9",
|
||||
@@ -497,7 +816,7 @@
|
||||
"source": [
|
||||
"### 1D block partition\n",
|
||||
"\n",
|
||||
"The following figure shows the portion of vector `u_new` is updated at each iteration by a particular process (CPU 3) left picture, and which entries of `u` are needed to update this data, right picture. We use analogous figures for the other partitions below.\n"
|
||||
"The following figure shows the portion of vector `u_new` that is updated at each iteration by a particular process (CPU 3) left picture, and which entries of `u` are needed to update this data, right picture. We use analogous figures for the other partitions below.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -594,6 +913,20 @@
|
||||
"- Communication/computation ratio is $O(1)$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "3d0693a7",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Summary\n",
|
||||
"\n",
|
||||
"|Partition | Messages <br> per iteration | Communication <br>per worker | Computation <br>per worker | Ratio communication/<br>computation |\n",
|
||||
"|---|---|---|---|---|\n",
|
||||
"| 1d block | 2 | O(N) | N²/P | O(P/N) |\n",
|
||||
"| 2d block | 4 | O(N/√P) | N²/P | O(√P/N) |\n",
|
||||
"| 2d cyclic | 4 |O(N²/P) | N²/P | O(1) |"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "850b1848",
|
||||
@@ -601,6 +934,8 @@
|
||||
"source": [
|
||||
"### Which partition is the best one?\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"- Both 1d and 2d block partitions are potentially scalable if $P<<N$\n",
|
||||
"- The 2d block partition is with the lowest communication complexity\n",
|
||||
"- The 1d block partition requires to send less messages (It can be useful if the fixed cost of sending a message is high)\n",
|
||||
@@ -609,299 +944,6 @@
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "f6b54b7b",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Implementation\n",
|
||||
"\n",
|
||||
"We consider the implementation using MPI. The programming model of MPI is generally better suited for data-parallel algorithms like this one than the task-based model provided by Distributed.jl. In any case, one can also implement it using Distributed, but it requires some extra effort to setup remote channels right for the communication between neighbor processes."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "1b3c8c05",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Ghost (aka halo) cells\n",
|
||||
"\n",
|
||||
"A usual way of implementing the Jacobi method and related algorithms is using so-called ghost cells. Ghost cells represent the missing data dependencies in the data owned by each process. After importing the appropriate values from the neighbor processes one can perform the usual sequential Jacobi update locally in the processes."
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {
|
||||
"fig13.png": {
|
||||
"image/png": "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 truncated
|
||||
}
|
||||
},
|
||||
"cell_type": "markdown",
|
||||
"id": "98e0eb71",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div>\n",
|
||||
"<img src=\"attachment:fig13.png\" align=\"left\" width=\"450\"/>\n",
|
||||
"</div>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0148f9b3",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Thus, the algorithm is usually implemented following two main phases at each iteration Jacobi:\n",
|
||||
"\n",
|
||||
"1. Fill the ghost entries with communications\n",
|
||||
"2. Do the Jacobi update sequentially at each process"
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {
|
||||
"fig15.png": {
|
||||
"image/png": "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 truncated
|
||||
}
|
||||
},
|
||||
"cell_type": "markdown",
|
||||
"id": "baccd833",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div>\n",
|
||||
"<img src=\"attachment:fig15.png\" align=\"left\" width=\"450\"/>\n",
|
||||
"</div>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "8ed4129c",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Code\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Take a look at the implementation below and try to understand it. Note that we have used MPIClustermanagers and Distributed just to run the MPI code on the notebook. When running it on a cluster, MPIClustermanagers and Distributed are not needed.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e15082fb",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"] add MPI MPIClusterManagers"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "a66cbf9a",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"using MPIClusterManagers \n",
|
||||
"using Distributed"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e0d63c6b",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"if procs() == workers()\n",
|
||||
" nw = 3\n",
|
||||
" manager = MPIWorkerManager(nw)\n",
|
||||
" addprocs(manager)\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "a0923606",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Test cell, remove me\n",
|
||||
"u = [-1, 0, 0, 0, 0, 1]\n",
|
||||
"view(u, 6:6)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "68851107",
|
||||
"metadata": {
|
||||
"code_folding": []
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"@mpi_do manager begin\n",
|
||||
" using MPI\n",
|
||||
" comm = MPI.Comm_dup(MPI.COMM_WORLD)\n",
|
||||
" nw = MPI.Comm_size(comm)\n",
|
||||
" iw = MPI.Comm_rank(comm)+1\n",
|
||||
" function jacobi_mpi(n,niters)\n",
|
||||
" if mod(n,nw) != 0\n",
|
||||
" println(\"n must be a multiple of nw\")\n",
|
||||
" MPI.Abort(comm,1)\n",
|
||||
" end\n",
|
||||
" n_own = div(n,nw)\n",
|
||||
" u = zeros(n_own+2)\n",
|
||||
" u[1] = -1\n",
|
||||
" u[end] = 1\n",
|
||||
" u_new = copy(u)\n",
|
||||
" for t in 1:niters\n",
|
||||
" reqs = MPI.Request[]\n",
|
||||
" # Exchange cell values with neighbors\n",
|
||||
" if iw != 1\n",
|
||||
" neig_rank = (iw-1)-1\n",
|
||||
" req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" req = MPI.Irecv!(view(u,1:1),comm,source=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" end\n",
|
||||
" if iw != nw\n",
|
||||
" neig_rank = (iw+1)-1\n",
|
||||
" s = n_own+1\n",
|
||||
" r = n_own+2\n",
|
||||
" req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" end\n",
|
||||
" MPI.Waitall(reqs)\n",
|
||||
" for i in 2:(n_own+1)\n",
|
||||
" u_new[i] = 0.5*(u[i-1]+u[i+1])\n",
|
||||
" end\n",
|
||||
" u, u_new = u_new, u\n",
|
||||
" end\n",
|
||||
" u\n",
|
||||
" @show u\n",
|
||||
" # Gather results in root process\n",
|
||||
" results = zeros(n+2)\n",
|
||||
" results[1] = -1\n",
|
||||
" results[n+2] = 1\n",
|
||||
" MPI.Gather!(view(u,2:n_own+1), view(results, 2:n+1), root=0, comm)\n",
|
||||
" if iw == 1\n",
|
||||
" @show results\n",
|
||||
" end \n",
|
||||
" end\n",
|
||||
" niters = 100\n",
|
||||
" load = 4\n",
|
||||
" n = load*nw\n",
|
||||
" jacobi_mpi(n,niters)\n",
|
||||
"end"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "eff25246",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div class=\"alert alert-block alert-success\">\n",
|
||||
"<b>Question:</b> How many messages per iteration are sent from a process away from the boundary?\n",
|
||||
"</div>\n",
|
||||
"\n",
|
||||
" a) 1\n",
|
||||
" b) 2\n",
|
||||
" c) 3\n",
|
||||
" d) 4\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "98bd9b5e",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"answer = \"x\" # replace x with a, b, c or d\n",
|
||||
"jacobi_2_check(answer)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "075dd6d8",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div class=\"alert alert-block alert-success\">\n",
|
||||
"<b>Question:</b> After the end of the for-loop (line 43), ...\n",
|
||||
"</div>\n",
|
||||
"\n",
|
||||
" a) each worker holds the complete solution.\n",
|
||||
" b) the root process holds the solution. \n",
|
||||
" c) the ghost cells contain redundant values. \n",
|
||||
" d) all ghost cells contain the initial values -1 and 1. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "c3b58002",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"answer = \"x\" # replace x with a, b, c or d\n",
|
||||
"jacobi_3_check(answer)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "4537661d",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div class=\"alert alert-block alert-success\">\n",
|
||||
"<b>Question:</b> In line 35 of the code, we wait for all receive and send requests. Is it possible to instead wait for just the receive requests?\n",
|
||||
"</div>\n",
|
||||
"\n",
|
||||
" \n",
|
||||
" a) No, because the send buffer might be overwritten if we don't wait for send requests.\n",
|
||||
" b) No, because MPI does not allow an asynchronous send without a Wait().\n",
|
||||
" c) Yes, because each send has a matching receive, so all requests are done when the receive requests return. \n",
|
||||
" d) Yes, because there are no writes to the send buffer in this iteration."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e16ea5eb",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"answer = \"x\" # replace x with a, b, c or d.\n",
|
||||
"jacobi_4_check(answer)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c9aa2901",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Latency hiding\n",
|
||||
"\n",
|
||||
"Can our implementation above be improved? Note that we only need communications to update the values at the boundary of the portion owned by each process. The other values (the one in green in the figure below) can be updated without communications. This provides the opportunity of overlapping the computation of the interior values (green cells in the figure) with the communication of the ghost values. This technique is called latency hiding, since we are hiding communication latency by overlapping it with communications that we need to do anyway.\n",
|
||||
"\n",
|
||||
"The modification of the implementation above to include latency hiding is leaved as an exercise (see below).\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"attachments": {
|
||||
"fig16.png": {
|
||||
"image/png": "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 truncated
|
||||
}
|
||||
},
|
||||
"cell_type": "markdown",
|
||||
"id": "7d66b1a2",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<div>\n",
|
||||
"<img src=\"attachment:fig16.png\" align=\"left\" width=\"450\"/>\n",
|
||||
"</div>\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "47643bf6",
|
||||
@@ -917,72 +959,7 @@
|
||||
"source": [
|
||||
"### Exercise 1\n",
|
||||
"\n",
|
||||
"Transform the following parallel implementation of the 1d Jacobi method (it is copied from above) to use latency hiding (overlap between computation of interior values and communication)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "66db180d",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"@mpi_do manager begin\n",
|
||||
" using MPI\n",
|
||||
" comm = MPI.Comm_dup(MPI.COMM_WORLD)\n",
|
||||
" nw = MPI.Comm_size(comm)\n",
|
||||
" iw = MPI.Comm_rank(comm)+1\n",
|
||||
" function jacobi_mpi(n,niters)\n",
|
||||
" if mod(n,nw) != 0\n",
|
||||
" println(\"n must be a multiple of nw\")\n",
|
||||
" MPI.Abort(comm,1)\n",
|
||||
" end\n",
|
||||
" n_own = div(n,nw)\n",
|
||||
" u = zeros(n_own+2)\n",
|
||||
" u[1] = -1\n",
|
||||
" u[end] = 1\n",
|
||||
" u_new = copy(u)\n",
|
||||
" for t in 1:niters\n",
|
||||
" reqs = MPI.Request[]\n",
|
||||
" # Exchange cell values with neighbors\n",
|
||||
" if iw != 1\n",
|
||||
" neig_rank = (iw-1)-1\n",
|
||||
" req = MPI.Isend(view(u,2:2),comm,dest=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" req = MPI.Irecv!(view(u,1:1),comm,source=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" end\n",
|
||||
" if iw != nw\n",
|
||||
" neig_rank = (iw+1)-1\n",
|
||||
" s = n_own+1\n",
|
||||
" r = n_own+2\n",
|
||||
" req = MPI.Isend(view(u,s:s),comm,dest=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" req = MPI.Irecv!(view(u,r:r),comm,source=neig_rank,tag=0)\n",
|
||||
" push!(reqs,req)\n",
|
||||
" end\n",
|
||||
" MPI.Waitall(reqs)\n",
|
||||
" for i in 2:(n_own+1)\n",
|
||||
" u_new[i] = 0.5*(u[i-1]+u[i+1])\n",
|
||||
" end\n",
|
||||
" u, u_new = u_new, u\n",
|
||||
" end\n",
|
||||
" u\n",
|
||||
" @show u\n",
|
||||
" # Gather results in root process\n",
|
||||
" results = zeros(n+2)\n",
|
||||
" results[1] = -1\n",
|
||||
" results[n+2] = 1\n",
|
||||
" MPI.Gather!(view(u,2:n_own+1), view(results, 2:n+1), root=0, comm)\n",
|
||||
" if iw == 1\n",
|
||||
" @show results\n",
|
||||
" end \n",
|
||||
" end\n",
|
||||
" niters = 100\n",
|
||||
" load = 4\n",
|
||||
" n = load*nw\n",
|
||||
" jacobi_mpi(n,niters)\n",
|
||||
"end"
|
||||
"Transform the parallel implementation of the 1d Jacobi method (function `jacopi_mpi`) to use latency hiding (overlap between computation of interior values and communication)."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -992,18 +969,10 @@
|
||||
"source": [
|
||||
"# License\n",
|
||||
"\n",
|
||||
"TODO: replace link to website\n",
|
||||
"\n",
|
||||
"This notebook is part of the course [Programming Large Scale Parallel Systems](http://localhost:8000/) at Vrije Universiteit Amsterdam and may be used under a [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/) license."
|
||||
"\n",
|
||||
"This notebook is part of the course [Programming Large Scale Parallel Systems](https://www.francescverdugo.com/XM_40017) at Vrije Universiteit Amsterdam and may be used under a [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/) license."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "3d72ff47",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
|
||||
Reference in new issue
Block a user