284 lines
8.2 KiB
Text
284 lines
8.2 KiB
Text
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namespace tf {
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/** @page kmeans k-means Clustering
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We study a fundamental clustering problem in unsupervised learning, <em>k-means clustering</em>.
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We will begin by discussing the problem formulation and then learn how to
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write a parallel k-means algorithm.
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@tableofcontents
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@section KMeansProblemFormulation Problem Formulation
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k-means clustering uses @em centroids,
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k different randomly-initiated points in the data, and assigns every data point to the nearest centroid. After every point has been assigned, the centroid is moved to the average of all of the points assigned to it.
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We describe the k-means algorithm in the following steps:
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<ul>
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<li>Step 1: initialize k random centroids</li>
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<li>Step 2: for every data point, find the nearest centroid (L2 distance or other measurements) and assign the point to it</li>
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<li>Step 3: for every centroid, move the centroid to the average of the points assigned to that centroid</li>
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<li>Step 4: go to Step 2 until converged (no more changes in the last few iterations) or maximum iterations reached
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</ul>
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The algorithm is illustrated as follows:
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@image html images/kmeans_1.png
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A sequential implementation of k-means is described as follows:
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@code{.cpp}
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// sequential implementation of k-means on a CPU
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// N: number of points
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// K: number of clusters
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// M: number of iterations
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// px/py: 2D point vector
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void kmeans_seq(
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int N, int K, int M, const std::vector<float>& px, const std::vector<float>& py
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) {
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std::vector<int> c(K);
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std::vector<float> sx(K), sy(K), mx(K), my(K);
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// initial centroids
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std::copy_n(px.begin(), K, mx.begin());
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std::copy_n(py.begin(), K, my.begin());
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// k-means iteration
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for(int m=0; m<M; m++) {
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// clear the storage
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std::fill_n(sx.begin(), K, 0.0f);
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std::fill_n(sy.begin(), K, 0.0f);
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std::fill_n(c.begin(), K, 0);
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// find the best k (cluster id) for each point
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for(int i=0; i<N; ++i) {
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float x = px[i];
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float y = py[i];
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float best_d = std::numeric_limits<float>::max();
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int best_k = 0;
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for (int k = 0; k < K; ++k) {
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const float d = L2(x, y, mx[k], my[k]);
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if (d < best_d) {
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best_d = d;
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best_k = k;
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}
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}
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sx[best_k] += x;
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sy[best_k] += y;
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c [best_k] += 1;
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}
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// update the centroid
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for(int k=0; k<K; k++) {
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const int count = max(1, c[k]); // turn 0/0 to 0/1
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mx[k] = sx[k] / count;
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my[k] = sy[k] / count;
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}
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}
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// print the k centroids found
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for(int k=0; k<K; ++k) {
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std::cout << "centroid " << k << ": " << std::setw(10) << mx[k] << ' '
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<< std::setw(10) << my[k] << '\n';
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}
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}
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@endcode
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@section ParallelKMeansUsingCPUs Parallel k-means using CPUs
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The second step of k-means algorithm, <em>assigning every point to the nearest centroid</em>,
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is highly parallelizable across individual points.
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We can create a @em parallel-for task to run parallel iterations.
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@code{.cpp}
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std::vector<int> best_ks(N); // nearest centroid of each point
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unsigned P = 12; // 12 partitioned tasks
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// update cluster
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taskflow.for_each_index(0, N, 1, [&](int i){
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float x = px[i];
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float y = py[i];
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float best_d = std::numeric_limits<float>::max();
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int best_k = 0;
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for (int k = 0; k < K; ++k) {
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const float d = L2(x, y, mx[k], my[k]);
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if (d < best_d) {
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best_d = d;
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best_k = k;
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}
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}
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best_ks[i] = best_k;
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});
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@endcode
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The third step of moving every centroid to the average of points is also parallelizable
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across individual centroids.
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However, since k is typically not large, one task of doing this update is sufficient.
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@code{.cpp}
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taskflow.emplace([&](){
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// sum of points
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for(int i=0; i<N; i++) {
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sx[best_ks[i]] += px[i];
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sy[best_ks[i]] += py[i];
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c [best_ks[i]] += 1;
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}
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// average of points
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for(int k=0; k<K; ++k) {
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auto count = max(1, c[k]); // turn 0/0 to 0/1
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mx[k] = sx[k] / count;
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my[k] = sy[k] / count;
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}
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});
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@endcode
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To describe @c M iterations, we create a condition task that loops the second step
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of the algorithm by @c M times.
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The return value of zero goes to the first successor which we will connect to the task
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of the second step later; otherwise, k-means completes.
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@code{.cpp}
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taskflow.emplace([m=0, M]() mutable {
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return (m++ < M) ? 0 : 1;
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});
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@endcode
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The entire code of CPU-parallel k-means is shown below.
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Here we use an additional storage, @c best_ks, to record the nearest centroid of a point at an iteration.
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@code{.cpp}
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// N: number of points
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// K: number of clusters
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// M: number of iterations
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// px/py: 2D point vector
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void kmeans_par(
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int N, int K, int M, cconst std::vector<float>& px, const std::vector<float>& py
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) {
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unsigned P = 12; // 12 partitions of the parallel-for graph
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tf::Executor executor;
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tf::Taskflow taskflow("K-Means");
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std::vector<int> c(K), best_ks(N);
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std::vector<float> sx(K), sy(K), mx(K), my(K);
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// initial centroids
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tf::Task init = taskflow.emplace([&](){
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for(int i=0; i<K; ++i) {
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mx[i] = px[i];
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my[i] = py[i];
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}
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}).name("init");
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// clear the storage
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tf::Task clean_up = taskflow.emplace([&](){
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for(int k=0; k<K; ++k) {
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sx[k] = 0.0f;
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sy[k] = 0.0f;
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c [k] = 0;
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}
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}).name("clean_up");
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// update cluster
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tf::Task pf = taskflow.for_each_index(0, N, 1, [&](int i){
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float x = px[i];
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float y = py[i];
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float best_d = std::numeric_limits<float>::max();
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int best_k = 0;
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for (int k = 0; k < K; ++k) {
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const float d = L2(x, y, mx[k], my[k]);
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if (d < best_d) {
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best_d = d;
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best_k = k;
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}
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}
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best_ks[i] = best_k;
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}).name("parallel-for");
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tf::Task update_cluster = taskflow.emplace([&](){
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for(int i=0; i<N; i++) {
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sx[best_ks[i]] += px[i];
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sy[best_ks[i]] += py[i];
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c [best_ks[i]] += 1;
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}
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for(int k=0; k<K; ++k) {
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auto count = max(1, c[k]); // turn 0/0 to 0/1
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mx[k] = sx[k] / count;
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my[k] = sy[k] / count;
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}
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}).name("update_cluster");
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// convergence check
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tf::Task condition = taskflow.emplace([m=0, M]() mutable {
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return (m++ < M) ? 0 : 1;
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}).name("converged?");
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init.precede(clean_up);
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clean_up.precede(pf);
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pf.precede(update_cluster);
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condition.precede(clean_up)
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.succeed(update_cluster);
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executor.run(taskflow).wait();
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}
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@endcode
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The taskflow consists of two parts, a @c clean_up task and a parallel-for graph.
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The former cleans up the storage @c sx, @c sy, and @c c that are used to average points for new centroids,
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and the later parallelizes the searching for nearest centroids across individual points using
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12 tasks (may vary depending on the machine).
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If the iteration count is smaller than @c M,
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the condition task returns 0 to let the execution path go back to @c clean_up.
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Otherwise, it returns 1 to stop (i.e., no successor tasks at index 1).
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The taskflow graph is illustrated below:
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<!-- @image html images/kmeans_2.svg width=100% -->
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@dotfile images/kmeans_2.dot
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The scheduler starts with @c init, moves on to @c clean_up, and then enters the
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parallel-for task @c parallel-for that spawns a subflow of 12 workers to perform
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parallel iterations.
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When @c parallel-for completes, it updates the cluster centroids and checks if
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they have converged through a condition task.
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If not, the condition task informs the scheduler to go back to @c clean_up and then
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@c parallel-for; otherwise, it returns a nominal index to stop the scheduler.
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@section KMeansBenchmarking Benchmarking
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Based on the discussion above, we compare the runtime of computing
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various k-means problem sizes between a sequential
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CPU and parallel CPUs on a machine of 12 Intel i7-8700 CPUs at 3.2 GHz.
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<div align="center">
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| N | K | M | CPU Sequential | CPU Parallel |
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| :-: | :-: | :-: | :-: | :-: |
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| 10 | 5 | 10 | 0.14 ms | 77 ms |
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| 100 | 10 | 100 | 0.56 ms | 86 ms |
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| 1000 | 10 | 1000 | 10 ms | 98 ms |
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| 10000 | 10 | 10000 | 1006 ms | 713 ms |
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| 100000 | 10 | 100000 | 102483 ms | 49966 ms |
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</div>
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When the number of points is larger than 10K,
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the parallel CPU implementation starts to outperform the sequential CPU
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implementation.
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*/
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}
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