12. MetropolisHastings.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using StatsBase\n",
    "#plotly()\n",
    "# using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "addprocs(\n",
    "    [(\"alanedelman@anubis.juliacomputing.io\", 10)],\n",
    "    exename = \"/home/andreasnoack/julia06/bin/julia\",\n",
    "    exeflags = \"--compilecache=no\",\n",
    "    dir = \"/home/alanedelman\"\n",
    ")\n",
    "nworkers()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Generates random data distributed proportionally to f(x) on [a,b]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "@everywhere begin\n",
    "\n",
    "using StatsBase,DistributedArrays\n",
    "\n",
    "function jump(f, x, a = -1.0, b = 1.0)\n",
    "    y = x + 0.1 * (rand()-.5)   \n",
    "    rand() < f(y)/f(x) && a<y<b ? y  : x\n",
    "end\n",
    "\n",
    "\n",
    "function walk(f, n=10_000_000, x = 0.0, a = -1.0, b = 1.0)\n",
    " data = Float64[]\n",
    "  h = Histogram(a:.01:b ,:left)\n",
    "  for i=1:n-1\n",
    "    x = jump(f,x)\n",
    "    push!(h,x)\n",
    "    push!(data,x)\n",
    "  end\n",
    "  return h,data \n",
    "end\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "@everywhere f(x)=x.^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41-element DistributedArrays.DArray{Float64,1,Array{Float64,1}}:\n",
       " -1.6147   \n",
       " -0.3551   \n",
       " -0.828194 \n",
       " -0.928045 \n",
       " -1.44511  \n",
       "  0.0834151\n",
       " -0.628575 \n",
       " -1.35503  \n",
       "  0.499463 \n",
       "  0.351816 \n",
       " -1.17443  \n",
       "  2.40879  \n",
       "  0.339591 \n",
       "  ⋮        \n",
       " -1.27994  \n",
       "  0.948762 \n",
       "  0.043453 \n",
       " -0.561867 \n",
       " -1.22878  \n",
       "  1.63525  \n",
       " -0.0249118\n",
       " -1.1641   \n",
       " -0.693643 \n",
       " -2.41449  \n",
       " -1.02006  \n",
       " -0.989073 "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = distribute( randn(41))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: both Interact and Base export \"select\"; uses of it in module Main must be qualified\n"
     ]
    }
   ],
   "source": [
    "using Colors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "100-element Array{ColorTypes.RGB{FixedPointNumbers.Normed{UInt8,8}},1}:\n",
       " RGB{N0f8}(0.0,0.0,0.0)      \n",
       " RGB{N0f8}(1.0,1.0,0.384)    \n",
       " RGB{N0f8}(1.0,0.624,1.0)    \n",
       " RGB{N0f8}(0.0,0.839,1.0)    \n",
       " RGB{N0f8}(0.843,0.267,0.0)  \n",
       " RGB{N0f8}(0.0,0.502,0.161)  \n",
       " RGB{N0f8}(0.0,0.373,0.835)  \n",
       " RGB{N0f8}(0.576,0.0,0.408)  \n",
       " RGB{N0f8}(1.0,0.796,0.71)   \n",
       " RGB{N0f8}(0.655,0.525,0.0)  \n",
       " RGB{N0f8}(0.0,1.0,0.773)    \n",
       " RGB{N0f8}(0.0,0.494,0.525)  \n",
       " RGB{N0f8}(0.325,0.259,0.0)  \n",
       " ⋮                           \n",
       " RGB{N0f8}(0.494,0.384,0.0)  \n",
       " RGB{N0f8}(0.094,0.365,0.365)\n",
       " RGB{N0f8}(0.0,0.545,0.455)  \n",
       " RGB{N0f8}(1.0,0.247,0.216)  \n",
       " RGB{N0f8}(0.737,0.62,1.0)   \n",
       " RGB{N0f8}(0.51,0.455,0.427) \n",
       " RGB{N0f8}(0.333,0.337,0.2)  \n",
       " RGB{N0f8}(0.329,0.741,1.0)  \n",
       " RGB{N0f8}(0.804,0.22,0.635) \n",
       " RGB{N0f8}(0.506,0.827,0.675)\n",
       " RGB{N0f8}(0.541,0.435,0.502)\n",
       " RGB{N0f8}(0.969,0.714,0.416)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = distinguishable_colors(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
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       "10×91 Array{ColorTypes.RGB{FixedPointNumbers.Normed{UInt8,8}},2}:\n",
       " RGB{N0f8}(0.0,0.0,0.0)  …  RGB{N0f8}(0.655,0.525,0.0)\n",
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     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#@manipulate for n=1:100\n",
    " # a = distribute( randn(n))\n",
    " # g(x) =length(x[1])\n",
    " # d[g.(a.indexes)]\n",
    "#end\n",
    "\n",
    "n=20\n",
    "\n",
    "d[hcat( [g.(distribute( randn(n)).indexes) for n=10:100]...)]\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element DistributedArrays.DArray{Float64,1,Array{Float64,1}}:\n",
       " -0.918824 \n",
       "  1.33757  \n",
       " -0.584494 \n",
       " -0.314915 \n",
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       "  0.483079 \n",
       "  0.905282 \n",
       " -0.0370876\n",
       " -0.960637 "
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     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "@DArray [randn() for i=1:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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       " ⋮                                                                                                                                                                                                                                                                                                                                                                \n",
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     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "D = @DArray [walk(f,10) for i=1:10*nworkers()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
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     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Array(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Tuple{UnitRange{Int64}},1}:\n",
       " (1:10,)  \n",
       " (11:20,) \n",
       " (21:30,) \n",
       " (31:40,) \n",
       " (41:50,) \n",
       " (51:60,) \n",
       " (61:70,) \n",
       " (71:80,) \n",
       " (81:90,) \n",
       " (91:100,)"
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     },
     "execution_count": 48,
     "metadata": {},
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    }
   ],
   "source": [
    "D.indexes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h, data = walk(f)\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "@everywhere begin\n",
    "    f(x) =  abs(x)\n",
    "end\n",
    "\n",
    "mid(r) = r[1:length(r) - 1] + 0.5 * step(r)\n",
    "@time h = walk(f)\n",
    "\n",
    "\n",
    "plot(h)\n",
    "x = mid(h.edges[1])\n",
    "y = h.weights \n",
    "plot!(x,y,label=\"observed\")\n",
    "Y = f.(x)\n",
    "plot!(x, (y'Y)/(Y'Y)*Y, color=\"red\", label=\"predicted\", width=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    " h, data = @parallel (x,y)->(merge(x[1],y[1]), vcat(x[2],y[2])) for p = 1:nworkers()\n",
    "  walk(f,100)\n",
    "end\n",
    "\n",
    "#=\n",
    "plot(h,label=false)\n",
    "x = mid(h.edges[1])\n",
    "y = h.weights \n",
    "plot!(x,y,label=\"observed\")\n",
    "Y = f.(x)\n",
    "plot!(x, (y'Y)/(Y'Y)*Y, color=\"red\", label=\"predicted\", width=2)\n",
    "=#\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "@parallel (vcat) for p=1:nworkers()\n",
    "   myid()\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "function foo(x)\n",
    "    34\n",
    "    45\n",
    "    return\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = nothing\n",
    "typeof(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sum(h.weights)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "length(h.edges[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "merge(h,h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nworkers()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h2 = Histogram(-1:.01:1 ,:left)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "vcat( rand(3), rand(4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x =[rand(3,3) for i=1:100_000];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "length(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = rand(3,3,100_000);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    " # @fetchfrom 3 localpart(dD) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element DistributedArrays.DArray{Float64,1,Array{Float64,1}}:\n",
       " 0.105518 \n",
       " 0.758881 \n",
       " 0.408368 \n",
       " 0.254824 \n",
       " 0.822384 \n",
       " 0.67095  \n",
       " 0.0221236\n",
       " 0.679513 \n",
       " 0.37678  \n",
       " 0.406047 "
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = drand(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×5 Array{Tuple{UnitRange{Int64},UnitRange{Int64}},2}:\n",
       " (1:50, 1:20)    (1:50, 21:40)    …  (1:50, 61:80)    (1:50, 81:100)  \n",
       " (51:100, 1:20)  (51:100, 21:40)     (51:100, 61:80)  (51:100, 81:100)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = drand(100,100)\n",
    "a.indexes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "search: \u001b[1md\u001b[22m\u001b[1mr\u001b[22m\u001b[1ma\u001b[22m\u001b[1mn\u001b[22m\u001b[1md\u001b[22m\u001b[1mn\u001b[22m \u001b[1md\u001b[22m\u001b[1mr\u001b[22m\u001b[1ma\u001b[22m\u001b[1mn\u001b[22m\u001b[1md\u001b[22m\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/markdown": [
       "```\n",
       " drandn(dims, ...)\n",
       "```\n",
       "\n",
       "Construct a distributed normal random array. Trailing arguments are the same as those accepted by `DArray`.\n"
      ],
      "text/plain": [
       "```\n",
       " drandn(dims, ...)\n",
       "```\n",
       "\n",
       "Construct a distributed normal random array. Trailing arguments are the same as those accepted by `DArray`.\n"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?drandn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: ignoring conflicting import of SPMD.recvfrom into Main\n"
     ]
    },
    {
     "ename": "LoadError",
     "evalue": "\u001b[91m\u001b[91m\u001b[91mBoundsError: attempt to access 0×0 Array{Int64,2} at index [2]\u001b[39m\nfoo_spmd(::DistributedArrays.DArray{Int64,2,Array{Int64,2}}, ::DistributedArrays.DArray{Any,1,Any}, ::Int64) at ./In[59]:18\nspmd_local(::DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}}, ::Tuple{Int64,Int64}, ::Bool) at /Users/alanedelman/.julia/v0.6/DistributedArrays/src/spmd.jl:228\n(::Base.Distributed.##135#136{DistributedArrays.SPMD.#spmd_local,Tuple{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}},Tuple{Int64,Int64},Bool},Array{Any,1}})() at ./distributed/remotecall.jl:314\nrun_work_thunk(::Base.Distributed.##135#136{DistributedArrays.SPMD.#spmd_local,Tuple{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}},Tuple{Int64,Int64},Bool},Array{Any,1}}, ::Bool) at ./distributed/process_messages.jl:56\n#remotecall_fetch#140(::Array{Any,1}, ::Function, ::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:339\nremotecall_fetch(::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:339\n#remotecall_fetch#144(::Array{Any,1}, ::Function, ::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\nremotecall_fetch(::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\n(::DistributedArrays.SPMD.##26#28{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}}})() at ./task.jl:335\u001b[39m\n#remotecall_fetch#140(::Array{Any,1}, ::Function, ::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:340\nremotecall_fetch(::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:339\n#remotecall_fetch#144(::Array{Any,1}, ::Function, ::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\nremotecall_fetch(::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\n(::DistributedArrays.SPMD.##26#28{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}}})() at ./task.jl:335\u001b[39m",
     "output_type": "error",
     "traceback": [
      "\u001b[91m\u001b[91m\u001b[91mBoundsError: attempt to access 0×0 Array{Int64,2} at index [2]\u001b[39m\nfoo_spmd(::DistributedArrays.DArray{Int64,2,Array{Int64,2}}, ::DistributedArrays.DArray{Any,1,Any}, ::Int64) at ./In[59]:18\nspmd_local(::DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}}, ::Tuple{Int64,Int64}, ::Bool) at /Users/alanedelman/.julia/v0.6/DistributedArrays/src/spmd.jl:228\n(::Base.Distributed.##135#136{DistributedArrays.SPMD.#spmd_local,Tuple{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}},Tuple{Int64,Int64},Bool},Array{Any,1}})() at ./distributed/remotecall.jl:314\nrun_work_thunk(::Base.Distributed.##135#136{DistributedArrays.SPMD.#spmd_local,Tuple{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}},Tuple{Int64,Int64},Bool},Array{Any,1}}, ::Bool) at ./distributed/process_messages.jl:56\n#remotecall_fetch#140(::Array{Any,1}, ::Function, ::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:339\nremotecall_fetch(::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:339\n#remotecall_fetch#144(::Array{Any,1}, ::Function, ::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\nremotecall_fetch(::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\n(::DistributedArrays.SPMD.##26#28{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}}})() at ./task.jl:335\u001b[39m\n#remotecall_fetch#140(::Array{Any,1}, ::Function, ::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:340\nremotecall_fetch(::Function, ::Base.Distributed.LocalProcess, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:339\n#remotecall_fetch#144(::Array{Any,1}, ::Function, ::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\nremotecall_fetch(::Function, ::Int64, ::Function, ::Vararg{Any,N} where N) at ./distributed/remotecall.jl:367\n(::DistributedArrays.SPMD.##26#28{DistributedArrays.SPMD.##25#27{#foo_spmd,Tuple{DistributedArrays.DArray{Int64,2,Array{Int64,2}},DistributedArrays.DArray{Any,1,Any},Int64}}})() at ./task.jl:335\u001b[39m",
      "",
      "Stacktrace:",
      " [1] \u001b[1msync_end\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./task.jl:287\u001b[22m\u001b[22m",
      " [2] \u001b[1mmacro expansion\u001b[22m\u001b[22m at \u001b[1m./task.jl:303\u001b[22m\u001b[22m [inlined]",
      " [3] \u001b[1m#spmd#24\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Array{Int64,1}, ::Void, ::Function, ::Function, ::DistributedArrays.DArray{Int64,2,Array{Int64,2}}, ::Vararg{Any,N} where N\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/alanedelman/.julia/v0.6/DistributedArrays/src/spmd.jl:243\u001b[22m\u001b[22m",
      " [4] \u001b[1mspmd\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Function, ::DistributedArrays.DArray{Int64,2,Array{Int64,2}}, ::Vararg{Any,N} where N\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m/Users/alanedelman/.julia/v0.6/DistributedArrays/src/spmd.jl:234\u001b[22m\u001b[22m",
      " [5] \u001b[1minclude_string\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::String, ::String\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./loading.jl:515\u001b[22m\u001b[22m"
     ]
    }
   ],
   "source": [
    "@everywhere importall DistributedArrays.SPMD\n",
    "\n",
    "d_in=d=DArray(I->fill(myid(), (map(length,I)...)), (nworkers(), 2), workers(), [nworkers(),1])\n",
    "d_out=ddata()\n",
    "\n",
    "# define the function everywhere\n",
    "@everywhere function foo_spmd(d_in, d_out, n)\n",
    "    pids = sort(vec(procs(d_in)))\n",
    "    pididx = findfirst(pids, myid())\n",
    "    mylp = d_in[:L]\n",
    "    localsum = 0\n",
    "\n",
    "    # Have each worker exchange data with its neighbors\n",
    "    n_pididx = pididx+1 > length(pids) ? 1 : pididx+1\n",
    "    p_pididx = pididx-1 < 1 ? length(pids) : pididx-1\n",
    "\n",
    "    for i in 1:n\n",
    "        sendto(pids[n_pididx], mylp[2])\n",
    "        sendto(pids[p_pididx], mylp[1])\n",
    "\n",
    "        mylp[2] = recvfrom(pids[p_pididx])\n",
    "        mylp[1] = recvfrom(pids[n_pididx])\n",
    "\n",
    "        barrier(;pids=pids)\n",
    "        localsum = localsum + mylp[1] + mylp[2]\n",
    "    end\n",
    "\n",
    "    # finally store the sum in d_out\n",
    "    d_out[:L] = localsum\n",
    "end\n",
    "\n",
    "# run foo_spmd on all workers\n",
    "spmd(foo_spmd, d_in, d_out, 10)\n",
    "\n",
    "# print values of d_in and d_out after the run\n",
    "println(d_in)\n",
    "println(d_out)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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