134 lines
21 KiB
XML
134 lines
21 KiB
XML
<?xml version='1.0' encoding='UTF-8' standalone='no'?>
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<doxygen xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="compound.xsd" version="1.9.1" xml:lang="en-US">
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<compounddef id="flipcoins" kind="page">
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<compoundname>flipcoins</compoundname>
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<title>Flip Coins</title>
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<tableofcontents>
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<tocsect>
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<name>Problem Formulation</name>
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<reference>flipcoins_1FlipCoinsProblemFormulation</reference>
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</tocsect>
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<tocsect>
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<name>Probabilistic Conditions</name>
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<reference>flipcoins_1FlipCoinsProbabilistic</reference>
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</tocsect>
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<tocsect>
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<name>Ternary Coins</name>
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<reference>flipcoins_1FlipCoinsTernaryCoins</reference>
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</tocsect>
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</tableofcontents>
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<briefdescription>
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</briefdescription>
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<detaileddescription>
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<para>We study dynamic control flow of non-determinism using conditional tasking. Non-deterministic control flow is a fundamental building block in many optimization and simulation algorithms that rely on stochastic convergence rules or probabilistic pruning.</para>
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<sect1 id="flipcoins_1FlipCoinsProblemFormulation">
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<title>Problem Formulation</title>
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<para>We have a fair binary coin and want to simulate its tosses. We flip the coin for five times. Apparently, the probability for the result to be all heads is 1/32. It is equivalently to say the expected number we need to toss for obtaining five heads is 32.</para>
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<para><dotfile name="/home/thuang295/Code/taskflow/doxygen/images/flipcoins_1.dot"></dotfile>
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</para>
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</sect1>
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<sect1 id="flipcoins_1FlipCoinsProbabilistic">
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<title>Probabilistic Conditions</title>
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<para>We use condition tasks to simulate the five coin tosses. We create five condition tasks each returning a random binary number. If the return is zero (head toss), the execution moves to the next condition task; or it (tail toss) goes back to the first condition task to start over the simulation.</para>
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<para><programlisting filename=".cpp"><codeline><highlight class="preprocessor">#include<sp/><<ref refid="taskflow_8hpp" kindref="compound">taskflow/taskflow.hpp</ref>></highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"></highlight><highlight class="keywordtype">int</highlight><highlight class="normal"><sp/>main()<sp/>{</highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Taskflow" kindref="compound">tf::Taskflow</ref><sp/>taskflow;</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Executor" kindref="compound">tf::Executor</ref><sp/>executor;</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="keywordtype">size_t</highlight><highlight class="normal"><sp/>rounds<sp/>=<sp/>10000;</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="keywordtype">size_t</highlight><highlight class="normal"><sp/>tosses<sp/>=<sp/>0;<sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="keywordtype">size_t</highlight><highlight class="normal"><sp/>total_tosses<sp/>=<sp/>0;<sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="keywordtype">double</highlight><highlight class="normal"><sp/>average_tosses<sp/>=<sp/>0.0;<sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>A<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/>tosses<sp/>=<sp/>0;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"init"</highlight><highlight class="normal">);<sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>B<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/>++tosses;<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%2;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-1"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>C<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%2;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-2"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>D<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%2;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-3"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>E<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%2;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-4"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>F<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%2;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-5"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="comment">//<sp/>reach<sp/>the<sp/>target;<sp/>record<sp/>the<sp/>number<sp/>of<sp/>tosses<sp/></highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>G<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/>total_tosses<sp/>+=<sp/>tosses;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"stop"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>A.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(B);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="comment">//<sp/>five<sp/>probabilistic<sp/>conditions</highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>B.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(C,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>C.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(D,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>D.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(E,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>E.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(F,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>F.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(G,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="comment">//<sp/>repeat<sp/>the<sp/>flip-coin<sp/>simulation<sp/>by<sp/>rounds<sp/>times</highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>executor.<ref refid="classtf_1_1Executor_1af15db5f7dde8e7ff1f86ef8fe825e9e2" kindref="member">run_n</ref>(taskflow,<sp/>rounds).wait();</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="comment">//<sp/>calculate<sp/>the<sp/>expected<sp/>number<sp/>of<sp/>tosses</highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>average_tosses<sp/>=<sp/>total_tosses<sp/>/<sp/>(double)rounds;</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/>assert(std::fabs(average_tosses-32.0)<1.0);</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/>0;</highlight></codeline>
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<codeline><highlight class="normal">}</highlight></codeline>
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</programlisting></para>
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<para>Running the taskflow by a fair number of times, the average tosses we have is close to 32. The taskflow diagram is depicted below.</para>
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<para><dotfile name="/home/thuang295/Code/taskflow/doxygen/images/flipcoins_2.dot"></dotfile>
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</para>
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<para>Although the execution of this taskflow is non-deterministic, its control flow can expand to a tree of tasks based on our scheduling rule for conditional tasking (see <ref refid="ConditionalTasking" kindref="compound">Conditional Tasking</ref>). Each path from the root to a leaf represents a result of five heads, and none of them can overlap at the same time (no task race). You must follow the same rule when creating a probabilistic framework using conditional tasking.</para>
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</sect1>
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<sect1 id="flipcoins_1FlipCoinsTernaryCoins">
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<title>Ternary Coins</title>
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<para>We can extend the binary coin example to a ternary case. Each condition task has one successor going back to the beginning and two successors moving to the next task. The expected number of tosses to reach five identical results is 3*3*3*3*3 = 243.</para>
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<para><programlisting filename=".cpp"><codeline><highlight class="normal"><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>A<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>(<sp/>[&](){<sp/>tosses<sp/>=<sp/>0;<sp/>}<sp/>)</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"init"</highlight><highlight class="normal">);<sp/></highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"></highlight><highlight class="comment">//<sp/>start<sp/>over<sp/>the<sp/>flip<sp/>again</highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>B<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/>++tosses;<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%3;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-1"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>C<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%3;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-2"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>D<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%3;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-3"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>E<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%3;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-4"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>F<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/></highlight><highlight class="keywordflow">return</highlight><highlight class="normal"><sp/><ref refid="cpp/numeric/random/rand" kindref="compound" external="/home/thuang295/Code/taskflow/doxygen/cppreference-doxygen-web.tag.xml">std::rand</ref>()%3;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"flip-coin-5"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"></highlight><highlight class="comment">//<sp/>reach<sp/>the<sp/>target;<sp/>record<sp/>the<sp/>number<sp/>of<sp/>tosses<sp/></highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"><ref refid="classtf_1_1Task" kindref="compound">tf::Task</ref><sp/>G<sp/>=<sp/>taskflow.<ref refid="classtf_1_1FlowBuilder_1a60d7a666cab71ecfa3010b2efb0d6b57" kindref="member">emplace</ref>([&](){<sp/>total_tosses<sp/>+=<sp/>tosses;<sp/>})</highlight></codeline>
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<codeline><highlight class="normal"><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/><sp/>.name(</highlight><highlight class="stringliteral">"stop"</highlight><highlight class="normal">);</highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal">A.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(B);</highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"></highlight><highlight class="comment">//<sp/>five<sp/>probabilistic<sp/>conditions</highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal">B.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(C,<sp/>B,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal">C.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(D,<sp/>B,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal">D.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(E,<sp/>B,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal">E.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(F,<sp/>B,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal">F.<ref refid="classtf_1_1Task_1a8c78c453295a553c1c016e4062da8588" kindref="member">precede</ref>(G,<sp/>B,<sp/>B);</highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"></highlight><highlight class="comment">//<sp/>repeat<sp/>the<sp/>flip-coin<sp/>simulation<sp/>by<sp/>rounds<sp/>times</highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal">executor.<ref refid="classtf_1_1Executor_1af15db5f7dde8e7ff1f86ef8fe825e9e2" kindref="member">run_n</ref>(taskflow,<sp/>rounds).wait();</highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal"></highlight><highlight class="comment">//<sp/>calculate<sp/>the<sp/>expected<sp/>number<sp/>of<sp/>tosses</highlight><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal">average_tosses<sp/>=<sp/>total_tosses<sp/>/<sp/>(double)rounds;</highlight></codeline>
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<codeline><highlight class="normal"></highlight></codeline>
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<codeline><highlight class="normal">assert(std::fabs(average_tosses-243.0)<1.0);</highlight></codeline>
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</programlisting></para>
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<para><dotfile name="/home/thuang295/Code/taskflow/doxygen/images/flipcoins_3.dot"></dotfile>
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</para>
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<para>Similarly, we can extend the probabilistic condition to any degree. </para>
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</sect1>
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</detaileddescription>
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<location file="doxygen/examples/flipcoins.dox"/>
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</compounddef>
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</doxygen>
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