{"author":{"avatar":"https://nguyencungthanh.github.io/animation_avatar%282%29.PNG","name":"Nguyen Cung Thanh","url":"https://nguyencungthanh.github.io/"},"feed_url":"https://nguyencungthanh.github.io/categories/projects/feed.json","home_page_url":"https://nguyencungthanh.github.io/categories/projects/","items":[{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003cp\u003e\u003ca href=\"/2026/05/26/batch-norm/\"  \u003eBatch Norm\u003c/a\u003e,\n\u003ca href=\"/2026/05/26/layer-norm/\"  \u003eLayer Norm\u003c/a\u003e\u003c/p\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: Neural%20Networks\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2026-05-26T00:00:00+00:00","date_published":"2026-05-26T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2026/05/26/neural-networks/","summary":"","tags":["Neural Networks"],"title":"Neural Networks","url":"https://nguyencungthanh.github.io/2026/05/26/neural-networks/"},{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003cp\u003eGiven $n$ points in the plane. We want to find a point that minimizes the sum of distance between this point and all of the given points. One of the easiest but efficient way to approach this problem is using \u003ccode\u003eLocal Search\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eLocal search is a heuristic method for solving computationally hard optimization problems. Indeed, sometimes we tackle an optimization problem that we cannot find the exactly minimization argument, local search is created to find a \u0026lsquo;good enough\u0026rsquo; solution for the problem. The main ideas of algorithm can be expressed as follow:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eInitialize an arbitrary configuration\u003c/li\u003e\n\u003cli\u003eConsider any configuration that adjacents to present configuration and update to the best configuration\u003c/li\u003e\n\u003cli\u003eIterate until fixed number of iteration or meet some conditions\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eBack to the initial problem, we can express it as follow:\u003c/p\u003e\n\u003cdiv class=\"post-callout\"\u003e\nGiven $n$ points $(x_i,y_i)$ on the plane. Find a point $(x, y)$ that minizes \n$$f(x, y) = \\sum_{i = 1}^{n} \\sqrt{(x-x_i)^2 + (y-y_i)^2}$$ \n\u003c/div\u003e\n\u003cp\u003eTaking the partial derivative of $f$ with respect to $x$ and $y$ yields:\n$$\\begin{cases}\n\\dfrac{\\partial f}{\\partial x} = \\sum_{i=1}^{n} \\dfrac{x-x_i}{\\sqrt{(x-x_i)^2 + (y-y_i)^2}} \\\\\n\\dfrac{\\partial f}{\\partial y} = \\sum_{i=1}^{n} \\dfrac{y-y_i}{\\sqrt{(x-x_i)^2 + (y-y_i)^2}}\n\\end{cases}$$\u003c/p\u003e\n\u003cp\u003eSet them equal to $0$ we will have\n$$x = \\dfrac{\\sum_{i=1}^{n} w_ix_i}{\\sum_{i=1}^n w_i} \\quad \\text{ and } \\quad y = \\dfrac{ \\sum_{i=1}^{n} w_iy_i}{\\sum_{i=1}^n w_i}$$\nwhere $$w_i =  \\dfrac{1}{\\sqrt{(x-x_i)^2 + (y-y_i)^2}} \\quad \\forall i = \\overline{1,n}$$\u003c/p\u003e\n\u003cp\u003eThis give our idea of update from the k-iteration point $(z_k,w_k)$ to the next point $(z_{k+1}, w_{k+1})$ equal to $$\\left(\\dfrac{\\sum_{i=1}^{n} w_{i,k}x_i}{\\sum_{i=1}^n w_{i,k}}, \\dfrac{\\sum_{i=1}^{n} w_{i,k}x_i}{\\sum_{i=1}^n w_{i,k}} \\right)$$\nwith $$w_{i,k} =  \\dfrac{1}{\\sqrt{(z_k-x_i)^2 + (w_k-y_i)^2}} \\quad \\forall i = \\overline{1,n}$$\nFollowing that idea, we can iterate 100 times and the condition to stop is the distance between the previous point and the new point is less than $10^{-6}$. The below C++ code is the example implementation code for that idea.\u003c/p\u003e\n\u003cdiv class=\"highlight\"\u003e\u003cpre tabindex=\"0\" class=\"chroma\"\u003e\u003ccode class=\"language-cpp\" data-lang=\"cpp\"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e  \u003cspan class=\"nf\"\u003edistance\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"k\"\u003econst\u003c/span\u003e \u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026gt;\u0026amp;\u003c/span\u003e \u003cspan class=\"n\"\u003ex\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"k\"\u003econst\u003c/span\u003e \u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026gt;\u0026amp;\u003c/span\u003e \u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e){\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e  \u003cspan class=\"n\"\u003eresult\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003esqrt\u003c/span\u003e\u003cspan class=\"p\"\u003e((\u003c/span\u003e\u003cspan class=\"n\"\u003ex\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003efirst\u003c/span\u003e\u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003efirst\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\u003cspan class=\"o\"\u003e*\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003ex\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003efirst\u003c/span\u003e\u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003efirst\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"o\"\u003e+\u003c/span\u003e \u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003ex\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003esecond\u003c/span\u003e\u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003esecond\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e\u003cspan class=\"o\"\u003e*\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003ex\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003esecond\u003c/span\u003e\u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003esecond\u003c/span\u003e\u003cspan class=\"p\"\u003e));\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"n\"\u003eresult\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003egeometric_median\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"k\"\u003econst\u003c/span\u003e \u003cspan class=\"n\"\u003evector\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026gt;\u0026gt;\u0026amp;\u003c/span\u003e \u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003emax_iteration\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e100\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"n\"\u003etol\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e1e-6\u003c/span\u003e\u003cspan class=\"p\"\u003e){\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003esize\u003c/span\u003e\u003cspan class=\"p\"\u003e();\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"n\"\u003esum_x\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e0.0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"n\"\u003esum_y\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e0.0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"c1\"\u003e// Find the centroid of n points and assign the initial point to the centroid\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c1\"\u003e\u003c/span\u003e    \u003cspan class=\"k\"\u003efor\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"o\"\u003e++\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e){\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003esum_x\u003c/span\u003e \u003cspan class=\"o\"\u003e+=\u003c/span\u003e \u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e].\u003c/span\u003e\u003cspan class=\"n\"\u003efirst\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003esum_y\u003c/span\u003e \u003cspan class=\"o\"\u003e+=\u003c/span\u003e \u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e].\u003c/span\u003e\u003cspan class=\"n\"\u003esecond\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003ecentroid\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\u003cspan class=\"n\"\u003esum_x\u003c/span\u003e\u003cspan class=\"o\"\u003e/\u003c/span\u003e\u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003esum_y\u003c/span\u003e\u003cspan class=\"o\"\u003e/\u003c/span\u003e\u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e};\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003ecur_point\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003ecentroid\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"c1\"\u003e// lists store the distance and weight between current point and n points\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c1\"\u003e\u003c/span\u003e    \u003cspan class=\"n\"\u003evector\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003edistances\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003evector\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003eweights\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003ewhile\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003emax_iteration\u003c/span\u003e\u003cspan class=\"o\"\u003e--\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e){\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"kt\"\u003ebool\u003c/span\u003e \u003cspan class=\"n\"\u003ecoincides\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"nb\"\u003efalse\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003ecoincide_idx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"mi\"\u003e1\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"k\"\u003efor\u003c/span\u003e \u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"o\"\u003e++\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"n\"\u003edistances\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003edistance\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003ecur_point\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e]);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"k\"\u003eif\u003c/span\u003e \u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003edistances\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e \u003cspan class=\"o\"\u003e==\u003c/span\u003e \u003cspan class=\"mf\"\u003e0.0\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e                \u003cspan class=\"n\"\u003ecoincides\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"nb\"\u003etrue\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e                \u003cspan class=\"n\"\u003ecoincide_idx\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e                \u003cspan class=\"k\"\u003ebreak\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"k\"\u003eif\u003c/span\u003e \u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003ecoincides\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ecoincide_idx\u003c/span\u003e\u003cspan class=\"p\"\u003e];\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"k\"\u003efor\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"o\"\u003e++\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e){\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"n\"\u003eweights\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e1.0\u003c/span\u003e\u003cspan class=\"o\"\u003e/\u003c/span\u003e\u003cspan class=\"n\"\u003edistances\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e];\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"c1\"\u003e// Compute new point by the formular: x = (sum w_i*x_i)/(sum w_i), y = (sum w_i*y_i)/(sum w_i)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c1\"\u003e\u003c/span\u003e        \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e  \u003cspan class=\"n\"\u003enumerator_x\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e0.0\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003enumerator_y\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e0.0\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003edenominator\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mf\"\u003e0.0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"k\"\u003efor\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"o\"\u003e++\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e){\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"n\"\u003enumerator_x\u003c/span\u003e \u003cspan class=\"o\"\u003e+=\u003c/span\u003e \u003cspan class=\"n\"\u003eweights\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003cspan class=\"o\"\u003e*\u003c/span\u003e\u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e].\u003c/span\u003e\u003cspan class=\"n\"\u003efirst\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"n\"\u003enumerator_y\u003c/span\u003e \u003cspan class=\"o\"\u003e+=\u003c/span\u003e \u003cspan class=\"n\"\u003eweights\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003cspan class=\"o\"\u003e*\u003c/span\u003e\u003cspan class=\"n\"\u003epoints\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e].\u003c/span\u003e\u003cspan class=\"n\"\u003esecond\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e            \u003cspan class=\"n\"\u003edenominator\u003c/span\u003e \u003cspan class=\"o\"\u003e+=\u003c/span\u003e \u003cspan class=\"n\"\u003eweights\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e];\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003enew_point\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\u003cspan class=\"n\"\u003enumerator_x\u003c/span\u003e\u003cspan class=\"o\"\u003e/\u003c/span\u003e\u003cspan class=\"n\"\u003edenominator\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003enumerator_y\u003c/span\u003e\u003cspan class=\"o\"\u003e/\u003c/span\u003e\u003cspan class=\"n\"\u003edenominator\u003c/span\u003e\u003cspan class=\"p\"\u003e};\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"c1\"\u003e// Compute the distance between new point and current point; if the result is smaller than tolerence then stop the iteration\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c1\"\u003e\u003c/span\u003e        \u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e  \u003cspan class=\"n\"\u003eshift\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003edistance\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003enew_point\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003ecur_point\u003c/span\u003e\u003cspan class=\"p\"\u003e);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"k\"\u003eif\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003eshift\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003etol\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"k\"\u003ebreak\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e        \u003cspan class=\"n\"\u003ecur_point\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003enew_point\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"n\"\u003ecur_point\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"nf\"\u003emain\u003c/span\u003e\u003cspan class=\"p\"\u003e()\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"n\"\u003eios_base\u003c/span\u003e\u003cspan class=\"o\"\u003e::\u003c/span\u003e\u003cspan class=\"n\"\u003esync_with_stdio\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003efalse\u003c/span\u003e\u003cspan class=\"p\"\u003e);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003ecin\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003etie\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003eNULL\u003c/span\u003e\u003cspan class=\"p\"\u003e);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"n\"\u003ecin\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"n\"\u003evector\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"n\"\u003epair\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e\u003cspan class=\"o\"\u003e\u0026gt;\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003epts\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003efor\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"kt\"\u003eint\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"n\"\u003ei\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003en\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e \u003cspan class=\"o\"\u003e++\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e){\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"kt\"\u003elong\u003c/span\u003e \u003cspan class=\"kt\"\u003edouble\u003c/span\u003e \u003cspan class=\"n\"\u003ex\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"n\"\u003ecin\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003ex\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026gt;\u0026gt;\u003c/span\u003e \u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"n\"\u003epts\u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"n\"\u003ei\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\u003cspan class=\"n\"\u003ex\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"n\"\u003ey\u003c/span\u003e\u003cspan class=\"p\"\u003e};\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003eauto\u003c/span\u003e \u003cspan class=\"n\"\u003eopt\u003c/span\u003e \u003cspan class=\"o\"\u003e=\u003c/span\u003e \u003cspan class=\"n\"\u003egeometric_median\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"n\"\u003epts\u003c/span\u003e\u003cspan class=\"p\"\u003e);\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"n\"\u003ecout\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003efixed\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003esetprecision\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"mi\"\u003e6\u003c/span\u003e\u003cspan class=\"p\"\u003e)\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e \u003cspan class=\"s\"\u003e\u0026#34;Optimal solution\u0026#34;\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e \u003cspan class=\"s\"\u003e\u0026#34;(\u0026#34;\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003eopt\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003efirst\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;, \u0026#34;\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e \u003cspan class=\"n\"\u003eopt\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"n\"\u003esecond\u003c/span\u003e \u003cspan class=\"o\"\u003e\u0026lt;\u0026lt;\u003c/span\u003e \u003cspan class=\"s\"\u003e\u0026#34;)\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e    \u003cspan class=\"k\"\u003ereturn\u003c/span\u003e \u003cspan class=\"mi\"\u003e0\u003c/span\u003e\u003cspan class=\"p\"\u003e;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\u003c/div\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: Local%20Search\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2026-01-17T00:00:00+00:00","date_published":"2026-01-17T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2026/01/17/local-search/","summary":"Given $n$ points in the plane. We want to find a point that minimizes the sum of distance between this point and all of the given points. One of the easiest but efficient way to approach this problem is using Local Search.\n","tags":["Heuristic Algorithm"],"title":"Local Search","url":"https://nguyencungthanh.github.io/2026/01/17/local-search/"}],"title":"Projects - Nguyen Cung Thanh","version":"https://jsonfeed.org/version/1"}