{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# TP4 : Programmation dynamique" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "readKnapInstance" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "\"\"\"Open and read a KnapFile.\n", "\n", "Args: \\\\\n", " - filename (String): the name of the file to read.\n", "\n", "Returns: \\\\\n", " - price (Vector{Integer}): prices of items to put in the KnapSack. \\\\\n", " - weight (Vector{Integer}): weights of items to put in the KnapSack. \\\\\n", " - capacity (Integer): the maximum capacity of the KnapSack.\n", "\"\"\"\n", "function readKnapInstance(filename)\n", " price = []\n", " weight = []\n", " capacity = -1\n", " open(filename) do f\n", " for i = 1:3\n", " tok = split(readline(f))\n", " if (tok[1] == \"ListPrices=\")\n", " for i = 2:(length(tok)-1)\n", " push!(price, parse(Int64, tok[i]))\n", " end\n", " elseif (tok[1] == \"ListWeights=\")\n", " for i = 2:(length(tok)-1)\n", " push!(weight, parse(Int64, tok[i]))\n", " end\n", " elseif (tok[1] == \"Capacity=\")\n", " capacity = parse(Int64, tok[2])\n", " else\n", " println(\"Unknown read :\", tok)\n", " end\n", " end\n", " end\n", " return price, weight, capacity\n", "end" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "analyseRecursive (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "function analyseRecursive(price, weight, i, j)\n", " if i > 0\n", " C1, sol1 = analyseRecursive(price, weight, i - 1, j)\n", " if j - weight[i] >= 0\n", " C2, sol2 = analyseRecursive(price, weight, i - 1, j - weight[i])\n", " C2 += price[i]\n", " if C1 >= C2\n", " res = C1\n", " sol = push!(sol1, 0)\n", " else\n", " res = C2\n", " sol = push!(sol2, 1)\n", " end\n", " else\n", " res = C1\n", " sol = push!(sol1, 0)\n", " end\n", " else\n", " res = 0\n", " sol = Vector{Int}()\n", " end\n", " \n", " return res, sol\n", "end" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "knapsack (generic function with 1 method)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# nice read : https://jarednielsen.com/dynamic-programming-memoization-tabulation/\n", "\n", "function analyseDynamique(prices, weights, capacity)\n", " n = length(prices)\n", " memo = zeros(UInt, n, capacity)\n", "\n", " for i in 2:n\n", " for j in 1:capacity\n", " if weights[i] > j\n", " memo[i, j] = memo[i-1, j]\n", " else\n", " memo[i, j] = max(memo[i-1, j], memo[i-1, j-weights[i]+1] + prices[i])\n", " end\n", " end\n", " end\n", "\n", " return memo\n", "end\n", "\n", "function knapsack(memo, weights, i, j)\n", " if i == 1\n", " return Vector{UInt}()\n", " end\n", " if memo[i, j] > memo[i-1, j]\n", " return push!(knapsack(memo, weights, i-1, j-weights[i]+1) ,i)\n", " else\n", " return knapsack(memo, weights, i-1, j)\n", " end\n", "end" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Solution récursive:\n", " 0.000466 seconds (2.16 k allocations: 78.172 KiB)\n", " * res = 2291\n", " * sol = [0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0]\n", "\n", "Solution dynamique:\n", " 0.005790 seconds (31.52 k allocations: 647.844 KiB)\n", " 0.000022 seconds (9 allocations: 256 bytes)\n", " * res = 2291\n", " * sol = UInt64[0x0000000000000003, 0x0000000000000006, 0x000000000000000f, 0x0000000000000011]\n" ] } ], "source": [ "prices, weights, capacity = readKnapInstance(\"test2.opb\")\n", "\n", "println(\"Solution récursive:\");\n", "@time res, sol = analyseRecursive(prices, weights, length(prices), capacity)\n", "println(\" * res = \", res);\n", "println(\" * sol = \", sol);\n", "\n", "println()\n", "\n", "println(\"Solution dynamique:\")\n", "@time memo = analyseDynamique(prices, weights, capacity)\n", "@time objects_taken = knapsack(memo, weights, length(prices), capacity)\n", "objects_taken_price = sum(prices[i] for i in objects_taken)\n", "println(\" * res = \", objects_taken_price)\n", "println(\" * sol = \", objects_taken)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Rapport\n", "\n", "Voici deux implémentation de l'algorithme du sac à dos 1-0.\n", "- La première implémentation est naïve et récursive.\n", "- La deuxième implémentation est dynamique.\n", "\n", "La programmation dynamique consiste à sauvegarder des résultats passés d'un calcul, pour ne pas avoir à les recalculer entièrement lorsque l'on retombe dessus. Dans notre cas, nous stockons les résultats précédents dans une matrice `mémo` de taille $n \\times \\text{capacity}$.\n", "\n", "À l'issue de benchmarks sur les deux implémentation, on remarque que l'algorithme récursif est plus rapide sur de petits jeux de données, et ce à cause de l'overhead causé par l'allocation de la matrice `mémo`. Cependant sur des exemples plus grands (comme `test4.opb`) on observe tout l'intéret de la programmation dynamique. En effet, alors que l'algorithme récursif peine à finir, l'algorithme dynamique est très rapide. Ce gain de vitesse est dû au fait que l'on a échangé de la complexité temporelle par de la complexité spatiale.\n", "\n", "Enfin, une propriété non négligeable de la programmation dynamique est que celle-ci permet le calcul parallel (via la matrice `memo`), ainsi (sur des jeux de données encore plus grand) la vitesse de l'algorithme n'est limité que par le nombre de ressources disponibles (contrairement à l'algorithme récursive et à celui du TP2-3)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# BONUS : Bellman Ford" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "\u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n", "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n", "\u001b[32m\u001b[1mPrecompiling\u001b[22m\u001b[39m project...\n", "\u001b[32m ✓ \u001b[39mTestOptinum\n", " 1 dependency successfully precompiled in 4 seconds (265 already precompiled, 2 skipped during auto due to previous errors)\n", "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n", "\u001b[32m\u001b[1mPrecompiling\u001b[22m\u001b[39m project...\n", "\u001b[32m ✓ \u001b[39mTestOptinum\n", " 1 dependency successfully precompiled in 3 seconds (265 already precompiled, 2 skipped during auto due to previous errors)\n", "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n", "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n", "\u001b[32m\u001b[1mPrecompiling\u001b[22m\u001b[39m project...\n", "\u001b[32m ✓ \u001b[39mTestOptinum\n", " 1 dependency successfully precompiled in 2 seconds (265 already precompiled, 2 skipped during auto due to previous errors)\n", "WARNING: using Graphs.weights in module Main conflicts with an existing identifier.\n" ] } ], "source": [ "using Pkg\n", "Pkg.add(\"Graphs\")\n", "Pkg.add(\"GraphPlot\")\n", "Pkg.add(\"SimpleWeightedGraphs\")\n", "using Graphs\n", "using GraphPlot\n", "using SimpleWeightedGraphs" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": "\n\n\n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n 3.0\n \n \n \n \n 5.0\n \n \n \n \n 4.0\n \n \n \n \n 2.0\n \n 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Measures.Measure}((-1.0cx, -0.05197295568420002cy), 0.04082482904638631w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}, Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, 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Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.3806194598998489cx, 0.716451934778696cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-1.0cx, -0.05197295568420002cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "g = SimpleWeightedDiGraph(6)\n", "add_edge!(g, 1, 2, 3)\n", "add_edge!(g, 1, 5, 5)\n", "add_edge!(g, 2, 3, 4)\n", "add_edge!(g, 3, 4, 2)\n", "add_edge!(g, 4, 6, 3)\n", "add_edge!(g, 5, 2, -1)\n", "add_edge!(g, 5, 4, 9)\n", "\n", "# g = SimpleWeightedDiGraph(5)\n", "# add_edge!(g, 1, 2, 9)\n", "# add_edge!(g, 1, 3, 3)\n", "# add_edge!(g, 2, 3, 6)\n", "# add_edge!(g, 2, 5, 2)\n", "# add_edge!(g, 3, 4, 1)\n", "# add_edge!(g, 3, 2, 2)\n", "# add_edge!(g, 4, 3, 2)\n", "# add_edge!(g, 4, 5, 2)\n", "\n", "nodelabel = 1:nv(g)\n", "edgelabel = edges(g).iter.is[3]\n", "gplot(g, nodelabel=nodelabel, edgelabel=edgelabel)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "([0.0, 3.0, 7.0, 9.0, 5.0, 12.0], [0, 1, 2, 3, 1, 4])" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm\n", "function bellmanFord(graph, source)\n", "\n", " # Step 1: initialize graph\n", " d = fill(Inf, nv(graph)); # distance array\n", " p = fill(0, nv(graph)); # predecessor array\n", " d[source] = 0; # source's distance to source is null\n", "\n", " # Step 2: relax edges repeatedly\n", " for _ in 2:nv(graph)\n", " # println(\"test: \", k)\n", " for (v, u, w) in edges(graph).iter\n", " # println(u, \" \", v, \" \", w)\n", " if d[u] + w < d[v]\n", " d[v] = d[u] + w\n", " p[v] = u\n", " end\n", " end\n", " end\n", "\n", " # Step 3: check for negative-weight cycles\n", " for (v, u, w) in edges(graph).iter\n", " if d[u] + w < d[v]\n", " error(\"Graph contains a negative-weight cycle\")\n", " end\n", " end\n", "\n", " return d, p\n", "end\n", "bellmanFord(g, 1)" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.7.0", "language": "julia", "name": "julia-1.7" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.7.0" }, "orig_nbformat": 4 }, "nbformat": 4, "nbformat_minor": 2 }