{"id":20355,"date":"2025-12-07T09:24:59","date_gmt":"2025-12-07T01:24:59","guid":{"rendered":"https:\/\/www.orczhou.com\/?p=20355"},"modified":"2025-12-16T14:51:48","modified_gmt":"2025-12-16T06:51:48","slug":"understanding-diskann-robustprune","status":"publish","type":"post","link":"https:\/\/www.orczhou.com\/index.php\/2025\/12\/understanding-diskann-robustprune\/","title":{"rendered":"\u7406\u89e3 DiskANN \u7684\u6838\u5fc3\u201cRobustPrune\u201d"},"content":{"rendered":"\n\n\n\n<p>DiskANN \u662f\u8f83\u4e3a\u5e38\u89c1\u7684\u5411\u91cf\u6570\u636e\u5e93\u641c\u7d22\u7b97\u6cd5\uff0c\u4f5c\u8005\u901a\u8fc7\u201c\u8d2a\u5a6a\u641c\u7d22\u201d\u548c\u201c\u5065\u58ee\u6027\u526a\u679d\u201d\uff08\u201cRobustPrune\u201d\uff09\u6765\u6784\u5efa\u4e00\u4e2a\u201c\u4f4e\u76f4\u5f84\u201d\u7684\u56fe\uff0c\u4ece\u800c\u5b9e\u73b0\u9ad8\u6027\u80fd\u7684\u5411\u91cf\u641c\u7d22\u3002\u8fd9\u91cc\u7684\u201c\u4f4e\u76f4\u5f84\u201d\u5f88\u597d\u7684\u4fdd\u969c\u4e86\u70b9\u4e0e\u70b9\u4e4b\u95f4\u975e\u5e38\u9ad8\u6548\uff08\u53ea\u9700\u8981\u5c11\u6570\u51e0\u6761\uff09\u5b8c\u6210\u641c\u7d22\u3002\u201cRobustPrune\u201d\u5219\u662f\u6784\u5efa\u4f4e\u76f4\u5f84\u7684\u201cVamana\u201d\u56fe\u7684\u5173\u952e\u6b65\u9aa4\u3002\u672c\u6587\u901a\u8fc7\u8ba1\u7b97\u63a8\u5bfc\u3001\u793a\u4f8b\u4e0e\u8bd5\u9a8c\u7684\u65b9\u5f0f\uff0c\u6765\u770b\u770b\u201cRobustPrune\u201d\u5982\u4f55\u5b9e\u73b0\u201c\u4f4e\u76f4\u5f84\u201d\u56fe\u7684\u6784\u5efa\uff0c\u800c\u8fd9\u4e5f\u6b63\u662f DiskANN \u7b97\u6cd5\u7684\u6838\u5fc3\u4e4b\u4e00\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. RobustPrune \u6982\u8ff0<\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u8bba\u6587\u4e2d\u7684\u539f\u59cb\u63cf\u8ff0\u5982\u53f3\u56fe\u3002<\/p>\n\n\n\n<p>\u53ef\u4ee5\u8fd9\u6837\u7406\u89e3\uff1a\u5728\u627e\u5230 \\(p \\) \u7684\u82e5\u5e72\u201c\u90bb\u5c45\u201d\u540e\uff08\u90bb\u5c45\u96c6\u5408\u8bb0\u4e3a \\(N_{out}(p) \\) \uff09\uff0c\u7531\u4e8e\u201c\u8fb9\u201d\uff08\u51fa\u5ea6\uff09\u6570\u91cf\u6709\u9650\uff08\u8fd9\u91cc\u662f \\(R \\) \uff09\uff0c\u6545\u9700\u8981\u8fdb\u884c\u201c\u526a\u679d\u201d\uff0c\u90a3\u4e48\u8fd9\u91cc\u7684\u201c\u526a\u679d\u201d\u65b9\u6cd5\u5373\u4e3a\u201cRobustPrune\u201d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5728 \\(N_{out}(p) \\) \u4e2d\u627e\u51fa\u8ddd\u79bb  \\(p \\) \u6700\u8fd1\u7684\u70b9\uff0c\u8bb0\u4e3a \\(p^{*} \\) <\/li>\n\n\n\n<li>\u9700\u8981\u201c\u526a\u679d\u201d\u65f6\uff0c\u5c06\u6ee1\u8db3\u4e0b\u5217\u6761\u4ef6\u7684\u70b9\u90fd\u201c\u526a\u6389\u201d\uff1a\n<ul class=\"wp-block-list\">\n<li>\\(\\alpha \\cdot d(p^{*},p&#8217;) \\le d(p,p&#8217;) \\)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"488\" height=\"482\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-30.png\" alt=\"\" class=\"wp-image-20376\" style=\"width:381px;height:auto\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-30.png 488w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-30-300x296.png 300w\" sizes=\"auto, (max-width: 488px) 100vw, 488px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:1em\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p>\u5728\u521d\u6b21\u770b\u5230\uff0c\u8868\u8fbe\u5f0f \\(\\alpha \\cdot d(p^{*},p&#8217;) \\le d(p,p&#8217;) \\) \u7684\u65f6\u5019\uff0c\u662f\u6bd4\u8f83\u96be\u7406\u89e3\u5176\u80cc\u540e\u7684\u7528\u610f\u7684\uff0c\u672c\u6587\u5219\u8f83\u4e3a\u8be6\u7ec6\u4ee5\u5206\u6790\u3001\u63a8\u5bfc\u3001\u201c\u8bd5\u9a8c\u201d\u7684\u65b9\u5f0f\u4ecb\u7ecd\u8be5\u516c\u5f0f\u5728\u56fe\u6784\u5efa\u8fc7\u7a0b\u4e2d\u6240\u53d6\u7684\u6548\u679c\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u4e8c\u7ef4\u7a7a\u95f4\u7684\u793a\u4f8b<\/h3>\n\n\n\n<p>\u8fd9\u91cc\u4ee5\u4e8c\u7ef4\u7a7a\u95f4\u4e3a\u4f8b\uff0c\u4ee5\u4e00\u4e2a\u5177\u4f53\u7684\u573a\u666f\u6765\u89c2\u5bdf RobustPrune \u7684\u6548\u679c\u3002\u5177\u4f53\u7684\uff0c\u5c06\u4e0a\u8ff0\u7684\u201c\u526a\u679d\u201d\u64cd\u4f5c\u63cf\u8ff0\u4e3a\u5982\u4e0b\u7684\u6570\u5b66\u5f62\u5f0f\uff1a<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2.1 \u201c\u526a\u679d\u201d\u533a\u57df\u8ba1\u7b97<\/h4>\n\n\n\n<p>\u5728\u4e8c\u7ef4\u5750\u6807\u7cfb\u4e2d\u6709 \\(P = (0,0) \\, A = (2,0) \\) \uff0c\u8003\u8651\u8be5\u96c6\u5408\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\mathscr{L} = \\{ X | \\frac{d(X,P)}{d(X,A)} \\le \\alpha \\quad \\text{and} \\quad d(X,A) < d(X,P) \\} \n$$\n\n<\/p>\n\n\n\n<p>\u5373\uff0c\u5728 RobustPrune \u4e2d\u843d\u5165\u8be5\u533a\u57df\u7684\u70b9\uff0c\u5c06\u88ab\u526a\u679d\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc \\(\\alpha \\) \u662f\u4e00\u4e2a\u5e38\u6570\uff0c\u4e14 \\(\\alpha \\gt 1 \\) \u3002 \u5047\u8bbe\uff0c \\(\\mathscr{L} \\) \u4e2d\u7684\u70b9 \\(X\\) \u5750\u6807\u662f \\((x,y) \\)\uff0c\u90a3\u4e48\u63a8\u5bfc \\(x \\, y \\) \u9700\u8981\u6ee1\u8db3\u600e\u6837\u7684\u9650\u5236\u6761\u4ef6\u3002<\/p>\n\n\n\n<p>\u5728\u4e0a\u8ff0\u7684\u201c\u6570\u5b66\u201d\u95ee\u9898\u4e2d\uff0c\u70b9 \\(P = (0,0) \\) \u5373\u4e3a\u8bba\u6587\u7b97\u6cd5\u4e2d\u7684 \\(p \\)\uff0c\u70b9 \\(A \\) \u5373\u4e3a\u8bba\u6587\u7b97\u6cd5\u4e2d\u8ddd\u79bb \\(p \\) \u70b9\u6700\u8fd1\u7684\u90bb\u5c45 \\(p^{*} \\)\u3002\u95ee\u9898\u4e2d\u7684 \\(x \\, y \\) \u9650\u5236\u63d0\u4ea4\uff0c\u5373\u4e3a\u8bba\u6587\u4e2d\u7684\u201c\u526a\u679d\u201d\u6761\u4ef6\u3002<\/p>\n\n\n\n<p>\u6839\u636e\u201c\u7b80\u5355\u201d\u7684\u6570\u5b66\u63a8\u5bfc\uff08\u8be6\u7ec6\u63a8\u5bfc\u53ef\u4ee5\u53c2\u8003\u9644\u5f55\uff0c\u4e3a\u4e86\u8fde\u7eed\u6027\u8fd9\u91cc\u6682\u4e0d\u8be6\u8ff0\uff09\uff0c\u6709\u5982\u4e0b\u7ed3\u679c\uff0c\u5373\u5750\u6807 \\((x , y) \\) \u9700\u6ee1\u8db3\u5982\u4e0b\u6761\u4ef6\uff1a<\/p>\n\n\n<p>$$<br \/>\n(x-\\frac{2\\alpha^2}{\\alpha^2-1})^2 + y^2 \\le \\frac{4\\alpha^2}{(\\alpha^2-1)^2}<br \/>\n$$<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u53ef\u4ee5\u770b\u5230\uff0c\u8fd9\u662f\u4e00\u4e2a\u4ee5 \\(\\frac{2\\alpha}{\\alpha^2 -1} \\) \u4e3a\u534a\u5f84\uff0c\u4ee5 \\((\\frac{2\\alpha^2}{\\alpha^2-1},0) \\) \u4e3a\u5706\u5fc3\u7684\u5706\u5185\u90e8\u533a\u57df\uff08\u5305\u62ec\u8fb9\u7f18\uff09\u3002<\/p>\n\n\n\n<p>\u66f4\u4e3a\u5177\u4f53\u7684\uff0c\u53d6 \\(\\alpha = 1.2 \\)\uff0c\u5219\u5728\u5750\u6807\u7cfb\u4e2d\uff0c\u8be5\u533a\u57df\u4e3a\u53f3\u56fe\u3002<\/p>\n\n\n\n<p>\u5373\uff0c\u5f53 \\(P \\) \u70b9\u3001 \\(A \\) \u70b9\u786e\u5b9a\u65f6\uff0c\u843d\u5165\u53f3\u4fa7\u56fe\u5f62\u4e2d\u9634\u5f71\u90e8\u5206\u7684\u70b9\uff0c\u5747\u4f1a\u88ab\u201c\u526a\u679d\u201d\u3002<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"498\" height=\"476\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-29.png\" alt=\"\" class=\"wp-image-20375\" style=\"width:377px;height:auto\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-29.png 498w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-29-300x287.png 300w\" sizes=\"auto, (max-width: 498px) 100vw, 498px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\">2.2 \u5b9e\u9645\u201c\u526a\u679d\u201d\u533a\u57df<\/h4>\n\n\n\n<p>\u66f4\u4e3a\u4e00\u822c\u7684\uff0c\u5982\u679c\u5728\u5e73\u9762\u4e2d\u4efb\u610f\u53bb\u4e00\u4e9b\u70b9\uff0c\u4e5f\u6709\u7c7b\u4f3c\u7684\u7ed3\u8bba\u3002\u5176\u5bf9\u5e94\u7684\u201c\u526a\u679d\u5706\u201d\uff08\u6709\u7684\u5730\u65b9\u4f1a\u79f0\u4e3a\u201c\u906e\u853d\u7a7a\u95f4\uff09\uff0c\u5219\u6709\u5982\u4e0b\u7684\u5f62\u5f0f\uff1a<\/p>\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u8003\u8651\u53f3\u56fe\u4e2d\uff0c \\(A \\) \u4e3a\u8ddd\u79bb \\(P \\) \u6700\u8fd1\u7684\u90bb\u5c45\uff0c\u90a3\u4e48\u770b\u770b\u70b9 \\(B \\)\u3001\\(C \\) \u662f\u5426\u4f1a\u88ab\u906e\u853d\u3002<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"948\" height=\"370\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/11\/image-3.png\" alt=\"\" class=\"wp-image-20663\" style=\"width:374px;height:auto\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/11\/image-3.png 948w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/11\/image-3-300x117.png 300w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/11\/image-3-768x300.png 768w\" sizes=\"auto, (max-width: 948px) 100vw, 948px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u4e0e\u4e0a\u8ff0\u63a8\u5bfc\u7c7b\u4f3c\u7684\uff0c\u53ef\u4ee5\u8ba1\u7b97\u51fa\u5bf9\u5e94\u7684\u201c\u526a\u679d\u5706\u201d\u5982\u53f3\u56fe\u9634\u5f71\u90e8\u5206\uff0c\u56e0\u4e3a \\(C \\) \u70b9\u6070\u597d\u843d\u5728\u8be5\u533a\u57df\uff08\u4e14\u53c8\u56e0\u4e3a\u90bb\u5c45\u6570\u91cf\u8d85\u8fc7\u6700\u5927\u51fa\u5ea6\uff09\uff0c\u6545 \\(C \\) \u5c06\u88ab\u4ece\u90bb\u5c45\u4e2d\u5220\u9664\u3002<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"692\" height=\"526\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/11\/image-4.png\" alt=\"\" class=\"wp-image-20664\" style=\"width:336px;height:auto\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/11\/image-4.png 692w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/11\/image-4-300x228.png 300w\" sizes=\"auto, (max-width: 692px) 100vw, 692px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>\u8fd9\u5c31\u662f RobustPrune \u5728\u4e8c\u7ef4\u7a7a\u95f4\u7684\u6548\u679c\u3002\u66f4\u8fdb\u4e00\u6b65\u7684\uff0c\u4e3a\u4ec0\u4e48\u8fd9\u6837\u8fdb\u884c\u526a\u679d\u5c31\u53ef\u4ee5\u4fdd\u969c\u201c\u4f4e\u76f4\u5f84\u201d\u5462\uff1f<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. \u201c\u526a\u679d\u201d\u3001\u4e0d\u201c\u526a\u679d\u201d\u7684\u5bf9\u6bd4<\/h3>\n\n\n\n<p>\u4e3a\u4e86\u8fdb\u4e00\u6b65\u52a0\u6df1\u5bf9 RobustPrune \u6548\u679c\u7684\u7406\u89e3\uff0c\u6211\u4eec\u8003\u8651\u8fd9\u6837\u7684\u95ee\u9898\uff1a\u5728\u4e00\u4e2a Vamana \u56fe\u7684\u6784\u5efa\u8fc7\u7a0b\u4e2d\uff0c\u4f7f\u7528 RobustPrune \u8fdb\u884c\u201c\u526a\u679d\u201d\u548c\u4e0d\u4f7f\u7528\u8be5\u65b9\u6cd5\u526a\u679d\uff0c\u6784\u5efa\u7684\u56fe\u4f1a\u6709\u4ec0\u4e48\u4e0d\u540c\uff1f<\/p>\n\n\n\n<p>\u5728\u8fdb\u884c\u6b63\u5f0f\u7684\u8bd5\u9a8c\u9a8c\u8bc1\u4e4b\u524d\uff0c\u6211\u4eec\u6709\u7406\u7531\u8fd9\u6837\u53bb\u8003\u8651\uff1a\u56e0\u4e3a\u201c\u51fa\u5ea6\u201d \\(R \\) \u662f\u56fa\u5b9a\u7684\u503c\uff0c\u5c06\u8ddd\u79bb \\(A \\) \u5f88\u8fd1\u7684\u70b9\u4ece\u5019\u9009\u90bb\u5c45\u4e2d\u5220\u9664\uff0c\u53ef\u4ee5\u907f\u514d\u67d0\u4e9b\u533a\u57df\u90bb\u5c45\u8fc7\u4e8e\u96c6\u4e2d\uff0c\u5e76\u5c06\u8be5\u4f4d\u7f6e\u9884\u7559\u7ed9\u8ddd\u79bb \\(P \\) \u8fdc\u4e00\u4e9b\u7684\u90bb\u5c45\uff0c\u8fd9\u53ef\u80fd\u4f1a\u6709\u5229\u4e8e\u964d\u4f4e\u56fe\u7684\u76f4\u5f84\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7406\u89e3\uff0c\u5373\u4e3a RobustPrune \u7684\u76f4\u89c9\u5efa\u7acb\u7684\u6bd4\u8f83\u6838\u5fc3\u7684\u7406\u89e3\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u66f4\u597d\u7684\u7406\u89e3\u4e0a\u8ff0\u5185\u5bb9\uff0c\u5e76\u9a8c\u8bc1\u4e0a\u8ff0\u201c\u76f4\u89c9\u201d\uff0c\u8fd9\u91cc\u901a\u8fc7\u7a0b\u5e8f\u968f\u673a\u751f\u6210<code>30<\/code>\u70b9\uff0c\u5e76\u5c1d\u8bd5\u6784\u5efa Vamana \u56fe\uff0c\u5728\u6784\u5efa\u8fc7\u7a0b\u4e2d\u5bf9\u6bd4\u4f7f\u7528 RobustPrune \u526a\u679d\u548c\u4e0d\u8fdb\u884c\u526a\u679d\uff0c\u770b\u770b\u4e24\u79cd\u65b9\u5f0f\u6784\u5efa\u7684\u56fe\u6709\u4ec0\u4e48\u5dee\u5f02\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">3.1 \u526a\u679d\u3001\u4e0d\u526a\u679d\u5bf9\u6bd4\u56fe<\/h4>\n\n\n\n<p>\u5177\u4f53\u7684\u5b9e\u73b0\u7a0b\u5e8f\u53c2\u8003\uff1a<a href=\"https:\/\/colab.research.google.com\/drive\/18du3ejwL71LA9sDwQUipu1f7g7M_nMZh#scrollTo=LC32TtcZ7MZs\">Vamana Graph.ipynb<\/a>\u3002\u7a0b\u5e8f\u4e2d\uff0c\u5148\u751f\u6210 30 \u4e2a\u968f\u673a\u7684\u70b9\uff0c\u5982 Figure 1\uff1b\u7136\u540e\u5206\u522b\u4f7f\u7528\u5e26\u6709\u201c\u526a\u679d\u201d\u3001\u6ca1\u6709\u201c\u526a\u679d\u201d\u7684\u7b97\u6cd5\uff0c\u751f\u6210\u56fe\uff0c\u5982 Figure 2\u3001Figure3\u3002<\/p>\n\n\n\n<figure data-wp-context=\"{&quot;imageId&quot;:&quot;69f3dbc0ddbef&quot;}\" data-wp-interactive=\"core\/image\" data-wp-key=\"69f3dbc0ddbef\" class=\"wp-block-image size-large wp-lightbox-container\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"334\" data-wp-class--hide=\"state.isContentHidden\" data-wp-class--show=\"state.isContentVisible\" data-wp-init=\"callbacks.setButtonStyles\" data-wp-on--click=\"actions.showLightbox\" data-wp-on--load=\"callbacks.setButtonStyles\" data-wp-on-window--resize=\"callbacks.setButtonStyles\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-1024x334.png\" alt=\"\" class=\"wp-image-20413\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-1024x334.png 1024w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-300x98.png 300w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-768x250.png 768w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-1536x501.png 1536w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31.png 1638w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><button\n\t\t\tclass=\"lightbox-trigger\"\n\t\t\ttype=\"button\"\n\t\t\taria-haspopup=\"dialog\"\n\t\t\taria-label=\"Enlarge\"\n\t\t\tdata-wp-init=\"callbacks.initTriggerButton\"\n\t\t\tdata-wp-on--click=\"actions.showLightbox\"\n\t\t\tdata-wp-style--right=\"state.imageButtonRight\"\n\t\t\tdata-wp-style--top=\"state.imageButtonTop\"\n\t\t>\n\t\t\t<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12\" height=\"12\" fill=\"none\" viewBox=\"0 0 12 12\">\n\t\t\t\t<path fill=\"#fff\" d=\"M2 0a2 2 0 0 0-2 2v2h1.5V2a.5.5 0 0 1 .5-.5h2V0H2Zm2 10.5H2a.5.5 0 0 1-.5-.5V8H0v2a2 2 0 0 0 2 2h2v-1.5ZM8 12v-1.5h2a.5.5 0 0 0 .5-.5V8H12v2a2 2 0 0 1-2 2H8Zm2-12a2 2 0 0 1 2 2v2h-1.5V2a.5.5 0 0 0-.5-.5H8V0h2Z\" \/>\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n\n\n\n<p>\u8fd9\u91cc\u53ef\u4ee5\u8f83\u4e3a\u76f4\u89c2\u7684\u770b\u5230\uff0c\u56e0\u4e3a\u8fd9\u91cc\u7684\u201c\u51fa\u5ea6\u201d \\(R = 3 \\) \u662f\u56fa\u5b9a\u7684\uff0c\u70b9\u201c3\u201d\u548c\u70b9\u201c22\u201d\uff0c\u5728\u526a\u679d\u548c\u4e0d\u526a\u679d\u7684\u60c5\u51b5\u4e0b\uff0c\u4e24\u4e2a\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u76f8\u5dee\u975e\u5e38\u5927\uff0c\u8fd9\u4e5f\u6700\u7ec8\u5bfc\u81f4\u56fe\u7684\u201c\u76f4\u5f84\u201d\u76f8\u5dee\u5f88\u5927\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. \u62c6\u89e3 RobustPrune \u201c\u526a\u679d\u201d\u8fc7\u7a0b<\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u8fd9\u91cc\u4ee5\u70b9\u201c3\u201d\uff0c\u8bb0\u4e3a \\(P_3 \\)\uff0c \u4e3a\u4f8b\uff0c\u770b\u770bRobustPrune \u201c\u526a\u679d\u201d\u662f\u5982\u4f55\u8fdb\u884c\u7684\u3002<\/p>\n\n\n\n<p>\u9996\u5148\uff0c\u5728\u8be5\u793a\u4f8b\u4e2d\u4f7f\u7528\u201c\u6700\u8fd1\u201d\u521d\u59cb\u5316\u7684\u65b9\u6cd5\uff08\u800c\u4e0d\u662f\u968f\u673a\u521d\u59cb\u5316\uff09\uff0c\u5373\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u70b9\uff0c\u4f8b\u5982\u8fd9\u91cc\u7684 \\(P_3 \\) \uff0c\u521d\u59cb\u5176\u201c\u51fa\u5ea6\u201d\uff08out degree\uff09\u5019\u9009\u90bb\u5c45\u4e3a\u8ddd\u79bb\u5176\u6700\u8fd1\u768410\u4e2a\u70b9\uff0c\u5373 <\/p>\n\n\n\n<p>$$  N_{out}(p) = \\{ P_{21}\\,P_{16}\\,P_5 \\, P_{22}\\, P_0 \\, P_{20}\\, P_{13}\\, P_{12}\\, P_8\\, P_{23} \\} $$<\/p>\n\n\n\n<p>\u8fd9\u91cc\u8003\u8651\u7684\u6b27\u5f0f\u8ddd\u79bb\uff0c\u6545\u6700\u8fd1\u7684\u70b9\u4e3a \\(P_{21} \\)\u3002<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"438\" height=\"442\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-32.png\" alt=\"\" class=\"wp-image-20418\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-32.png 438w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-32-297x300.png 297w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-32-150x150.png 150w\" sizes=\"auto, (max-width: 438px) 100vw, 438px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h4 class=\"wp-block-heading has-text-align-center\">4.1 <strong>\u8003\u8651<code>RobustPrune<\/code>\u526a\u679d<\/strong><\/h4>\n\n\n\n<p>\u5728\u8fdb\u884c<code>RobustPrune<\/code>\u526a\u679d\u65f6\uff0c\u70b9 \\(P_5 \\, P_{16} \\)\u5904\u4e8e\u70b9\\(P_{21} \\)\u7684\u906e\u853d\u533a\u57df\uff0c\u6545\u8fdb\u884c\u526a\u679d\u3002\u6700\u7ec8\u4fdd\u7559\u4e86\u70b9\uff1a\\( N_{out}(p) = \\{ P_{21} \\, P_{22}\\, P_0  \\} \\)<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"438\" height=\"444\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-33.png\" alt=\"\" class=\"wp-image-20423\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-33.png 438w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-33-296x300.png 296w\" sizes=\"auto, (max-width: 438px) 100vw, 438px\" \/><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h4 class=\"wp-block-heading has-text-align-center\">4.2 <strong>\u4e0d\u8003\u8651\u526a\u679d<\/strong><\/h4>\n\n\n\n<p>\u4e0d\u8003\u8651\u201c\u526a\u679d\u201d\uff0c\u53c8\u56e0\u4e3a\u8fd9\u91cc\u6700\u5927\u51fa\u5ea6\\(R=3 \\)\uff0c\u6545\u4fdd\u7559\u8ddd\u79bb \\(P_3 \\) \u6700\u8fd1\u7684\u4e09\u4e2a\u70b9\uff0c\u5219\u4e3a\uff1a\\( N_{out}(p) = \\{ P_{5} \\, P_{16}\\, P_{21}  \\} \\)<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"438\" height=\"442\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-34.png\" alt=\"\" class=\"wp-image-20424\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-34.png 438w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-34-297x300.png 297w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-34-150x150.png 150w\" sizes=\"auto, (max-width: 438px) 100vw, 438px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\">4.3 \u526a\u679d\u533a\u57df<\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u5728\u4e0a\u8ff0\u7684\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7ed8\u5236\u8003\u8651 \\(P_3 \\) \u65f6\uff0c\u53d6 \\(\\alpha = 1.2 \\)\u65f6\uff0c\u526a\u679d\u6240\u6d89\u53ca\u7684\u906e\u853d\u533a\u57df\u5982\u4e0b\uff0c\u5373\u843d\u5230\u8be5\u533a\u57df\u5185\u7684\u70b9\uff08\u9664\u4e86\u6700\u8fd1\u7684 \\(P_{21} \\) \uff09\uff0c\u90fd\u5c06\u88ab\u526a\u679d\uff0c\u5982\u53f3\u56fe\u3002<\/p>\n\n\n\n<p>\u5e76\u4e14\uff0c\u56e0\u4e3a\u6700\u5927\u51fa\u5ea6 \\(R = 3 \\)\uff0c\u5728\u5269\u4f59\u7684\u70b9\u4e2d\uff0c\u518d\u989d\u5916\u9009\u53d6\u4e24\u4e2a\u6700\u8fd1\u7684\u70b9\u4fdd\u7559\uff0c\u8fd9\u91cc\u5373\u4e3a\uff1a\\(\\{ P_0 \\, P_{22} \\} \\)\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5\uff0c\u7ecf\u8fc7<strong><code>RobustPrune<\/code><\/strong>\u526a\u679d\uff0c\u6700\u7ec8 \\(\\{ P_3 \\} \\) \u4fdd\u7559\u7684\u201c\u51fa\u5ea6\u201d\u90bb\u5c45\u70b9\u96c6\u4e3a<\/p>\n\n\n<p>$$<br \/>\nN_{out}(P_3) = \\{ P_{21} \\, P_{0} \\, P_{22} \\}<br \/>\n$$<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"855\" height=\"834\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-37.png\" alt=\"\" class=\"wp-image-20439\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-37.png 855w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-37-300x293.png 300w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-37-768x749.png 768w\" sizes=\"auto, (max-width: 855px) 100vw, 855px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>\u53ef\u4ee5\u770b\u5230\uff0c\u7ecf\u8fc7\u526a\u679d\uff0c\u5728\u6784\u5efa \\(P_3 \\) \u7684\u51fa\u5ea6\u90bb\u5c45\u65f6\uff0c\u5728\u6700\u5927\u51fa\u5ea6\u9650\u5236\u4e3a \\(R=3 \\) \u65f6\uff0c\u5f88\u597d\u7684\u907f\u514d\u4e86\u56fe\u90bb\u5c45\u7684\u201c\u805a\u96c6\u201d\uff0c\u4ece\u800c\u6700\u7ec8\u51cf\u4f4e\u4e86\u6574\u4e2a\u56fe\u7684\u76f4\u5f84\uff0c\u5c31\u6709\u4e86\u4e0a\u8ff0\u7684\u6700\u7ec8\u6784\u5efa\u6548\u679c\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"334\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-1024x334.png\" alt=\"\" class=\"wp-image-20413\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-1024x334.png 1024w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-300x98.png 300w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-768x250.png 768w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31-1536x501.png 1536w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2025\/10\/image-31.png 1638w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">5. \u6700\u540e<\/h3>\n\n\n\n<p>\u8fd9\u91cc\u7ed9\u51fa\u7684\u793a\u4f8b\u662f\u4e8c\u7ef4\u7684\uff0c\u5728\u8fd9\u4e2a\u573a\u666f\u4e0b\uff0c\u9ad8\u7ef4\u6269\u5c55\u5e76\u6ca1\u6709\u592a\u5927\u7684\u4e0d\u540c\u3002\u4ece\u6784\u5efa\u201c\u76f4\u89c9\u201d\u7684\u89d2\u5ea6\uff0c\u8fd9\u91cc\u7ed9\u51fa\u793a\u4f8b\u5df2\u7ecf\u8db3\u591f\u3002\u5bf9\u4e8e\u66f4\u9ad8\u7ef4\u7684\u573a\u666f\uff0c\u5982\u679c\u611f\u5174\u8da3\u7684\uff0c\u53ef\u4ee5\u81ea\u5df1\u8fdb\u884c\u63a8\u5bfc\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u9644\u5f55\uff1a\u6570\u5b66\u63a8\u5bfc<\/h3>\n\n\n\n<p>\u8fd9\u91cc\uff0c\\(P = (0,0) \\, A = (2,0) \\)\uff0c\u518d\u6839\u636e\u526a\u679d\u516c\u5f0f\u6709\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\n\\frac{d(X,P)}{d(X,A)} &#038; \\ge \\alpha  \\\\<br \/>\n\\frac{\\sqrt{x^2+y^2}}{\\sqrt{(x-2)^2+y^2}} &#038; \\ge \\alpha \\\\<br \/>\n\\frac{x^2+y^2}{(x-2)^2+y^2} &#038; \\ge \\alpha^2 \\\\<br \/>\nx^2+y^2 &#038; \\ge \\alpha^2((x-2)^2+y^2) \\\\<br \/>\nx^2+y^2  &#038; \\ge \\alpha^2x^2 -4\\alpha^2 x + 4\\alpha^2 + \\alpha^2 y^2 \\\\<br \/>\n0 &#038; \\ge x^2 &#8211; \\frac{4\\alpha^2 x}{(\\alpha^2-1)} + \\frac{4\\alpha^2}{(\\alpha^2-1)} + y^2 \\\\<br \/>\n0 &#038; \\ge (x- \\frac{2\\alpha^2}{(\\alpha^2-1)})^2 &#8211; (\\frac{2\\alpha^2}{(\\alpha^2-1)})^2 + \\frac{4\\alpha^2}{(\\alpha^2-1)} + y^2 \\\\<br \/>\n(x- \\frac{2\\alpha^2}{(\\alpha^2-1)})^2 + y^2 &#038; \\le (\\frac{2\\alpha^2}{(\\alpha^2-1)})^2 &#8211; \\frac{4\\alpha^2}{(\\alpha^2-1)}\\\\<br \/>\n(x- \\frac{2\\alpha^2}{(\\alpha^2-1)})^2 + y^2 &#038; \\le (\\frac{2\\alpha}{\\alpha^2-1})^2  \\quad \\blacksquare<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>DiskANN 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