{"id":14939,"date":"2024-01-06T20:53:35","date_gmt":"2024-01-06T17:23:35","guid":{"rendered":"https:\/\/rasanegar.com\/blog\/%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5\/"},"modified":"2024-01-06T20:53:35","modified_gmt":"2024-01-06T17:23:35","slug":"%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5","status":"publish","type":"post","link":"https:\/\/rasanegaar.com\/blog\/%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5\/","title":{"rendered":"\u0646\u0645\u0648\u062f\u0627\u0631\u0647\u0627 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 &#8211; \u062a\u0626\u0648\u0631\u06cc \u0648 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc &#8211; \u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a\u0627\u0646 \u067e\u0648\u0634\u0627"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">\u0633\u0631\u0641\u0635\u0644\u0647\u0627\u06cc \u0645\u0637\u0644\u0628<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/rasanegaar.com\/blog\/%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5\/#%d8%ad%d8%af%d8%a7%d9%82%d9%84_%d8%af%d8%b1%d8%ae%d8%aa_%d9%be%d9%88%d8%b4%d8%a7_%da%86%db%8c%d8%b3%d8%aa%d8%9f\" >\u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627 \u0686\u06cc\u0633\u062a\u061f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/rasanegaar.com\/blog\/%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5\/#%d8%aa%d9%88%d8%b6%db%8c%d8%ad_%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%da%a9%d8%b1%d9%88%d8%b3%da%a9%d8%a7%d9%84\" >\u062a\u0648\u0636\u06cc\u062d \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/rasanegaar.com\/blog\/%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5\/#%d8%b1%d9%88%d8%b4_%d9%be%db%8c%d8%a7%d8%af%d9%87_%d8%b3%d8%a7%d8%b2%db%8c_%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%da%a9%d8%b1%d9%88%d8%b3%da%a9%d8%a7%d9%84_%d8%af%d8%b1_%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\" >\u0631\u0648\u0634 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/rasanegaar.com\/blog\/%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5\/#%d9%be%db%8c%da%86%db%8c%d8%af%da%af%db%8c_%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%da%a9%d8%b1%d9%88%d8%b3%da%a9%d8%a7%d9%84_%da%86%db%8c%d8%b3%d8%aa%d8%9f\" >\u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u0686\u06cc\u0633\u062a\u061f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/rasanegaar.com\/blog\/%d9%86%d9%85%d9%88%d8%af%d8%a7%d8%b1%d9%87%d8%a7-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d8%aa%d8%a6%d9%88%d8%b1%db%8c-%d9%88-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-5\/#%d9%86%d8%aa%db%8c%d8%ac%d9%87\" >\u0646\u062a\u06cc\u062c\u0647<\/a><\/li><\/ul><\/nav><\/div>\n<span class=\"span-reading-time rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">\u0632\u0645\u0627\u0646 \u0644\u0627\u0632\u0645 \u0628\u0631\u0627\u06cc \u0645\u0637\u0627\u0644\u0639\u0647: <\/span> <span class=\"rt-time\"> 9<\/span> <span class=\"rt-label rt-postfix\">\u062f\u0642\u06cc\u0642\u0647<\/span><\/span><p> <br \/>\n<\/p>\n<div><noscript><\/noscript><\/p>\n<p>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u06cc\u06a9\u06cc \u0627\u0632 \u0633\u0647 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0639\u0631\u0648\u0641 \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 \u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627 (MST) \u062f\u0631 \u0646\u0645\u0648\u062f\u0627\u0631 \u0627\u0633\u062a.<\/p>\n<p>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u06cc\u06a9 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u062d\u0631\u06cc\u0635\u0627\u0646\u0647 \u0627\u0633\u062a \u06a9\u0647 \u0628\u0627 \u06cc\u0627\u0641\u062a\u0646 \u0628\u0647\u06cc\u0646\u0647 \u0647\u0627\u06cc \u06a9\u0648\u0686\u06a9 \u0648 \u0645\u062d\u0644\u06cc \u0648 \u062a\u0631\u06a9\u06cc\u0628 \u0622\u0646\u0647\u0627\u060c \u0631\u0627\u0647 \u062d\u0644 \u0628\u0647\u06cc\u0646\u0647 \u062c\u0647\u0627\u0646\u06cc \u0631\u0627 \u067e\u06cc\u062f\u0627 \u0645\u06cc \u06a9\u0646\u062f.  \u0639\u0644\u0627\u0648\u0647 \u0628\u0631 \u0627\u06cc\u0646\u060c \u0647\u0646\u0648\u0632 \u0647\u0645 \u0628\u0633\u06cc\u0627\u0631 \u0645\u0641\u06cc\u062f \u0648 \u0628\u0647 \u0637\u0648\u0631 \u06af\u0633\u062a\u0631\u062f\u0647 \u0627\u06cc \u06af\u0633\u062a\u0631\u0634 \u06cc\u0627\u0641\u062a\u0647 \u0627\u0633\u062a.<\/p>\n<blockquote>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u062f\u0631\u0633\u060c \u0631\u0648\u0634 \u0639\u0645\u0644\u06a9\u0631\u062f \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u0631\u0627 \u0628\u0647 \u062a\u0635\u0648\u06cc\u0631 \u0645\u06cc \u06a9\u0634\u06cc\u0645 \u0648 \u062a\u0648\u0636\u06cc\u062d \u0645\u06cc \u062f\u0647\u06cc\u0645 \u0631\u0648\u06cc \u06cc\u06a9 \u0645\u062b\u0627\u0644 \u0639\u0645\u0644\u06cc\u060c \u0648 \u0633\u067e\u0633 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u062f\u0642\u06cc\u0642 \u0622\u0646 \u0631\u0627 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0628\u0647 \u0634\u0645\u0627 \u0627\u0631\u0627\u0626\u0647 \u0645\u06cc \u062f\u0647\u062f.<\/p>\n<\/blockquote>\n<p>\u0645\u0627 \u0642\u0628\u0644\u0627\u064b \u0645\u0648\u0636\u0648\u0639\u0627\u062a \u0632\u06cc\u0627\u062f\u06cc \u0631\u0627 \u062f\u0631 \u0645\u0648\u0631\u062f \u0646\u0645\u0648\u062f\u0627\u0631\u0647\u0627 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u067e\u0648\u0634\u0634 \u062f\u0627\u062f\u0647 \u0627\u06cc\u0645.  \u0627\u06af\u0631 \u0639\u0644\u0627\u0642\u0647 \u0645\u0646\u062f \u0628\u0647 \u06a9\u0633\u0628 \u0627\u0637\u0644\u0627\u0639\u0627\u062a \u0628\u06cc\u0634\u062a\u0631 \u062f\u0631 \u0645\u0648\u0631\u062f \u0645\u0648\u0636\u0648\u0639\u0627\u062a \u06cc\u0627 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u062e\u0627\u0635 \u0647\u0633\u062a\u06cc\u062f\u060c \u062d\u062a\u0645\u0627 \u0628\u0627\u06cc\u062f \u0628\u0631\u062e\u06cc \u0627\u0632 \u0622\u0646\u0647\u0627 \u0631\u0627 \u0645\u0637\u0627\u0644\u0639\u0647 \u06a9\u0646\u06cc\u062f:<\/p>\n<div class=\"alert alert-note\">\n<div class=\"flex\">\n<div class=\"flex-shrink-0 mr-3\"><\/div>\n<p><strong>\u062a\u0648\u062c\u0647 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f:<\/strong> \u062f\u0631 \u0627\u06cc\u0646 \u062f\u0631\u0633\u060c \u0645\u0627 \u0641\u0642\u0637 \u0645\u0641\u0627\u0647\u06cc\u0645\u06cc \u0631\u0627 \u062a\u0648\u0636\u06cc\u062d \u062e\u0648\u0627\u0647\u06cc\u0645 \u062f\u0627\u062f \u06a9\u0647 \u0628\u0631\u0627\u06cc \u062f\u0631\u06a9 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 MST \u062f\u0631 \u06cc\u06a9 \u0646\u0645\u0648\u062f\u0627\u0631 \u0636\u0631\u0648\u0631\u06cc \u0647\u0633\u062a\u0646\u062f.<br \/>\u062a\u0645\u0627\u0645 \u0645\u0641\u0627\u0647\u06cc\u0645 \u062f\u06cc\u06af\u0631 \u062f\u0631 \u062f\u0631\u0633 \u062a\u0648\u0636\u06cc\u062d \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a <em>&#8220;\u0646\u0638\u0631\u06cc\u0647 \u06af\u0631\u0627\u0641 \u0648 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0645\u0631\u062a\u0628\u0637 \u0628\u0627 \u06af\u0631\u0627\u0641&#8221;<\/em>\u060c \u067e\u0633 \u062d\u062a\u0645\u0627\u064b \u0628\u0627\u06cc\u062f \u0627\u0632 \u0642\u0628\u0644 \u0622\u0646 \u0631\u0627 \u0628\u062e\u0648\u0627\u0646\u06cc\u062f.<\/p>\n<\/p><\/div><\/div>\n<h3 id=\"whatisaminimumspanningtree\"><span class=\"ez-toc-section\" id=\"%d8%ad%d8%af%d8%a7%d9%82%d9%84_%d8%af%d8%b1%d8%ae%d8%aa_%d9%be%d9%88%d8%b4%d8%a7_%da%86%db%8c%d8%b3%d8%aa%d8%9f\"><\/span>\u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627 \u0686\u06cc\u0633\u062a\u061f<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0628\u0647 \u0632\u0628\u0627\u0646 \u0633\u0627\u062f\u0647\u060c <strong>\u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627<\/strong> \u062f\u0631\u062e\u062a\u06cc \u0627\u0633\u062a \u06a9\u0647 \u0627\u0632 \u06cc\u06a9 \u06af\u0631\u0627\u0641 \u0648\u0632\u0646 \u062f\u0627\u0631 \u0648 \u0628\u062f\u0648\u0646 \u062c\u0647\u062a \u0633\u0627\u062e\u062a\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646:<\/p>\n<ul>\n<li><strong>\u0647\u0645\u0647 \u06af\u0631\u0647 \u0647\u0627 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f<\/strong> (\u0628\u0647 \u0622\u0646 \u0631\u0626\u0648\u0633 \u0646\u06cc\u0632 \u06af\u0641\u062a\u0647 \u0645\u06cc \u0634\u0648\u062f)<\/li>\n<li>\u062f\u0627\u0631\u062f <strong>\u0628\u062f\u0648\u0646 \u0686\u0631\u062e\u0647<\/strong><\/li>\n<li>\u062f\u0627\u0631\u062f <strong>\u06a9\u0648\u0686\u06a9\u062a\u0631\u06cc\u0646 \u0645\u062c\u0645\u0648\u0639 \u0648\u0632\u0646\u0647\u0627\u06cc \u0644\u0628\u0647 \u0645\u0645\u06a9\u0646 \u0627\u0633\u062a<\/strong><\/li>\n<\/ul>\n<p>\u0628\u06cc\u0627\u06cc\u06cc\u062f \u0627\u06cc\u0646 \u0631\u0627 \u0628\u0631\u0631\u0633\u06cc \u06a9\u0646\u06cc\u0645 <em>\u062f\u0648\u0633\u062a\u0627\u0646\u0647 \u0648 \u063a\u06cc\u0631 \u0631\u0633\u0645\u06cc<\/em> \u062a\u0639\u0631\u06cc\u0641\u06cc \u0628\u0631\u0627\u06cc \u0631\u0648\u0634\u0646 \u06a9\u0631\u062f\u0646 \u062a\u0645\u0627\u0645 \u062a\u0635\u0648\u0631\u0627\u062a \u063a\u0644\u0637 \u0645\u0645\u06a9\u0646 \u062f\u0631 \u0645\u0648\u0631\u062f \u0634\u0631\u0627\u06cc\u0637 \u062a\u0639\u0631\u06cc\u0641 \u0634\u062f\u0647.<\/p>\n<p>\u0627\u0648\u0644\u06cc\u0646 \u0648\u0627\u0642\u0639\u06cc\u062a \u0645\u0647\u0645 \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0634\u0645\u0627 \u0641\u0642\u0637 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f MST \u0631\u0627 \u0628\u0633\u0627\u0632\u06cc\u062f \u0631\u0648\u06cc \u0622 <strong>\u0646\u0645\u0648\u062f\u0627\u0631 \u0648\u0632\u0646 \u062f\u0627\u0631 \u0648 \u0628\u062f\u0648\u0646 \u062c\u0647\u062a<\/strong>.  \u0627\u06cc\u0646 \u0628\u062f\u0627\u0646 \u0645\u0639\u0646\u0627\u0633\u062a \u06a9\u0647 \u0647\u0631 \u06cc\u0627\u0644 \u0646\u0645\u0648\u062f\u0627\u0631 \u062f\u0627\u0631\u0627\u06cc \u0648\u0632\u0646 \u0639\u062f\u062f\u06cc \u0627\u0633\u062a \u0648 \u0634\u0645\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0647\u0631 \u06cc\u0627\u0644 \u0631\u0627 \u0627\u0632 \u0647\u0631 \u062f\u0648 \u062c\u0647\u062a \u0637\u06cc \u06a9\u0646\u06cc\u062f (\u0628\u0631\u062e\u0644\u0627\u0641 \u0646\u0645\u0648\u062f\u0627\u0631\u0647\u0627\u06cc \u062c\u0647\u062a \u062f\u0627\u0631).<\/p>\n<p>MST <strong>\u0647\u0645\u0647 \u06af\u0631\u0647 \u0647\u0627 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f<\/strong> \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631\u060c \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0639\u0646\u06cc \u06a9\u0647 \u0634\u0645\u0627 \u0628\u0627\u06cc\u062f \u0628\u062a\u0648\u0627\u0646\u06cc\u062f \u0628\u0647 \u0647\u0631 \u062a\u06a9\u06cc \u062f\u0633\u062a\u0631\u0633\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f node \u0627\u0632 \u0647\u0631 \u062f\u06cc\u06af\u0631\u06cc node \u062f\u0631 \u0646\u0645\u0648\u062f\u0627\u0631  \u0628\u062f\u06cc\u0647\u06cc \u0627\u0633\u062a \u06a9\u0647 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0628\u0631\u062e\u06cc \u0627\u0632 \u0644\u0628\u0647 \u0647\u0627 \u0631\u0627 \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631 \u062d\u0630\u0641 \u06a9\u0646\u06cc\u062f.  \u062f\u0631 \u0648\u0627\u0642\u0639\u060c \u062d\u0630\u0641 \u06cc\u0627\u0644\u200c\u0647\u0627\u06cc \u063a\u06cc\u0631\u0636\u0631\u0648\u0631\u06cc \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631\u060c \u0645\u062c\u0645\u0648\u0639 \u0648\u0632\u0646\u200c\u0647\u0627\u06cc \u0644\u0628\u0647 \u0631\u0627 \u06a9\u0627\u0647\u0634 \u0645\u06cc\u200c\u062f\u0647\u062f\u060c \u06a9\u0647 \u0634\u0627\u0646\u0633 \u062a\u0628\u062f\u06cc\u0644 \u0634\u062f\u0646 \u062f\u0631\u062e\u062a \u062a\u0648\u0644\u06cc\u062f \u0634\u062f\u0647 \u0628\u0647 MST \u0631\u0627 \u0627\u0641\u0632\u0627\u06cc\u0634 \u0645\u06cc\u200c\u062f\u0647\u062f.<\/p>\n<p>MST \u0628\u0627\u06cc\u062f \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f <strong>\u06a9\u0648\u0686\u06a9\u062a\u0631\u06cc\u0646 \u0645\u062c\u0645\u0648\u0639 \u0648\u0632\u0646 \u0644\u0628\u0647 \u0647\u0627\u06cc \u0645\u0645\u06a9\u0646<\/strong>.  \u06af\u0627\u0647\u06cc \u0627\u0648\u0642\u0627\u062a \u0686\u0646\u062f\u06cc\u0646 \u062f\u0631\u062e\u062a \u0628\u0627 \u0645\u062c\u0645\u0648\u0639 \u0648\u0632\u0646 \u0644\u0628\u0647 \u0647\u0627\u06cc \u06cc\u06a9\u0633\u0627\u0646 \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f.  \u0627\u06af\u0631 \u0627\u06cc\u0646 \u0645\u062c\u0645\u0648\u0639 \u06a9\u0648\u0686\u06a9\u062a\u0631\u06cc\u0646 \u0628\u0627\u0634\u062f\u060c \u0647\u0631 \u06cc\u06a9 \u0627\u0632 \u0622\u0646\u0647\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f MST \u0628\u0627\u0634\u062f.<\/p>\n<p>\u0622\u062e\u0631\u06cc\u0646 \u0648\u0627\u0642\u0639\u06cc\u062a \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u062e\u0648\u062f \u062a\u0648\u0636\u06cc\u062d\u06cc \u0627\u0633\u062a.  \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0627\u062d\u062a\u0645\u0627\u0644\u0627 \u0645\u06cc \u062f\u0627\u0646\u06cc\u062f\u060c \u062f\u0631\u062e\u062a\u0627\u0646 \u0646\u0648\u0639 \u062e\u0627\u0635\u06cc \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631 \u0647\u0633\u062a\u0646\u062f.  \u0622\u0646\u0647\u0627 \u0628\u0647 \u0632\u0628\u0627\u0646 \u0633\u0627\u062f\u0647 \u0639\u0628\u0627\u0631\u062a\u0646\u062f \u0627\u0632 <em>\u0646\u0645\u0648\u062f\u0627\u0631\u0647\u0627\u06cc \u0645\u062a\u0635\u0644 \u0628\u062f\u0648\u0646 \u0686\u0631\u062e\u0647<\/em>\u0648 MST \u0646\u0648\u0639\u06cc \u062f\u0631\u062e\u062a \u0627\u0633\u062a \u06a9\u0647 \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0639\u0646\u06cc \u0627\u0633\u062a <strong>\u0646\u0628\u0627\u06cc\u062f \u0647\u06cc\u0686 \u0686\u0631\u062e\u0647 \u0627\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f<\/strong>.<\/p>\n<div class=\"alert alert-note\">\n<div class=\"flex\">\n<div class=\"flex-shrink-0 mr-3\"><\/div>\n<p><strong>\u062f\u0627\u0631\u0627\u06cc\u06cc \u0645\u0647\u0645:<\/strong> \u0627\u06af\u0631 \u06cc\u06a9 \u0646\u0645\u0648\u062f\u0627\u0631 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f <code>n<\/code> \u06af\u0631\u0647 \u0647\u0627\u060c \u0647\u0631 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627 (\u0627\u0632 \u062c\u0645\u0644\u0647 MST) \u062f\u0627\u0631\u062f <code>n-1<\/code> \u0644\u0628\u0647 \u0647\u0627.<\/p>\n<\/p><\/div><\/div>\n<h3 id=\"kruskalsalgorithmexplained\"><span class=\"ez-toc-section\" id=\"%d8%aa%d9%88%d8%b6%db%8c%d8%ad_%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%da%a9%d8%b1%d9%88%d8%b3%da%a9%d8%a7%d9%84\"><\/span>\u062a\u0648\u0636\u06cc\u062d \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u062f\u0631 \u06a9\u0646\u0627\u0631 Prim \u0648 Bor\u016fvka\u060c \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u06cc\u06a9\u06cc \u0627\u0632 \u0645\u0639\u0631\u0648\u0641\u200c\u062a\u0631\u06cc\u0646 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u200c\u0647\u0627 \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 \u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627 \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631\u0647\u0627\u06cc \u0648\u0632\u0646\u200c\u062f\u0627\u0631 \u0648 \u0628\u062f\u0648\u0646 \u062c\u0647\u062a \u0627\u0633\u062a.  \u062f\u0631 \u0627\u06cc\u0646 \u062f\u0631\u0633\u060c \u0627\u0632 \u0647\u0645\u0627\u0646 \u0646\u0645\u0648\u062f\u0627\u0631\u06cc \u06a9\u0647 \u062f\u0631 \u062f\u0631\u0633 \u062a\u0648\u0636\u06cc\u062d \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Bor\u016fvka \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-1.png\" alt=\"\" title=\"\"><\/p>\n<div class=\"alert alert-note\">\n<div class=\"flex\">\n<div class=\"flex-shrink-0 mr-3\"><\/div>\n<p><strong>\u062a\u0648\u062c\u0647 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f:<\/strong> \u0641\u0642\u0637 \u062f\u0631 \u0627\u06cc\u0646 \u0628\u062e\u0634 \u0647\u0631 \u06a9\u062f\u0627\u0645 \u0631\u0627 \u0639\u0644\u0627\u0645\u062a \u06af\u0630\u0627\u0631\u06cc \u06a9\u0631\u062f\u0647 \u0627\u06cc\u0645 node \u0628\u0627 \u062d\u0631\u0641 \u0628\u0647 \u062c\u0627\u06cc \u0639\u062f\u062f \u0628\u0631\u0627\u06cc \u062e\u0648\u0627\u0646\u0627\u06cc\u06cc \u0628\u0647\u062a\u0631.<\/p>\n<\/p><\/div><\/div>\n<p>\u0627\u06cc\u062f\u0647 \u067e\u0634\u062a \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u0628\u0633\u06cc\u0627\u0631 \u0633\u0627\u062f\u0647 \u0627\u0633\u062a\u060c \u0627\u0633\u0627\u0633\u0627\u064b \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f <strong>\u062f\u0648 \u0642\u062f\u0645<\/strong> \u06a9\u0647 \u062a\u0627 \u0632\u0645\u0627\u0646\u06cc \u06a9\u0647 MST \u0631\u0627 \u0628\u062f\u0633\u062a \u0622\u0648\u0631\u06cc\u062f \u062a\u06a9\u0631\u0627\u0631 \u0645\u06cc \u06a9\u0646\u06cc\u062f:<\/p>\n<ol>\n<li>\n<p><strong>\u062a\u0645\u0627\u0645 \u0644\u0628\u0647 \u0647\u0627 \u0631\u0627 \u0628\u0631 \u0627\u0633\u0627\u0633 \u0648\u0632\u0646 \u0622\u0646\u0647\u0627 \u0628\u0647 \u062a\u0631\u062a\u06cc\u0628 \u0635\u0639\u0648\u062f\u06cc \u0645\u0631\u062a\u0628 \u06a9\u0646\u06cc\u062f<\/strong><\/p>\n<\/li>\n<li>\n<p><strong>\u0644\u0628\u0647 \u0631\u0627 \u0628\u0627 \u06a9\u0645\u062a\u0631\u06cc\u0646 \u0648\u0632\u0646 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0646\u06cc\u062f \u0648 \u0633\u0639\u06cc \u06a9\u0646\u06cc\u062f \u0622\u0646 \u0631\u0627 \u0628\u0647 \u062f\u0631\u062e\u062a \u0627\u0636\u0627\u0641\u0647 \u06a9\u0646\u06cc\u062f<\/strong><\/p>\n<ul>\n<li><strong>\u0627\u06af\u0631 \u06cc\u06a9 \u0686\u0631\u062e\u0647 \u062a\u0634\u06a9\u06cc\u0644 \u0645\u06cc \u062f\u0647\u062f\u060c \u0627\u0632 \u0622\u0646 \u0644\u0628\u0647 \u0628\u06af\u0630\u0631\u06cc\u062f<\/strong><\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>\u0627\u06cc\u0646 \u0645\u0631\u0627\u062d\u0644 \u0631\u0627 \u062a\u06a9\u0631\u0627\u0631 \u06a9\u0646\u06cc\u062f \u062a\u0627 \u0632\u0645\u0627\u0646\u06cc \u06a9\u0647 \u06cc\u06a9 \u062f\u0631\u062e\u062a \u0645\u062a\u0635\u0644 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f \u06a9\u0647 \u0647\u0645\u0647 \u06af\u0631\u0647 \u0647\u0627 \u0631\u0627 \u067e\u0648\u0634\u0634 \u062f\u0647\u062f<\/strong><\/p>\n<\/li>\n<\/ol>\n<p>\u0628\u0631\u0627\u06cc \u0627\u06cc\u0646 \u0646\u0645\u0648\u062f\u0627\u0631 \u062e\u0627\u0635\u060c \u0644\u06cc\u0633\u062a \u0645\u0631\u062a\u0628 \u0634\u062f\u0647 \u0627\u0632 \u062a\u0645\u0627\u0645 \u06cc\u0627\u0644 \u0647\u0627 \u0628\u0647 \u0634\u06a9\u0644 \u0632\u06cc\u0631 \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f:<\/p>\n<table class=\"table table-striped\">\n<tr>\n<th>\u062d\u0627\u0634\u06cc\u0647\u060c \u063a\u06cc\u0631\u0645\u062a\u0645\u0631\u06a9\u0632<\/th>\n<th>\u0648\u0632\u0646<\/th>\n<\/tr>\n<tr>\n<td>EF<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>CF<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>DE<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>FH<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>AB<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>EH<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>GI<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>DG<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>AC<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>BD<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>\u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u062b\u0627\u0644<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>\u0642\u0628\u0644 \u0627\u0632 \u0645\u06cc\u0644\u0627\u062f \u0645\u0633\u06cc\u062d<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>\u0633\u0644\u0627\u0645<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td>EI<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td>BF<\/td>\n<td>20<\/td>\n<\/tr>\n<\/table>\n<p>\u0644\u0628\u0647 \u0647\u0627 <code>EF<\/code> \u0648 <code>CF<\/code> \u0648\u0632\u0646 \u06cc\u06a9\u0633\u0627\u0646\u06cc \u062f\u0627\u0631\u0646\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u06cc\u06a9\u06cc \u0627\u0632 \u0622\u0646\u0647\u0627 \u0631\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0644\u0628\u0647 \u0634\u0631\u0648\u0639 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0646\u06cc\u062f.  \u0645\u0627 \u0641\u0631\u0636 \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u06a9\u0647 \u0634\u0645\u0627 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0631\u062f\u0647 \u0627\u06cc\u062f <code>EF<\/code>.  \u062f\u0631 \u0627\u06cc\u0646 \u0644\u062d\u0638\u0647 \u062f\u0631\u062e\u062a\u06cc \u06a9\u0647 \u0628\u0631\u0627\u06cc \u0633\u0627\u062e\u062a MST \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u062f \u062e\u0627\u0644\u06cc \u0627\u0633\u062a.  \u062d\u0627\u0644\u0627 \u0633\u0639\u06cc \u06a9\u0646\u06cc\u062f \u0627\u0636\u0627\u0641\u0647 \u06a9\u0646\u06cc\u062f <code>EF<\/code> \u0628\u0647 \u062f\u0631\u062e\u062a \u0648 \u0628\u0631\u0631\u0633\u06cc \u06a9\u0646\u06cc\u062f \u06a9\u0647 \u0622\u06cc\u0627 \u06cc\u06a9 \u0686\u0631\u062e\u0647 \u062a\u0634\u06a9\u06cc\u0644 \u0645\u06cc \u062f\u0647\u062f.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-2.png\" alt=\"\" title=\"\"><\/p>\n<p>\u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0645\u06cc \u0628\u06cc\u0646\u06cc\u062f\u060c \u0627\u0636\u0627\u0641\u0647 \u06a9\u0631\u062f\u0646 <code>EF<\/code> \u0647\u06cc\u0686 \u0686\u0631\u062e\u0647 \u0627\u06cc \u062a\u0634\u06a9\u06cc\u0644 \u0646\u0645\u06cc \u062f\u0647\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0622\u0646 \u0631\u0627 \u062f\u0631 \u062f\u0631\u062e\u062a \u0646\u06af\u0647 \u062f\u0627\u0631\u06cc\u062f.  \u0633\u067e\u0633 \u0627\u06cc\u0646 \u0645\u0631\u0627\u062d\u0644 \u0631\u0627 \u0628\u0631\u0627\u06cc \u0647\u0631 \u0644\u0628\u0647 \u062f\u0631 \u0644\u06cc\u0633\u062a \u0645\u0631\u062a\u0628 \u0634\u062f\u0647 \u062a\u06a9\u0631\u0627\u0631 \u0645\u06cc \u06a9\u0646\u06cc\u062f.  \u0627\u0644\u0628\u062a\u0647 \u0627\u0628\u062a\u062f\u0627 \u0644\u0628\u0647 \u0631\u0627 \u0628\u0627 \u062d\u062f\u0627\u0642\u0644 \u0648\u0632\u0646 \u0686\u06a9 \u0645\u06cc \u06a9\u0646\u06cc\u062f \u0648 \u0647\u0645\u06cc\u0646\u0637\u0648\u0631 \u0631\u0648\u06cc.<\/p>\n<p>\u0645\u0648\u0642\u0639\u06cc\u062a \u062c\u0627\u0644\u0628 \u0632\u0645\u0627\u0646\u06cc \u0631\u062e \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0645\u06cc \u062e\u0648\u0627\u0647\u06cc\u062f \u0622\u0646 \u0631\u0627 \u0627\u0636\u0627\u0641\u0647 \u06a9\u0646\u06cc\u062f <code>EH<\/code> \u0628\u0647 \u062f\u0631\u062e\u062a:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-3.png\" alt=\"\" title=\"\"><\/p>\n<p>\u0627\u0636\u0627\u0641\u0647 \u06a9\u0631\u062f\u0646 <code>EH<\/code> \u0628\u0647 \u062f\u0631\u062e\u062a \u06cc\u06a9 \u0686\u0631\u062e\u0647 \u0627\u06cc\u062c\u0627\u062f \u0645\u06cc \u06a9\u0646\u062f &#8211; <code>EFH<\/code>.  \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646\u060c \u0634\u0645\u0627 \u0628\u0627\u06cc\u062f \u062d\u0630\u0641 \u06a9\u0646\u06cc\u062f <code>EH<\/code> \u0627\u0632 \u062f\u0631\u062e\u062a\u060c \u0648 \u0628\u0631\u0631\u0633\u06cc \u06a9\u0646\u06cc\u062f \u06a9\u0647 \u0622\u06cc\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0644\u0628\u0647 \u0628\u0639\u062f\u06cc \u0631\u0627 \u0628\u0627 \u062d\u062f\u0627\u0642\u0644 \u0648\u0632\u0646 \u0627\u0636\u0627\u0641\u0647 \u06a9\u0646\u06cc\u062f &#8211; <code>GI<\/code>.  \u0628\u0639\u062f \u0627\u0632 \u0627\u06cc\u0646\u06a9\u0647 \u0627\u0636\u0627\u0641\u0647 \u06a9\u0631\u062f\u06cc\u062f <code>GI<\/code> \u0628\u0647 \u062f\u0631\u062e\u062a\u060c \u0647\u0645\u0647 \u06af\u0631\u0647 \u0647\u0627 \u0628\u0627\u0632\u062f\u06cc\u062f \u0645\u06cc \u0634\u0648\u0646\u062f:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-4.png\" alt=\"\" title=\"\"><\/p>\n<p>\u0627\u06af\u0631\u0686\u0647 \u0634\u0645\u0627 \u0627\u0632 \u0647\u0645\u0647 \u06af\u0631\u0647 \u0647\u0627 \u0628\u0627\u0632\u062f\u06cc\u062f \u06a9\u0631\u062f\u06cc\u062f\u060c \u062f\u0631\u062e\u062a \u0647\u0646\u0648\u0632 \u0645\u062a\u0635\u0644 \u0646\u0634\u062f\u0647 \u0627\u0633\u062a.  \u0634\u0645\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0645\u062a\u0648\u062c\u0647 \u0633\u0647 \u0645\u062a\u0645\u0627\u06cc\u0632 \u0634\u0648\u06cc\u062f <em>\u062f\u0631\u062e\u062a\u0627\u0646 \u0641\u0631\u0639\u06cc<\/em> &#8211; <code>{A, B}<\/code>\u060c <code>{C, D, E, F, H}<\/code>\u060c \u0648 <code>{G, I}<\/code>.  \u0622\u0646 \u062f\u0631\u062e\u062a\u0627\u0646 \u0641\u0631\u0639\u06cc \u0627\u0644\u0641 \u0631\u0627 \u062a\u0634\u06a9\u06cc\u0644 \u0645\u06cc \u062f\u0647\u0646\u062f <strong>\u062d\u062f\u0627\u0642\u0644 \u062c\u0646\u06af\u0644 \u067e\u0648\u0634\u0627<\/strong> \u06a9\u0647 \u062a\u0645\u0627\u0645 \u06af\u0631\u0647 \u0647\u0627\u06cc \u06af\u0631\u0627\u0641 \u0627\u0648\u0644\u06cc\u0647 \u0631\u0627 \u067e\u0648\u0634\u0634 \u0645\u06cc \u062f\u0647\u062f \u0627\u0645\u0627 MST \u0646\u06cc\u0633\u062a.  MST <em>\u0628\u0627\u06cc\u062f<\/em> \u062a\u0645\u0627\u0645 \u06af\u0631\u0647 \u0647\u0627\u06cc \u0646\u0645\u0648\u062f\u0627\u0631 \u0627\u0648\u0644\u06cc\u0647 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u06a9\u0646\u06cc\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0628\u0627\u06cc\u062f \u0628\u0647 \u0627\u0641\u0632\u0648\u062f\u0646 \u06cc\u0627\u0644 \u0647\u0627 \u0627\u062f\u0627\u0645\u0647 \u062f\u0647\u06cc\u062f \u062a\u0627 \u062c\u0646\u06af\u0644 \u0628\u0647 \u06cc\u06a9 \u062f\u0631\u062e\u062a \u0645\u062a\u0635\u0644 \u062a\u0628\u062f\u06cc\u0644 \u0634\u0648\u062f.  \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Kruskal \u062a\u0636\u0645\u06cc\u0646 \u0645\u06cc \u06a9\u0646\u062f \u06a9\u0647 \u0622\u0646 \u0644\u0628\u0647 \u0647\u0627\u06cc \u0627\u0636\u0627\u0641\u0647 \u0634\u062f\u0647 \u062d\u062f\u0627\u0642\u0644 \u0648\u0632\u0646 \u0645\u0645\u06a9\u0646 \u0631\u0627 \u062f\u0627\u0631\u0646\u062f.<\/p>\n<div class=\"alert alert-note\">\n<div class=\"flex\">\n<div class=\"flex-shrink-0 mr-3\"><\/div>\n<p><strong>\u062a\u0648\u062c\u0647 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f:<\/strong> \u0627\u0633\u0627\u0633\u0627\u064b\u060c \u0645\u06cc\u200c\u062a\u0648\u0627\u0646\u06cc\u062f \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u0631\u0627 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u06cc \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u062f \u06a9\u0647 \u0634\u06a9\u0644 \u0645\u06cc\u200c\u06af\u06cc\u0631\u062f <strong>\u062f\u0631\u062e\u062a\u0627\u0646 \u0641\u0631\u0639\u06cc<\/strong> \u0648 \u0622\u0646\u0647\u0627 \u0631\u0627 \u0628\u0627 \u0644\u0628\u0647 \u0647\u0627\u06cc \u062d\u062f\u0627\u0642\u0644 \u0648\u0632\u0646 \u0645\u0645\u06a9\u0646 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f.  \u0628\u0647 \u0627\u06cc\u0646 \u062a\u0631\u062a\u06cc\u0628\u060c MST \u0631\u0627 \u0627\u06cc\u062c\u0627\u062f \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<\/p><\/div><\/div>\n<p>\u0686\u0646\u062f \u0642\u062f\u0645 \u067e\u0627\u06cc\u06cc\u0646 \u062a\u0631 \u0627\u0632 \u062e\u0637\u060c \u067e\u0633 \u0627\u0632 \u0627\u0636\u0627\u0641\u0647 \u06a9\u0631\u062f\u0646 \u0644\u0628\u0647 \u0647\u0627 <code>DG<\/code> \u0648 <code>AC<\/code> \u0628\u0647 \u062c\u0646\u06af\u0644\u06cc \u06a9\u0647 \u062f\u0631 \u062a\u0635\u0648\u06cc\u0631 \u0642\u0628\u0644\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a\u060c \u0622\u0646 \u062c\u0646\u06af\u0644 \u0628\u0647 \u06cc\u06a9 \u062f\u0631\u062e\u062a \u0645\u062a\u0635\u0644 \u062a\u0628\u062f\u06cc\u0644 \u0645\u06cc \u0634\u0648\u062f.  \u062f\u0631 \u062d\u0642\u06cc\u0642\u062a\u060c <strong>\u0628\u0647 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627\u06cc \u06a9\u0648\u0686\u06a9 \u06af\u0631\u0627\u0641 \u0627\u0648\u0644\u06cc\u0647 \u062a\u0628\u062f\u06cc\u0644 \u0645\u06cc \u0634\u0648\u062f.<\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-mst.png\" alt=\"\" title=\"\"><\/p>\n<p>\u0627\u06cc\u0646 process \u0648\u0627\u0642\u0639\u0627\u064b \u0645\u06cc \u062a\u0648\u0627\u0646 \u0627\u0632 \u0637\u0631\u06cc\u0642 \u06cc\u06a9 \u0627\u0646\u06cc\u0645\u06cc\u0634\u0646 \u0642\u062f\u0631\u062f\u0627\u0646\u06cc \u06a9\u0631\u062f:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-6.gif\" alt=\"\" title=\"\"><\/p>\n<h3 id=\"howtoimplementkruskalsalgorithminpython\"><span class=\"ez-toc-section\" id=\"%d8%b1%d9%88%d8%b4_%d9%be%db%8c%d8%a7%d8%af%d9%87_%d8%b3%d8%a7%d8%b2%db%8c_%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%da%a9%d8%b1%d9%88%d8%b3%da%a9%d8%a7%d9%84_%d8%af%d8%b1_%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\"><\/span>\u0631\u0648\u0634 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u0628\u062e\u0634 \u0627\u0632 \u0647\u0645\u0627\u0646 \u0646\u0645\u0648\u062f\u0627\u0631 \u0642\u0633\u0645\u062a \u0642\u0628\u0644 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645.  \u062a\u0646\u0647\u0627 \u062a\u0641\u0627\u0648\u062a \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0645\u0627 \u0647\u0631 \u06a9\u062f\u0627\u0645 \u0631\u0627 \u0639\u0644\u0627\u0645\u062a \u06af\u0630\u0627\u0631\u06cc \u0645\u06cc \u06a9\u0646\u06cc\u0645 node \u0628\u0627 \u06cc\u06a9 \u0639\u062f\u062f (\u0634\u0631\u0648\u0639 \u0627\u0632 \u0635\u0641\u0631) \u0628\u0647 \u062c\u0627\u06cc \u062d\u0631\u0641 \u0628\u0631\u0627\u06cc \u0627\u062c\u0631\u0627\u06cc \u0633\u0627\u062f\u0647 \u062a\u0631.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-7.png\" alt=\"\" title=\"\"><\/p>\n<h4 id=\"implementingagraphclass\">\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0627\u0644\u0641 <code>Graph<\/code> \u06a9\u0644\u0627\u0633<\/h4>\n<p>\u0627\u0648\u0644\u06cc\u0646 \u0686\u06cc\u0632\u06cc \u06a9\u0647 \u0628\u0627\u06cc\u062f \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u062f \u06cc\u06a9 \u0627\u0633\u062a <strong><code>Graph<\/code>  \u06a9\u0644\u0627\u0633<\/strong>.  \u0627\u06cc\u0646 \u06cc\u06a9 \u0646\u0645\u0648\u062f\u0627\u0631 \u0631\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u0648 \u062a\u0645\u0627\u0645 \u0631\u0648\u0634 \u0647\u0627\u06cc\u06cc \u0631\u0627 \u06a9\u0647 \u0645\u0645\u06a9\u0646 \u0627\u0633\u062a \u062f\u0631 \u0647\u0646\u06af\u0627\u0645 \u06a9\u0627\u0631 \u0628\u0627 \u0646\u0645\u0648\u062f\u0627\u0631\u0647\u0627 \u0628\u0647 \u0622\u0646 \u0646\u06cc\u0627\u0632 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f\u060c \u062a\u0639\u0631\u06cc\u0641 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-class\"><span class=\"hljs-keyword\">class<\/span> <span class=\"hljs-title\">Graph<\/span>:<\/span>\n    <span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">__init__<\/span>(<span class=\"hljs-params\">self, num_of_nodes<\/span>):<\/span>\n        self.m_num_of_nodes = num_of_nodes\n        self.m_graph = ()\n\n    <span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">add_edge<\/span>(<span class=\"hljs-params\">self, node1, node2, weight<\/span>):<\/span>\n        self.m_graph.append((node1, node2, weight))\n<\/code><\/pre>\n<p>\u0627\u06cc\u0646 \u06cc\u06a9 \u0627\u062c\u0631\u0627\u06cc \u0646\u0633\u0628\u062a\u0627\u064b \u0639\u0645\u0648\u0645\u06cc \u0627\u0632 a \u0627\u0633\u062a <code>Graph<\/code> \u06a9\u0644\u0627\u0633 \u0627\u0645\u0627 \u0686\u0646\u062f \u062a\u0631\u0641\u0646\u062f \u0628\u0631\u0627\u06cc \u0633\u0627\u0632\u06af\u0627\u0631\u06cc \u0628\u0627 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Kruskal \u062e\u0648\u0627\u0647\u062f \u062f\u0627\u0634\u062a.  \u0645\u0627 \u0622\u0646 \u062a\u0631\u0641\u0646\u062f\u0647\u0627 \u0631\u0627 \u062f\u0631 \u0632\u06cc\u0631\u0628\u062e\u0634 \u0647\u0627\u06cc \u0628\u0639\u062f\u06cc \u0645\u0639\u0631\u0641\u06cc \u062e\u0648\u0627\u0647\u06cc\u0645 \u06a9\u0631\u062f.<\/p>\n<p>\u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0627\u062d\u062a\u0645\u0627\u0644\u0627\u064b \u0627\u0632 \u0642\u0628\u0644 \u0645\u06cc \u062f\u0627\u0646\u06cc\u062f \u060c <code>__init__<\/code> \u0628\u0647 \u0637\u0648\u0631 \u0645\u0648\u062b\u0631 \u0633\u0627\u0632\u0646\u062f\u0647 \u0647\u0631 \u06a9\u0644\u0627\u0633 \u0645\u0639\u06cc\u0646 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0627\u0633\u062a.  \u062f\u0631 \u0627\u06cc\u0646 \u062d\u0627\u0644\u062a \u060c \u0634\u0645\u0627 \u0628\u0627 \u0641\u0631\u0627\u062e\u0648\u0627\u0646\u06cc \u0646\u0645\u0648\u062f\u0627\u0631 \u0645\u06cc \u0633\u0627\u0632\u06cc\u062f <code>Graph(num_of_nodes)<\/code>.  \u062a\u0646\u0647\u0627 \u062f\u0648 \u0645\u0642\u062f\u0627\u0631 \u06a9\u0647 \u062f\u0631 a \u0630\u062e\u06cc\u0631\u0647 \u0645\u06cc \u0634\u0648\u0646\u062f <code>Graph<\/code> \u06a9\u0644\u0627\u0633 Nuber \u06af\u0631\u0647 \u0647\u0627 \u062f\u0631 \u0646\u0645\u0648\u062f\u0627\u0631 \u0627\u0633\u062a (<code>m_num_of_nodes<\/code>) \u0648 \u0645\u062c\u0645\u0648\u0639\u0647 \u0627\u06cc \u0627\u0632 \u0644\u0628\u0647 \u0647\u0627 (<code>m_graph<\/code>).  \u0622\u0646 \u0622\u0631\u0627\u06cc\u0647 \u0627\u0632 \u06cc\u0627\u0644 \u0647\u0627 \u0633\u0627\u062e\u062a\u0627\u0631 \u062f\u0627\u062f\u0647 \u0627\u06cc \u0627\u0633\u062a \u06a9\u0647 \u0628\u0631\u0627\u06cc \u0630\u062e\u06cc\u0631\u0647 \u0646\u0645\u0648\u062f\u0627\u0631 \u062f\u0631 \u0627\u06cc\u0646 \u0645\u062b\u0627\u0644 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u062f.<\/p>\n<p>\u0647\u0631 \u06cc\u0627\u0644 \u0647\u0631 \u06af\u0631\u0627\u0641 \u0648\u0632\u0646\u06cc \u062f\u0642\u06cc\u0642\u0627\u064b \u062f\u0648 \u06af\u0631\u0647 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f \u0648 \u0648\u0632\u0646 \u062e\u0627\u0635\u06cc \u0628\u0647 \u0622\u0646 \u0627\u062e\u062a\u0635\u0627\u0635 \u062f\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.  \u0631\u0627 <code>add_edge(node1, node2, weight)<\/code> \u0631\u0648\u0634 \u0622\u0646 \u062e\u0627\u0635\u06cc\u062a \u0631\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f.  \u0644\u0628\u0647 \u0631\u0627 \u0628\u0647 a \u0627\u0636\u0627\u0641\u0647 \u0645\u06cc \u06a9\u0646\u062f <code>Graph<\/code> \u0634\u06cc\u0621 \u0628\u0647 \u0634\u06a9\u0644 \u0632\u06cc\u0631 &#8211; <code>(node1, node2, weight)<\/code>.  \u0627\u06cc\u0646 \u0633\u0627\u062f\u0647 \u062a\u0631\u06cc\u0646 \u0648 \u0628\u0633\u06cc\u0627\u0631 \u0645\u0648\u062b\u0631\u062a\u0631\u06cc\u0646 \u0631\u0627\u0647 \u0628\u0631\u0627\u06cc \u0646\u0645\u0627\u06cc\u0634 \u06af\u0631\u0627\u0641 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u0627\u0633\u062a.<\/p>\n<p>\u0639\u0644\u0627\u0648\u0647 \u0628\u0631 \u0633\u0627\u0632\u0646\u062f\u0647 \u0648 <code>addEgde()<\/code>\u060c \u0686\u0646\u062f \u0631\u0648\u0634 \u062f\u06cc\u06af\u0631 \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f \u06a9\u0647 \u0628\u0627\u06cc\u062f \u0622\u0646\u0647\u0627 \u0631\u0627 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u062f.  \u0628\u0631\u0627\u06cc \u062f\u0631\u06a9 \u0686\u06af\u0648\u0646\u06af\u06cc \u0648 \u0686\u0631\u0627\u06cc\u06cc\u060c \u0627\u0628\u062a\u062f\u0627 \u0628\u0631\u062e\u06cc \u0627\u0632 \u0632\u0645\u06cc\u0646\u0647 \u0647\u0627\u06cc \u06a9\u0644\u06cc\u062f\u06cc \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0631\u0627 \u062a\u0648\u0636\u06cc\u062d \u0645\u06cc \u062f\u0647\u06cc\u0645 \u062a\u0627 \u0642\u0628\u0644 \u0627\u0632 \u0645\u0639\u0631\u0641\u06cc \u0622\u0646 \u0631\u0648\u0634 \u0647\u0627\u06cc \u062c\u062f\u06cc\u062f\u060c \u062f\u0627\u0646\u0634 \u067e\u0627\u06cc\u0647 \u0627\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f.  \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646\u060c \u0628\u06cc\u0627\u06cc\u06cc\u062f \u0634\u0631\u0648\u0639 \u06a9\u0646\u06cc\u0645.<\/p>\n<h4 id=\"auxiliaryarraysparentandsubtree_sizes\">\u0622\u0631\u0627\u06cc\u0647 \u0647\u0627\u06cc \u06a9\u0645\u06a9\u06cc &#8211; <code>parent<\/code> \u0648 <code>subtree_sizes<\/code><\/h4>\n<p>\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u062e\u0648\u062f \u0628\u0627\u06cc\u062f \u06a9\u0627\u0631 \u0646\u0633\u0628\u062a\u0627\u064b \u0633\u0627\u062f\u0647 \u0627\u06cc \u0628\u0627\u0634\u062f.  \u0641\u0642\u0637 \u0686\u0646\u062f \u0646\u06a9\u062a\u0647 \u06a9\u0644\u06cc\u062f\u06cc \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f \u06a9\u0647 \u0628\u0627\u06cc\u062f \u062f\u0631 \u0646\u0638\u0631 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f.  \u0627\u0648\u0644 \u0627\u0632 \u0647\u0645\u0647\u060c \u0628\u0627 \u0641\u0631\u0636 \u0627\u06cc\u0646\u06a9\u0647 \u0627\u0632 \u0642\u0628\u0644 \u0641\u0647\u0631\u0633\u062a\u06cc \u0627\u0632 \u0644\u0628\u0647 \u0647\u0627 \u062f\u0627\u0631\u06cc\u062f (\u06a9\u0647 \u0628\u0647 \u0631\u0648\u0634\u06cc \u06a9\u0647 \u0642\u0628\u0644\u0627\u064b \u062a\u0648\u0636\u06cc\u062d \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a)\u060c \u0628\u0627\u06cc\u062f <strong>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc<\/strong> \u0627\u0632 \u06a9\u0648\u0686\u06a9\u062a\u0631\u06cc\u0646 \u0648\u0632\u0646 \u0644\u0628\u0647 \u062a\u0627 \u0628\u0632\u0631\u06af\u062a\u0631\u06cc\u0646.<\/p>\n<p>\u067e\u0633 \u0627\u0632 \u0622\u0646\u060c \u0634\u0645\u0627 \u0628\u0627\u06cc\u062f <strong>\u062f\u0648 \u0622\u0631\u0627\u06cc\u0647 \u06a9\u0645\u06a9\u06cc \u0631\u0627 \u0627\u0648\u0644\u06cc\u0647 \u0648 \u062d\u0641\u0638 \u06a9\u0646\u06cc\u062f<\/strong> &#8211; <code>parent<\/code> \u0648 <code>subtree_sizes<\/code>.  \u0631\u0648\u0634 \u0633\u0627\u062e\u062a \u0622\u0646\u0647\u0627 \u0627\u0632 \u0627\u0647\u0645\u06cc\u062a \u0628\u0627\u0644\u0627\u06cc\u06cc \u0628\u0631\u062e\u0648\u0631\u062f\u0627\u0631 \u0627\u0633\u062a\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0628\u0627 \u062f\u0642\u062a \u062f\u0646\u0628\u0627\u0644 \u06a9\u0646\u06cc\u062f.  \u0647\u0631 \u062f\u0648\u06cc \u0622\u0646\u0647\u0627 \u062f\u0627\u0631\u0646\u062f <em>\u0627\u0646\u062f\u0627\u0632\u0647 \u0627\u06cc \u06a9\u0647 \u0628\u0627 \u062a\u0639\u062f\u0627\u062f \u06af\u0631\u0647 \u0647\u0627 \u062f\u0631 \u0646\u0645\u0648\u062f\u0627\u0631 \u0627\u0648\u0644\u06cc\u0647 \u0645\u0637\u0627\u0628\u0642\u062a \u062f\u0627\u0631\u062f<\/em> (\u06cc\u0639\u0646\u06cc \u0627\u06af\u0631 \u0646\u0645\u0648\u062f\u0627\u0631 \u0627\u0648\u0644\u06cc\u0647 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f <code>n<\/code> \u06af\u0631\u0647 \u0647\u0627 \u060c \u0622\u0646 \u062f\u0648 \u0622\u0631\u0627\u06cc\u0647 <code>n<\/code> \u0639\u0646\u0627\u0635\u0631).<\/p>\n<p>\u0628\u0647 \u0627\u06cc\u0646 \u062a\u0631\u062a\u06cc\u0628\u060c \u0647\u0631 \u0634\u0627\u062e\u0635 \u0627\u0632 \u0622\u0646 \u0622\u0631\u0627\u06cc\u0647 \u0647\u0627 \u0628\u0647 \u0637\u0648\u0631 \u0645\u0633\u062a\u0642\u06cc\u0645 \u062f\u0642\u06cc\u0642\u0627\u064b \u0628\u0627 \u06cc\u06a9 \u0645\u0648\u0631\u062f \u0645\u0637\u0627\u0628\u0642\u062a \u062f\u0627\u0631\u062f node \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631  \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u062b\u0627\u0644\u060c \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0628\u0647 \u062a\u06a9 \u062a\u06a9 \u0627\u0637\u0644\u0627\u0639\u0627\u062a\u06cc \u06a9\u0647 \u0645\u0645\u06a9\u0646 \u0627\u0633\u062a \u062f\u0631 \u0645\u0648\u0631\u062f \u0622\u0646 \u0646\u06cc\u0627\u0632 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f \u062f\u0633\u062a\u0631\u0633\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f node <code>3<\/code>  \u0628\u0627 \u062a\u0645\u0627\u0633 <code>parents(3)<\/code> \u0648 <code>subtree_sizes(3)<\/code>.<\/p>\n<p>\u062d\u0627\u0644\u0627 \u06a9\u0647 \u0641\u0647\u0645\u06cc\u062f\u06cc <em>\u0686\u06af\u0648\u0646\u0647<\/em> \u0628\u0631\u0627\u06cc \u0633\u0627\u062e\u062a \u0627\u06cc\u0646 \u062f\u0648 \u0622\u0631\u0627\u06cc\u0647\u060c \u0627\u062c\u0627\u0632\u0647 \u062f\u0647\u06cc\u062f \u062a\u0648\u0636\u06cc\u062d \u062f\u0647\u06cc\u0645 \u06a9\u0647 \u0686\u0631\u0627 \u0627\u0635\u0644\u0627\u064b \u0628\u0647 \u0622\u0646\u0647\u0627 \u0646\u06cc\u0627\u0632 \u062f\u0627\u0631\u06cc\u0645.  \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0642\u0628\u0644\u0627\u064b \u0628\u06cc\u0627\u0646 \u06a9\u0631\u062f\u06cc\u0645\u060c \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Kruskal \u0628\u0647 \u0637\u0648\u0631 \u0645\u0648\u062b\u0631 \u0627\u06cc\u062c\u0627\u062f \u0645\u06cc \u06a9\u0646\u062f <em>\u0645\u062c\u0645\u0648\u0639\u0647 \u0627\u06cc \u0627\u0632 \u0632\u06cc\u0631 \u062f\u0631\u062e\u062a\u0627\u0646<\/em> (\u06cc\u0627 \u06cc\u06a9 \u062c\u0646\u06af\u0644) \u0648 \u0622\u0646\u0647\u0627 \u0631\u0627 \u0628\u0627 \u0644\u0628\u0647 \u0647\u0627\u06cc \u062d\u062f\u0627\u0642\u0644 \u0648\u0632\u0646 \u0645\u0645\u06a9\u0646 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f.  \u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u0634\u0645\u0627 \u0647\u0631 \u0641\u0631\u062f\u06cc \u0631\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0645\u06cc \u06af\u06cc\u0631\u06cc\u062f node \u062f\u0631\u062e\u062a\u06cc \u0645\u062c\u0632\u0627 \u0628\u0627\u0634\u062f \u0648 \u0634\u0631\u0648\u0639 \u0628\u0647 \u0627\u062a\u0635\u0627\u0644 \u0622\u0646\u0647\u0627 \u06a9\u0646\u062f.  \u0634\u0645\u0627 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u0647\u0627 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u06cc\u062f \u062a\u0627 \u0632\u0645\u0627\u0646\u06cc \u06a9\u0647 \u06cc\u06a9 \u062f\u0631\u062e\u062a \u0628\u062f\u0633\u062a \u0622\u0648\u0631\u06cc\u062f \u06a9\u0647 \u062a\u0645\u0627\u0645 \u06af\u0631\u0647 \u0647\u0627\u06cc \u06af\u0631\u0627\u0641 \u0627\u0648\u0644\u06cc\u0647 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u0622\u0646\u062c\u0627\u0633\u062a \u06a9\u0647 <code>parent<\/code> \u0622\u0631\u0627\u06cc\u0647 \u062f\u0631 \u062c\u0627\u06cc \u062e\u0648\u062f \u0642\u0631\u0627\u0631 \u0645\u06cc \u06af\u06cc\u0631\u062f.  \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0645\u06cc \u062f\u0627\u0646\u06cc\u062f\u060c \u0634\u0627\u062e\u0635 \u0647\u0627\u06cc \u0622\u0646 \u0622\u0631\u0627\u06cc\u0647 \u0646\u0634\u0627\u0646 \u062f\u0647\u0646\u062f\u0647 \u06af\u0631\u0647 \u0647\u0627 \u0647\u0633\u062a\u0646\u062f (\u0645\u062b\u0644\u0627\u064b \u0634\u0627\u062e\u0635 <code>3<\/code> \u0627\u0634\u0627\u0631\u0647 \u062f\u0627\u0631\u062f \u0628\u0647 node <code>3<\/code>  \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631).  \u062f\u0631 <code>parent<\/code> \u0622\u0631\u0627\u06cc\u0647\u060c \u0634\u0645\u0627 \u0627\u0637\u0644\u0627\u0639\u0627\u062a \u0645\u0631\u0628\u0648\u0637 \u0628\u0647 \u0622\u0646 \u0631\u0627 \u0646\u06af\u0647 \u0645\u06cc \u062f\u0627\u0631\u06cc\u062f node \u0645\u062a\u0639\u0644\u0642 \u0628\u0647 \u06a9\u062f\u0627\u0645 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u0627\u0633\u062a  \u0627\u0632 \u0622\u0646\u062c\u0627\u06cc\u06cc \u06a9\u0647 \u0634\u0645\u0627 \u0647\u0631 \u06cc\u06a9 \u0631\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0645\u06cc \u06af\u06cc\u0631\u06cc\u062f node \u0628\u0631\u0627\u06cc \u0627\u06cc\u0646\u06a9\u0647 \u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u06cc\u06a9 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u062c\u062f\u0627\u06af\u0627\u0646\u0647 \u0628\u0627\u0634\u062f <code>parent<\/code> \u0622\u0631\u0627\u06cc\u0647 \u0628\u0647 \u0634\u06a9\u0644 \u0632\u06cc\u0631 \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-8.png\" alt=\"\" title=\"\"><\/p>\n<p>\u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0645\u06cc \u0628\u06cc\u0646\u06cc\u062f\u060c \u0647\u0631 \u06a9\u062f\u0627\u0645 node \u0648\u0627\u0644\u062f \u0628\u0631\u0627\u06cc \u062e\u0648\u062f\u0634 \u0627\u0633\u062a \u0648 \u062f\u0631\u062e\u062a \u062d\u0627\u0635\u0644 \u062e\u0627\u0644\u06cc \u0627\u0633\u062a.  \u0647\u0646\u06af\u0627\u0645\u06cc \u06a9\u0647 \u0644\u0628\u0647 \u0631\u0627 \u0628\u0627 \u062d\u062f\u0627\u0642\u0644 \u0648\u0632\u0646 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0631\u062f\u06cc\u062f \u0648 \u0622\u0646 \u0631\u0627 \u0628\u0647 \u062f\u0631\u062e\u062a \u062d\u0627\u0635\u0644 \u0627\u0636\u0627\u0641\u0647 \u06a9\u0631\u062f\u06cc\u062f\u060c \u062f\u0631 \u0648\u0627\u0642\u0639 \u0634\u0645\u0627 <em>\u062f\u0648 \u062a\u0627 \u0627\u0632 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u0647\u0627\u06cc \u0634\u0631\u0648\u0639 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0648\u0635\u0644 \u06a9\u0646\u06cc\u062f<\/em>.<\/p>\n<p>\u062f\u0631 \u0646\u0645\u0648\u062f\u0627\u0631 \u0645\u062b\u0627\u0644\u060c \u0644\u0628\u0647 \u0627\u06cc \u0628\u0627 \u062d\u062f\u0627\u0642\u0644 \u0648\u0632\u0646 \u0644\u0628\u0647 \u0627\u06cc \u0627\u0633\u062a \u06a9\u0647 \u06af\u0631\u0647 \u0647\u0627 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f <code>5<\/code> \u0648 <code>2<\/code>\u060c \u06a9\u0647 \u0647\u0631 \u062f\u0648 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u06af\u0631\u0627\u0641 \u0627\u0648\u0644\u06cc\u0647 \u0647\u0633\u062a\u0646\u062f.  \u0627\u062a\u0635\u0627\u0644 \u0622\u0646\u0647\u0627 \u06cc\u06a9 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u062c\u062f\u06cc\u062f \u0648 \u0628\u0632\u0631\u06af\u062a\u0631 \u0631\u0627 \u062a\u0634\u06a9\u06cc\u0644 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0634\u0627\u0645\u0644 \u06af\u0631\u0647 \u0647\u0627 \u0627\u0633\u062a <code>2<\/code> \u0648 <code>5<\/code>.  \u0631\u0627 <code>parent<\/code> \u0622\u0631\u0627\u06cc\u0647 \u0622\u0646 \u0631\u0627 \u0628\u0627 \u0627\u062e\u062a\u0635\u0627\u0635 \u062f\u0627\u062f\u0646 \u0645\u0646\u0639\u06a9\u0633 \u0645\u06cc \u06a9\u0646\u062f node <code>2<\/code>  \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u067e\u062f\u0631 \u0648 \u0645\u0627\u062f\u0631 node \u0627\u0632 node <code>5<\/code>.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-9.png\" alt=\"\" title=\"\"><\/p>\n<p>\u0627\u06cc\u0646 \u06a9\u0627\u0631 \u062a\u0627 \u0632\u0645\u0627\u0646\u06cc \u0627\u062f\u0627\u0645\u0647 \u0645\u06cc \u06cc\u0627\u0628\u062f \u06a9\u0647 \u0628\u0647 \u062c\u0627\u06cc \u0686\u0646\u062f\u06cc\u0646 \u062f\u0631\u062e\u062a \u0641\u0631\u0639\u06cc \u06a9\u0648\u0686\u06a9\u062a\u0631\u060c \u06cc\u06a9 \u062f\u0631\u062e\u062a \u0648\u0627\u062d\u062f \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f.<\/p>\n<p>\u0628\u0631\u0627\u06cc \u06a9\u0645\u06a9 \u0628\u0647 \u062d\u0641\u0638 <code>parent<\/code> \u0648 <code>subtree_sizes<\/code> \u0622\u0631\u0627\u06cc\u0647 \u0647\u0627 \u0628\u0647 \u0631\u0648\u0634\u06cc \u062a\u0648\u0635\u06cc\u0641 \u0634\u062f\u0647\u060c \u0628\u0627\u06cc\u062f \u0631\u0648\u0634 \u0647\u0627\u06cc \u0632\u06cc\u0631 \u0631\u0627 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u062f<em>n <code>Graph<\/code> \u06a9\u0644\u0627\u0633<\/em>:<\/p>\n<pre><code class=\"hljs\">\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">find_subtree<\/span>(<span class=\"hljs-params\">self, parent, i<\/span>):<\/span>\n    <span class=\"hljs-keyword\">if<\/span> parent(i) == i:\n        <span class=\"hljs-keyword\">return<\/span> i\n    <span class=\"hljs-keyword\">return<\/span> self.find_subtree(parent, parent(i))\n\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">connect_subtrees<\/span>(<span class=\"hljs-params\">self, parent, subtree_sizes, x, y<\/span>):<\/span>\n    xroot = self.find_subtree(parent, x)\n    yroot = self.find_subtree(parent, y)\n    <span class=\"hljs-keyword\">if<\/span> subtree_sizes(xroot) &lt; subtree_sizes(yroot):\n        parent(xroot) = yroot\n    <span class=\"hljs-keyword\">elif<\/span> subtree_sizes(xroot) &gt; subtree_sizes(yroot):\n        parent(yroot) = xroot\n    <span class=\"hljs-keyword\">else<\/span>:\n        parent(yroot) = xroot\n        subtree_sizes(xroot) += <span class=\"hljs-number\">1<\/span>\n<\/code><\/pre>\n<p>\u0631\u0627 <code>find_subtree()<\/code> \u0631\u0648\u0634 \u0628\u0647 \u0635\u0648\u0631\u062a \u0628\u0627\u0632\u06af\u0634\u062a\u06cc \u062f\u0631 \u0631\u0627 \u062c\u0633\u062a\u062c\u0648 \u0645\u06cc \u06a9\u0646\u062f <code>parent<\/code>\u0622\u0631\u0627\u06cc\u0647 \u0648 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a\u06cc \u0631\u0627 \u067e\u06cc\u062f\u0627 \u0645\u06cc \u06a9\u0646\u062f \u06a9\u0647 \u062d\u0627\u0648\u06cc node <code>i<\/code>.<\/p>\n<p>\u0631\u0627 <code>connect_subtrees()<\/code> \u0631\u0648\u0634 \u062f\u0648 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f.  \u06cc\u06a9 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u0634\u0627\u0645\u0644 node <code>x<\/code>  \u0648 \u062f\u06cc\u06af\u0631\u06cc \u0634\u0627\u0645\u0644 node <code>y<\/code>.  \u0627\u0628\u062a\u062f\u0627 \u0622\u0646 \u062f\u0648 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u0631\u0627 \u067e\u06cc\u062f\u0627 \u0645\u06cc \u06a9\u0646\u062f\u060c \u0633\u067e\u0633 \u0627\u0646\u062f\u0627\u0632\u0647 \u0622\u0646\u0647\u0627 \u0631\u0627 \u0645\u0642\u0627\u06cc\u0633\u0647 \u0645\u06cc \u06a9\u0646\u062f \u0648 \u0632\u06cc\u0631\u062f\u0631\u062e\u062a \u06a9\u0648\u0686\u06a9\u062a\u0631 \u0631\u0627 \u0628\u0647 \u062f\u0631\u062e\u062a \u0628\u0632\u0631\u06af\u062a\u0631 \u0645\u062a\u0635\u0644 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u06a9\u0647 \u0627\u062c\u0631\u0627\u06cc \u0622\u0646 \u0631\u0627 \u06a9\u062f\u0646\u0648\u06cc\u0633\u06cc \u0648 \u062a\u0648\u0636\u06cc\u062d \u062f\u0627\u062f\u06cc\u0645 <code>Graph<\/code> \u06a9\u0644\u0627\u0633 \u062f\u0631 \u06a9\u0646\u0627\u0631 \u062a\u0645\u0627\u0645 \u0645\u062a\u062f\u0647\u0627\u06cc \u0627\u0636\u0627\u0641\u06cc\u060c \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u0645 \u0646\u06af\u0627\u0647\u06cc \u0628\u0647 \u0622\u0646 \u0628\u06cc\u0646\u062f\u0627\u0632\u06cc\u0645 <strong>\u0631\u0648\u0634 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644<\/strong> \u062e\u0648\u062f<\/p>\n<h4 id=\"kruskalsmstalgorithm\">\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 MST \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644<\/h4>\n<p>\u0628\u0627 \u062a\u0645\u0627\u0645 \u0631\u0648\u0634 \u0647\u0627\u06cc \u062f\u06cc\u06af\u0631 \u0627\u0632 a <code>Graph<\/code> \u06a9\u0644\u0627\u0633 \u062a\u0648\u0636\u06cc\u062d \u062f\u0627\u062f\u060c \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Kruskal \u0628\u0627\u06cc\u062f \u0646\u0633\u0628\u062a\u0627\u064b \u0622\u0633\u0627\u0646 \u0628\u0627\u0634\u062f.  \u0645\u0627 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0631\u062f\u0647 \u0627\u06cc\u0645 \u06a9\u0647 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0631\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0631\u0648\u0634 \u062f\u0631 a \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u0645 <code>Graph<\/code> \u06a9\u0644\u0627\u0633:<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">kruskals_mst<\/span>(<span class=\"hljs-params\">self<\/span>):<\/span>\n    \n    result = ()\n    \n    \n    i = <span class=\"hljs-number\">0<\/span>\n    \n    e = <span class=\"hljs-number\">0<\/span>\n\n    \n    self.m_graph = <span class=\"hljs-built_in\">sorted<\/span>(self.m_graph, key=<span class=\"hljs-keyword\">lambda<\/span> item: item(<span class=\"hljs-number\">2<\/span>))\n    \n    \n    parent = ()\n    subtree_sizes = ()\n\n    \n    <span class=\"hljs-keyword\">for<\/span> node <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">range<\/span>(self.m_num_of_nodes):\n        parent.append(node)\n        subtree_sizes.append(<span class=\"hljs-number\">0<\/span>)\n\n    \n    \n    \n    <span class=\"hljs-keyword\">while<\/span> e &lt; (self.m_num_of_nodes - <span class=\"hljs-number\">1<\/span>):\n        \n        node1, node2, weight = self.m_graph(i)\n        i = i + <span class=\"hljs-number\">1<\/span>\n\n        x = self.find_subtree(parent, node1)\n        y = self.find_subtree(parent, node2)\n\n        <span class=\"hljs-keyword\">if<\/span> x != y:\n            e = e + <span class=\"hljs-number\">1<\/span>\n            result.append((node1, node2, weight))\n            self.connect_subtrees(parent, subtree_sizes, x, y)\n    \n    \n    <span class=\"hljs-keyword\">for<\/span> node1, node2, weight <span class=\"hljs-keyword\">in<\/span> result:\n        <span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-string\">\"%d - %d: %d\"<\/span> % (node1, node2, weight))\n<\/code><\/pre>\n<p>\u062e\u0648\u062f \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u062a\u0645\u0627\u0645 \u0631\u0648\u0634 \u0647\u0627\u06cc \u062a\u0648\u0635\u06cc\u0641 \u0634\u062f\u0647 \u0642\u0628\u0644\u06cc \u0631\u0627 \u062a\u0631\u06a9\u06cc\u0628 \u0645\u06cc \u06a9\u0646\u062f.  \u0627\u0648\u0644 \u0627\u0632 \u0647\u0645\u0647\u060c \u0644\u06cc\u0633\u062a\u06cc \u0627\u0632 \u0644\u0628\u0647 \u0647\u0627 \u0631\u0627 \u0628\u0631 \u0627\u0633\u0627\u0633 \u0648\u0632\u0646 \u0622\u0646\u0647\u0627 \u0645\u0631\u062a\u0628 \u0645\u06cc \u06a9\u0646\u062f \u0648 \u0645\u0642\u062f\u0627\u0631\u062f\u0647\u06cc \u0627\u0648\u0644\u06cc\u0647 \u0645\u06cc \u06a9\u0646\u062f <code>parent<\/code> \u0648 <code>subtree_sizes<\/code> \u0622\u0631\u0627\u06cc\u0647 \u0647\u0627  \u0633\u067e\u0633 \u0631\u0648\u06cc \u0644\u06cc\u0633\u062a \u0645\u0631\u062a\u0628 \u0634\u062f\u0647 \u0644\u0628\u0647 \u0647\u0627 \u062a\u06a9\u0631\u0627\u0631 \u0645\u06cc \u0634\u0648\u062f\u060c \u0622\u0646\u0647\u0627 \u0631\u0627 \u06cc\u06a9\u06cc \u06cc\u06a9\u06cc \u0627\u0646\u062a\u062e\u0627\u0628 \u0645\u06cc \u06a9\u0646\u062f \u0648 \u062f\u0631 \u0635\u0648\u0631\u062a \u0627\u0645\u06a9\u0627\u0646 \u0628\u0647 \u062f\u0631\u062e\u062a \u062d\u0627\u0635\u0644 \u0627\u0636\u0627\u0641\u0647 \u0645\u06cc \u06a9\u0646\u062f.  \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0632\u0645\u0627\u0646\u06cc \u0645\u062a\u0648\u0642\u0641 \u0645\u06cc \u0634\u0648\u062f \u06a9\u0647 \u062a\u0639\u062f\u0627\u062f \u0644\u0628\u0647 \u0647\u0627\u06cc \u062f\u0631\u062e\u062a \u062d\u0627\u0635\u0644 \u0628\u0631\u0627\u0628\u0631 \u0628\u0627\u0634\u062f <code>(num_of_nodes - 1)<\/code>.  \u062f\u0631\u062e\u062a \u0628\u0647 \u062f\u0633\u062a \u0622\u0645\u062f\u0647 \u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627 \u0627\u0633\u062a \u06a9\u0647 \u0645\u0627 \u062f\u0631 \u062a\u0644\u0627\u0634 \u0628\u0631\u0627\u06cc \u0633\u0627\u062e\u062a\u0646 \u0622\u0646 \u0628\u0648\u062f\u0647 \u0627\u06cc\u0645.<\/p>\n<p>\u0628\u06cc\u0627\u06cc\u06cc\u062f \u0627\u06cc\u0646 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0631\u0627 \u0622\u0632\u0645\u0627\u06cc\u0634 \u06a9\u0646\u06cc\u0645 \u0631\u0648\u06cc \u0646\u0645\u0648\u062f\u0627\u0631 \u0645\u062b\u0627\u0644 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0634\u062f\u0647 \u0642\u0628\u0644\u06cc<\/p>\n<h4 id=\"testingkruskalsalgorithmontheexamplegraph\">\u0622\u0632\u0645\u0627\u06cc\u0634 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u0631\u0648\u06cc \u0646\u0645\u0648\u062f\u0627\u0631 \u0646\u0645\u0648\u0646\u0647<\/h4>\n<p>\u0627\u0648\u0644 \u0627\u0632 \u0647\u0645\u0647\u060c \u0634\u0645\u0627 \u0628\u0627\u06cc\u062f \u0627\u06cc\u062c\u0627\u062f \u06a9\u0646\u06cc\u062f <code>Graph<\/code> \u0634\u06cc\u0626\u06cc \u06a9\u0647 \u0646\u0645\u0648\u062f\u0627\u0631 \u0645\u062b\u0627\u0644 \u0634\u0645\u0627 \u0631\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f:<\/p>\n<pre><code class=\"hljs\">\nexample_graph = Graph(<span class=\"hljs-number\">9<\/span>)\n<\/code><\/pre>\n<p>\u0633\u067e\u0633 \u062a\u0645\u0627\u0645 \u06af\u0631\u0647 \u0647\u0627 \u0631\u0627 \u0627\u0632 \u06af\u0631\u0647 \u0645\u062b\u0627\u0644 \u0628\u0647 \u06af\u0631\u0647 \u0627\u0636\u0627\u0641\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u062f <code>example_graph<\/code> \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0631\u062f\u0646 <code>add_edge()<\/code> \u0631\u0648\u0634:<\/p>\n<pre><code class=\"hljs\">example_graph.add_edge(<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">4<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">7<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">11<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">9<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">20<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">1<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">6<\/span>, <span class=\"hljs-number\">6<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">3<\/span>, <span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">2<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">6<\/span>, <span class=\"hljs-number\">10<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">8<\/span>, <span class=\"hljs-number\">15<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">7<\/span>, <span class=\"hljs-number\">5<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">1<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">7<\/span>, <span class=\"hljs-number\">3<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">6<\/span>, <span class=\"hljs-number\">8<\/span>, <span class=\"hljs-number\">5<\/span>)\nexample_graph.add_edge(<span class=\"hljs-number\">7<\/span>, <span class=\"hljs-number\">8<\/span>, <span class=\"hljs-number\">12<\/span>)\n<\/code><\/pre>\n<p>\u0648 \u062f\u0631 \u0646\u0647\u0627\u06cc\u062a \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0631\u0627 \u0627\u062c\u0631\u0627 \u06a9\u0646\u06cc\u062f:<\/p>\n<pre><code class=\"hljs\">example_graph.kruskals_mst()\n<\/code><\/pre>\n<p>\u06a9\u0647 \u062e\u0631\u0648\u062c\u06cc \u0632\u06cc\u0631 \u0631\u0627 \u0628\u0647 \u0647\u0645\u0631\u0627\u0647 \u062e\u0648\u0627\u0647\u062f \u062f\u0627\u0634\u062a:<\/p>\n<pre><code class=\"hljs\">2 - 5: 1\n4 - 5: 1\n3 - 4: 2\n5 - 7: 3\n0 - 1: 4\n6 - 8: 5\n3 - 6: 6\n0 - 2: 7\n<\/code><\/pre>\n<p>\u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0645\u06cc \u0628\u06cc\u0646\u06cc\u062f\u060c \u0627\u06cc\u0646 \u062e\u0631\u0648\u062c\u06cc MST \u0631\u0627 \u062a\u0648\u0635\u06cc\u0641 \u0645\u06cc \u06a9\u0646\u062f \u06a9\u0647 \u0647\u0645\u0627\u0646 \u0686\u06cc\u0632\u06cc \u0627\u0633\u062a \u06a9\u0647 \u062f\u0631 \u0628\u062e\u0634 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a. <em>\u062a\u0648\u0636\u06cc\u062d \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644<\/em>.  \u0647\u0631 \u0631\u062f\u06cc\u0641 \u0627\u0632 \u062e\u0631\u0648\u062c\u06cc \u06cc\u06a9 \u06cc\u0627\u0644 \u0631\u0627 \u0628\u0647 \u0634\u06a9\u0644 \u0632\u06cc\u0631 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f: <code>node1 - node2: weight<\/code>.  \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f MST \u0633\u0627\u062e\u062a\u0647 \u0634\u062f\u0647 \u0631\u0627 \u062f\u0631 \u062a\u0635\u0648\u06cc\u0631 \u0632\u06cc\u0631 \u0645\u0634\u0627\u0647\u062f\u0647 \u06a9\u0646\u06cc\u062f.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/graphs-in-python-minimum-spanning-trees-kruskals-algorithm-10.png\" alt=\"\" title=\"\"><\/p>\n<h3 id=\"whatisthecomplexityofkruskalsalgorithm\"><span class=\"ez-toc-section\" id=\"%d9%be%db%8c%da%86%db%8c%d8%af%da%af%db%8c_%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%da%a9%d8%b1%d9%88%d8%b3%da%a9%d8%a7%d9%84_%da%86%db%8c%d8%b3%d8%aa%d8%9f\"><\/span>\u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u0686\u06cc\u0633\u062a\u061f<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0628\u06cc\u0627\u06cc\u06cc\u062f \u0641\u0631\u0636 \u06a9\u0646\u06cc\u0645 \u06a9\u0647 \u0634\u0645\u0627 \u06cc\u06a9 \u0646\u0645\u0648\u062f\u0627\u0631 \u0628\u0627 <code>E<\/code> \u0644\u0628\u0647 \u0647\u0627 \u0648 <code>N<\/code> \u06af\u0631\u0647 \u0647\u0627  MST \u0631\u0627 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Kruskal \u062f\u0631 \u0632\u0645\u0627\u0646 \u067e\u06cc\u062f\u0627 \u062e\u0648\u0627\u0647\u06cc\u062f \u06a9\u0631\u062f <strong><code>O(E log E)<\/code><\/strong>\u060c \u06a9\u0647 \u0645\u0639\u0627\u062f\u0644 \u0627\u0633\u062a <strong><code>O(E log N)<\/code><\/strong>.<\/p>\n<h3 id=\"conclusion\"><span class=\"ez-toc-section\" id=\"%d9%86%d8%aa%db%8c%d8%ac%d9%87\"><\/span>\u0646\u062a\u06cc\u062c\u0647<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06a9\u0631\u0648\u0633\u06a9\u0627\u0644 \u06cc\u06a9\u06cc \u0627\u0632 \u067e\u0631\u06a9\u0627\u0631\u0628\u0631\u062f\u062a\u0631\u06cc\u0646 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u200c\u0647\u0627 \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 \u062d\u062f\u0627\u0642\u0644 \u062f\u0631\u062e\u062a \u067e\u0648\u0634\u0627 \u062f\u0631 \u0646\u0645\u0648\u062f\u0627\u0631 \u062f\u0631 \u06a9\u0646\u0627\u0631 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Prim \u0648 Bor\u016fvka \u0627\u0633\u062a.  \u0647\u0631 \u06cc\u06a9 \u0627\u0632 \u0622\u0646\u0647\u0627 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u06cc\u06a9\u0633\u0627\u0646\u06cc \u062f\u0627\u0631\u0646\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0627\u06cc\u0646 \u0634\u0645\u0627 \u0647\u0633\u062a\u06cc\u062f \u06a9\u0647 \u062a\u0635\u0645\u06cc\u0645 \u0645\u06cc \u06af\u06cc\u0631\u06cc\u062f \u0627\u0632 \u06a9\u062f\u0627\u0645 \u06cc\u06a9 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u06cc\u062f.<\/p>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u060c \u0645\u0627 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 Kruskal \u0631\u0627 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647 \u0627\u06cc\u0645 \u0631\u0648\u06cc \u06cc\u06a9 \u0645\u062b\u0627\u0644 \u0639\u0645\u0644\u06cc \u0648 \u0627\u062c\u0631\u0627\u06cc \u0648\u0627\u0642\u0639\u06cc \u0628\u0647 \u0634\u0645\u0627 \u062f\u0627\u062f\u0647 \u0627\u0633\u062a \u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0627\u0632 \u0622\u0646 \u062f\u0631 \u067e\u0631\u0648\u0698\u0647 \u0647\u0627\u06cc \u062e\u0648\u062f \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u06cc\u062f \u0648 \u0631\u0648\u0634 \u0639\u0645\u0644\u06a9\u0631\u062f \u0622\u0646 \u0631\u0627 \u062f\u0631\u06a9 \u06a9\u0646\u06cc\u062f.<\/p>\n<\/div>\n<p><script>\n                        !function(f,b,e,v,n,t,s)\n                        {if(f.fbq)return;n=f.fbq=function(){n.callMethod?\n                        n.callMethod.apply(n,arguments):n.queue.push(arguments)};\n                        if(!f._fbq)f._fbq=n;n.push=n;n.loaded=!0;n.version='2.0';\n                        n.queue=();t=b.createElement(e);t.async=!0;\n                        t.src=v;s=b.getElementsByTagName(e)(0);\n                        s.parentNode.insertBefore(t,s)}(window, document,'script',\n                        'https:\/\/connect.facebook.net\/en_US\/fbevents.js');\n                        fbq('init', 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