{"id":15345,"date":"2024-01-11T07:05:12","date_gmt":"2024-01-11T03:35:12","guid":{"rendered":"https:\/\/rasanegar.com\/blog\/%d9%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/"},"modified":"2024-01-11T07:05:12","modified_gmt":"2024-01-11T03:35:12","slug":"%d9%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86","status":"publish","type":"post","link":"https:\/\/rasanegaar.com\/blog\/%d9%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/","title":{"rendered":"\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646"},"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%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/#%d9%85%d8%b9%d8%b1%d9%81%db%8c\" >\u0645\u0639\u0631\u0641\u06cc<\/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%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/#%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_%d8%b4%d9%85%d8%a7%d8%b1%d8%b4\" >\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634<\/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%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/#%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_radix\" >\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc Radix<\/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%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/#%d8%b1%d9%88%d8%b4_%d9%be%db%8c%d8%a7%d8%af%d9%87_%d8%b3%d8%a7%d8%b2%db%8c_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3_%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 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \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-5\" href=\"https:\/\/rasanegaar.com\/blog\/%d9%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/#%d9%be%db%8c%da%86%db%8c%d8%af%da%af%db%8c_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3\" >\u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/rasanegaar.com\/blog\/%d9%85%d8%b1%d8%aa%d8%a8-%d8%b3%d8%a7%d8%b2%db%8c-%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\/#%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\"> 6<\/span> <span class=\"rt-label rt-postfix\">\u062f\u0642\u06cc\u0642\u0647<\/span><\/span><p> <br \/>\n<\/p>\n<div><noscript><\/noscript><\/p>\n<h3 id=\"introduction\"><span class=\"ez-toc-section\" id=\"%d9%85%d8%b9%d8%b1%d9%81%db%8c\"><\/span>\u0645\u0639\u0631\u0641\u06cc<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>\u0631\u06cc\u0634\u0647<\/strong> (\u06cc\u0627 <strong>\u067e\u0627\u06cc\u0647<\/strong>) \u0627\u0633\u062a <em>\u062a\u0639\u062f\u0627\u062f \u0627\u0631\u0642\u0627\u0645<\/em> \u0628\u0631\u0627\u06cc \u0646\u0645\u0627\u06cc\u0634 \u0627\u0639\u062f\u0627\u062f \u062f\u0631 a \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f <em>\u0633\u06cc\u0633\u062a\u0645 \u0627\u0639\u062f\u0627\u062f \u0645\u0648\u0642\u0639\u06cc\u062a\u06cc<\/em>.  \u0628\u0631\u0627\u06cc <em>\u0633\u06cc\u0633\u062a\u0645 \u0628\u0627\u06cc\u0646\u0631\u06cc<\/em>\u060c \u0631\u06cc\u0634\u0647 \u0627\u0633\u062a <em>2<\/em> (\u0641\u0642\u0637 \u0627\u0632 \u062f\u0648 \u0631\u0642\u0645 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u062f &#8211; 0 \u0648 1).  \u0628\u0631\u0627\u06cc <em>\u0633\u06cc\u0633\u062a\u0645 \u0627\u0639\u0634\u0627\u0631\u06cc<\/em>\u060c \u0631\u06cc\u0634\u0647 \u0627\u0633\u062a <em>10<\/em> (\u0627\u0632 \u062f\u0647 \u0631\u0642\u0645 \u0628\u0631\u0627\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0646 \u0647\u0645\u0647 \u0627\u0639\u062f\u0627\u062f &#8211; \u0627\u0632 0 \u062a\u0627 9 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u062f).<\/p>\n<blockquote>\n<p>\u06cc\u06a9 \u0633\u06cc\u0633\u062a\u0645 \u0627\u0639\u062f\u0627\u062f \u0645\u0648\u0642\u0639\u06cc\u062a\u06cc\u060c \u0628\u0647 \u0632\u0628\u0627\u0646 \u0633\u0627\u062f\u0647\u060c \u06cc\u06a9 \u0633\u06cc\u0633\u062a\u0645 \u0646\u0648\u0634\u062a\u0646 \u0627\u0639\u062f\u0627\u062f \u0627\u0633\u062a \u06a9\u0647 \u0648\u0632\u0646 (\u06cc\u0627 \u0645\u0642\u062f\u0627\u0631) \u06cc\u06a9 \u0631\u0642\u0645 \u0628\u0627 \u0645\u0648\u0642\u0639\u06cc\u062a \u0622\u0646 \u062a\u0639\u06cc\u06cc\u0646 \u0645\u06cc \u0634\u0648\u062f.  \u0645\u062b\u0644\u0627 \u062f\u0631 \u0634\u0645\u0627\u0631\u0647 <code>123<\/code>\u060c <code>1<\/code> \u0627\u0631\u0632\u0634 \u0628\u06cc\u0634\u062a\u0631\u06cc \u062f\u0627\u0631\u062f <code>3<\/code> \u0632\u06cc\u0631\u0627 \u062f\u0631 \u0645\u0648\u0642\u0639\u06cc\u062a\u06cc \u0642\u0631\u0627\u0631 \u062f\u0627\u0631\u062f \u06a9\u0647 \u0646\u0634\u0627\u0646 \u062f\u0647\u0646\u062f\u0647 \u0635\u062f\u0647\u0627 \u0627\u0633\u062a \u0648 <code>2<\/code> \u062f\u0631 \u062f\u0647 \u0627\u0633\u062a.<\/p>\n<\/blockquote>\n<p><strong>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647<\/strong> \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u0628\u0631\u0627\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0648\u0627\u0698\u06af\u0627\u0646\u06cc \u0628\u0633\u06cc\u0627\u0631\u06cc \u0627\u0632 \u0627\u0646\u0648\u0627\u0639 \u062f\u0627\u062f\u0647 \u0647\u0627 &#8211; \u0627\u0639\u062f\u0627\u062f \u0635\u062d\u06cc\u062d\u060c \u06a9\u0644\u0645\u0627\u062a\u060c \u0627\u06cc\u0645\u06cc\u0644 \u0647\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0634\u0648\u062f\u060c \u0627\u0645\u0627 \u0639\u0645\u062f\u062a\u0627\u064b \u0628\u0631\u0627\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0645\u062c\u0645\u0648\u0639\u0647 \u0647\u0627\u06cc\u06cc \u0627\u0632 <em>\u0627\u0639\u062f\u0627\u062f \u0635\u062d\u06cc\u062d<\/em> \u0648 <em>\u0631\u0634\u062a\u0647 \u0647\u0627\u06cc<\/em> (\u06a9\u0647 \u0628\u0647 \u06a9\u0644\u06cc\u062f\u0647\u0627\u06cc \u0639\u062f\u062f \u0635\u062d\u06cc\u062d \u0645\u0646\u0627\u0633\u0628 \u0646\u06af\u0627\u0634\u062a \u0645\u06cc \u0634\u0648\u0646\u062f).<\/p>\n<p>\u0627\u06cc\u0646 \u06cc\u06a9 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u063a\u06cc\u0631\u0645\u0642\u0627\u06cc\u0633\u0647\u200c\u0627\u06cc \u0627\u0633\u062a\u060c \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0639\u0646\u06cc \u06a9\u0647 \u0645\u062c\u0645\u0648\u0639\u0647\u200c\u0627\u06cc \u0631\u0627 \u0628\u0627 \u0645\u0642\u0627\u06cc\u0633\u0647 \u0639\u0646\u0627\u0635\u0631 \u062c\u062f\u0627\u06af\u0627\u0646\u0647 \u0622\u0646 \u0645\u0631\u062a\u0628 \u0646\u0645\u06cc\u200c\u06a9\u0646\u062f\u060c \u0628\u0644\u06a9\u0647 \u0627\u0632 \u0645\u0627\u0647\u06cc\u062a \u0630\u0627\u062a\u06cc \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc\u06cc \u06a9\u0647 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0645\u06cc\u200c\u06a9\u0646\u062f \u0628\u0631\u0627\u06cc \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0633\u0631\u06cc\u0639\u200c\u062a\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc\u200c\u06a9\u0646\u062f &#8211; \u062f\u0627\u062f\u0647\u200c\u0647\u0627 \u0631\u0627 \u0628\u0631\u0627\u0633\u0627\u0633 \u062f\u0627\u062f\u0647\u200c\u0647\u0627 \u0645\u0631\u062a\u0628 \u0645\u06cc\u200c\u06a9\u0646\u062f. \u0631\u0648\u06cc \u0622\u0646\u0647\u0627 <em>\u0631\u06cc\u0634\u0647<\/em>.<\/p>\n<blockquote>\n<p>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0645\u0642\u0627\u06cc\u0633\u0647 \u0627\u06cc \u062f\u0627\u0631\u0627\u06cc \u0628\u0647\u062a\u0631\u06cc\u0646 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0632\u0645\u0627\u0646\u06cc \u0647\u0633\u062a\u0646\u062f <em>O(nlogn)<\/em>\u060c \u06a9\u0647 \u0646\u0633\u0628\u062a\u0627\u064b \u0628\u062f\u062a\u0631 \u0627\u0632 <em>\u0632\u0645\u0627\u0646 \u0627\u062c\u0631\u0627\u06cc \u062e\u0637\u06cc<\/em> (<em>O(n+k)<\/em>) \u0627\u0632 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u063a\u06cc\u0631 \u0645\u0642\u0627\u06cc\u0633\u0647 \u0627\u06cc.<\/p>\n<\/blockquote>\n<p>\u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u062b\u0627\u0644\u060c \u0627\u062c\u0627\u0632\u0647 \u062f\u0647\u06cc\u062f <code>n<\/code> \u062a\u0639\u062f\u0627\u062f \u0639\u0646\u0627\u0635\u0631\u06cc \u0628\u0627\u0634\u062f \u06a9\u0647 \u0628\u0627\u06cc\u062f \u0645\u0631\u062a\u0628 \u0634\u0648\u0646\u062f\u060c \u0648 <code>k<\/code> \u0645\u062d\u062f\u0648\u062f\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0645\u062c\u0627\u0632 \u0639\u0646\u0635\u0631 \u0627\u0633\u062a.<\/p>\n<p><em>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634<\/em> (\u06cc\u06a9 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u063a\u06cc\u0631 \u0645\u0642\u0627\u06cc\u0633\u0647 \u0627\u06cc \u0645\u062d\u0628\u0648\u0628) \u062f\u0627\u0631\u0627\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0627\u0633\u062a <code>O(n+k)<\/code> \u0648\u0642\u062a\u06cc \u06a9\u0647 <code>k<\/code> \u062f\u0631 \u0645\u062d\u062f\u0648\u062f\u0647 \u0627\u0633\u062a <code>1..n<\/code>.  \u0627\u0645\u0627\u060c \u0627\u06af\u0631 \u0639\u0646\u0627\u0635\u0631 \u0627\u0632 <code>1..n\u00b2<\/code>\u060c \u0633\u067e\u0633 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0627\u0641\u0632\u0627\u06cc\u0634 \u0645\u06cc \u06cc\u0627\u0628\u062f <code>O(n\u00b2)<\/code>\u060c \u06a9\u0647 \u0628\u062f\u062a\u0631 \u0627\u0632 \u0647\u0631 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0645\u0642\u0627\u06cc\u0633\u0647 \u0627\u06cc \u0627\u0633\u062a.<\/p>\n<p>Sort Count <em>\u067e\u062a\u0627\u0646\u0633\u06cc\u0644<\/em> \u0628\u0647 \u0637\u0648\u0631 \u0642\u0627\u0628\u0644 \u062a\u0648\u062c\u0647\u06cc \u0633\u0631\u06cc\u0639\u062a\u0631 \u0627\u0632 \u0633\u0627\u06cc\u0631 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0645\u0642\u0627\u06cc\u0633\u0647 \u0627\u06cc \u0631\u0627\u06cc\u062c \u0627\u0633\u062a\u060c \u0647\u0631\u0686\u0646\u062f\u060c \u062a\u0646\u0647\u0627 \u062f\u0631 \u0635\u0648\u0631\u062a\u06cc \u06a9\u0647 \u0634\u0631\u0637 \u062e\u0627\u0635\u06cc \u0628\u0631\u0622\u0648\u0631\u062f\u0647 \u0634\u0648\u062f.<\/p>\n<p>\u0627\u06cc\u062f\u0647 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc Radix \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 <strong>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0631\u0627 \u0627\u0631\u062a\u0642\u0627 \u062f\u0647\u06cc\u062f \u062a\u0627 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0632\u0645\u0627\u0646\u06cc \u062e\u0637\u06cc \u0631\u0627 \u062d\u0641\u0638 \u06a9\u0646\u062f<\/strong> \u062d\u062a\u06cc \u0627\u06af\u0631 \u062f\u0627\u0645\u0646\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0639\u0646\u0627\u0635\u0631 \u0628\u0647 \u0634\u062f\u062a \u0627\u0632 \u062a\u0639\u062f\u0627\u062f \u0639\u0646\u0627\u0635\u0631 \u0628\u06cc\u0634\u062a\u0631 \u0628\u0627\u0634\u062f.<\/p>\n<p>\u062f\u0631 \u062d\u0642\u06cc\u0642\u062a\u060c <em>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647<\/em> \u0630\u0627\u062a\u0627\u064b \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u062f <em>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634<\/em> \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0632\u06cc\u0631\u0631\u0648\u0627\u0644 \u0627\u0635\u0644\u06cc\u060c \u0628\u0627 \u0686\u0646\u062f \u062a\u0631\u0641\u0646\u062f \u0628\u0631\u0627\u06cc \u063a\u0644\u0628\u0647 \u0628\u0631 \u0645\u0634\u06a9\u0644\u0627\u062a\u06cc \u06a9\u0647 \u0628\u0627 \u0627\u0641\u0632\u0627\u06cc\u0634 \u062f\u0627\u0645\u0646\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0639\u0646\u0627\u0635\u0631 \u0627\u06cc\u062c\u0627\u062f \u0645\u06cc \u0634\u0648\u062f.<\/p>\n<h3 id=\"countingsortalgorithm\"><span class=\"ez-toc-section\" id=\"%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_%d8%b4%d9%85%d8%a7%d8%b1%d8%b4\"><\/span>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0628\u0631\u0627\u06cc \u0627\u06cc\u0646\u06a9\u0647 \u0628\u062a\u0648\u0627\u0646\u06cc\u0645 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u0631\u0627 \u062f\u0631\u06a9 \u06a9\u0646\u06cc\u0645\u060c \u0627\u0628\u062a\u062f\u0627 \u0628\u0627\u06cc\u062f \u0628\u0647 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0628\u067e\u0631\u062f\u0627\u0632\u06cc\u0645\u060c \u0622\u0646 \u0631\u0627 \u067e\u06cc\u0627\u062f\u0647\u200c\u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u0645 \u0648 \u0628\u0627 \u0627\u0641\u0632\u0627\u06cc\u0634 \u062a\u0639\u062f\u0627\u062f \u0639\u0646\u0627\u0635\u0631\u060c \u0633\u0642\u0648\u0637 \u0631\u0627 \u0645\u0634\u0627\u0647\u062f\u0647 \u06a9\u0646\u06cc\u0645.<\/p>\n<h4 id=\"whyusecountingsortintheradixsort\">\u0686\u0631\u0627 \u0627\u0632 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u062f\u0631 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u06cc\u0645\u061f<\/h4>\n<p>\u0634\u0645\u0627\u0631\u0634 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc <em>\u067e\u0627\u06cc\u062f\u0627\u0631<\/em>\u060c <em>\u063a\u06cc\u0631 \u062a\u0637\u0628\u06cc\u0642\u06cc<\/em> \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc\u060c \u0648 \u0639\u0645\u062f\u062a\u0627 \u0628\u0631\u0627\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0622\u0631\u0627\u06cc\u0647 \u0647\u0627\u06cc \u0639\u062f\u062f \u0635\u062d\u06cc\u062d \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f.  \u0647\u0645\u0647 \u0627\u06cc\u0646 \u0648\u06cc\u0698\u06af\u06cc \u0647\u0627 \u0628\u0631\u0627\u06cc \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0622\u0646 \u062f\u0631 Radix Sort \u0645\u0647\u0645 \u0647\u0633\u062a\u0646\u062f.  \u0634\u0645\u0627 <em>\u0645\u06cc \u062a\u0648\u0627\u0646<\/em> \u0627\u0632 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u200c\u0647\u0627\u06cc \u062f\u06cc\u06af\u0631 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0632\u06cc\u0631\u0628\u0631\u0646\u0627\u0645\u0647 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u06cc\u062f\u060c \u062a\u0627 \u0632\u0645\u0627\u0646\u06cc \u06a9\u0647 \u0627\u06cc\u0646 \u0648\u06cc\u0698\u06af\u06cc\u200c\u0647\u0627 \u0631\u0627 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u0646\u062f\u060c \u0647\u0631\u0686\u0646\u062f\u060c \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0637\u0628\u06cc\u0639\u06cc\u200c\u062a\u0631\u06cc\u0646 \u062a\u0637\u0627\u0628\u0642 \u0627\u0633\u062a.<\/p>\n<p>\u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u0646\u06cc\u0627\u0632 \u0628\u0647 \u062d\u0641\u0638 \u062a\u0631\u062a\u06cc\u0628 \u0646\u0633\u0628\u06cc \u0639\u0646\u0627\u0635\u0631 \u0628\u0627 \u0645\u0642\u0627\u062f\u06cc\u0631 \u06a9\u0644\u06cc\u062f\u06cc \u06cc\u06a9\u0633\u0627\u0646 \u062f\u0631 \u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc \u062f\u0627\u0631\u062f\u060c \u062f\u0631 \u062d\u0627\u0644\u06cc \u06a9\u0647 \u0627\u0631\u0642\u0627\u0645 \u0627\u0631\u0632\u0634 \u0645\u06a9\u0627\u0646\u06cc \u06cc\u06a9\u0633\u0627\u0646 \u0631\u0627 \u0645\u0631\u062a\u0628 \u0645\u06cc\u200c\u06a9\u0646\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646\u060c \u0632\u06cc\u0631\u0628\u0631\u0646\u0627\u0645\u0647 \u0627\u0635\u0644\u06cc \u0645\u0627 \u0637\u0628\u0642 \u062a\u0639\u0631\u06cc\u0641 \u0628\u0627\u06cc\u062f \u0646\u0648\u0639\u06cc \u0628\u0627\u0634\u062f. <strong>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u067e\u0627\u06cc\u062f\u0627\u0631:<\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/1704944110_653_radix-sort-in-python-01.png\" alt=\"\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647 \u062f\u0631-python-01.png\" title=\"\"><\/p>\n<p><strong>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0647\u0627\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u063a\u06cc\u0631 \u0645\u0642\u0627\u06cc\u0633\u0647 \u0627\u06cc<\/strong> \u0628\u0647 \u0637\u0648\u0631 \u06a9\u0644\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u062e\u0637\u06cc \u062f\u0627\u0631\u0646\u062f\u060c \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062a\u0623\u062b\u06cc\u0631 \u06a9\u0645\u062a\u0631\u06cc \u062e\u0648\u0627\u0647\u0646\u062f \u062f\u0627\u0634\u062a \u0631\u0648\u06cc \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc Radix.<\/p>\n<h4 id=\"howdoesthecountingsortwork\">\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0686\u06af\u0648\u0646\u0647 \u06a9\u0627\u0631 \u0645\u06cc \u06a9\u0646\u062f\u061f<\/h4>\n<p>\u0628\u06cc\u0627\u06cc\u06cc\u062f \u0646\u06af\u0627\u0647\u06cc \u0628\u0647 \u06cc\u06a9 \u0622\u0631\u0627\u06cc\u0647 \u0639\u062f\u062f \u0635\u062d\u06cc\u062d \u0645\u0631\u062a\u0628\u200c\u0646\u0634\u062f\u0647 \u0628\u06cc\u0646\u062f\u0627\u0632\u06cc\u0645 \u06a9\u0647 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0645\u06cc\u200c\u06a9\u0646\u06cc\u0645:<\/p>\n<pre><code class=\"hljs\">I = (2, 2, 0, 6, 1, 9, 9, 7)\n<\/code><\/pre>\n<blockquote>\n<p>\u0634\u0645\u0627\u0631\u0634 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0628\u0631 \u0627\u0633\u0627\u0633 <em>\u0634\u0645\u0627\u0631\u0634 \u062a\u0639\u062f\u0627\u062f \u0639\u0646\u0627\u0635\u0631<\/em>\u060c \u06a9\u0647 \u0645\u062a\u0646\u0627\u0633\u0628 \u0627\u0633\u062a <em>\u0645\u0642\u062f\u0627\u0631 \u06a9\u0644\u06cc\u062f\u06cc \u0645\u062a\u0645\u0627\u06cc\u0632<\/em>\u060c \u0648 \u0633\u067e\u0633 \u0645\u0648\u0642\u0639\u06cc\u062a \u0647\u0627\u06cc \u0647\u0631 \u06a9\u0644\u06cc\u062f \u0631\u0627 \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<\/blockquote>\n<p>\u0627\u0648\u0644 \u0627\u0632 \u0647\u0645\u0647\u060c \u0645\u0627 \u062d\u062f\u0627\u06a9\u062b\u0631 \u0639\u0646\u0635\u0631 \u0631\u0627 \u062f\u0631 \u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc \u067e\u06cc\u062f\u0627 \u062e\u0648\u0627\u0647\u06cc\u0645 \u06a9\u0631\u062f &#8211; <code>max = 9<\/code>.<\/p>\n<p>\u0633\u067e\u0633\u060c \u06cc\u06a9 \u0622\u0631\u0627\u06cc\u0647 \u06a9\u0645\u06a9\u06cc \u0628\u0627 \u0622\u0646 \u0627\u06cc\u062c\u0627\u062f \u0645\u06cc \u06a9\u0646\u06cc\u0645 <code>max+1<\/code> \u0639\u0646\u0627\u0635\u0631.  \u0627\u06cc\u0646 \u0627\u0633\u062a <em>\u0622\u0631\u0627\u06cc\u0647 \u0634\u0645\u0627\u0631\u0634<\/em> (<code>C<\/code>) \u06a9\u0647 \u0628\u0631\u0627\u06cc \u0630\u062e\u06cc\u0631\u0647 \u062a\u0639\u062f\u0627\u062f \u062f\u0641\u0639\u0627\u062a \u0647\u0631 \u0639\u0646\u0635\u0631 \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc<\/em>.<\/p>\n<p>\u062f\u0631 \u0627\u0628\u062a\u062f\u0627\u060c \u0647\u0645\u0647 \u0634\u0645\u0627\u0631\u0634 \u0647\u0627 \u0628\u0647 0 \u0645\u0642\u062f\u0627\u0631\u062f\u0647\u06cc \u0645\u06cc \u0634\u0648\u0646\u062f:<\/p>\n<pre><code class=\"hljs\">C = (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>) \n\n<\/code><\/pre>\n<p><em><strong>\u062d\u0627\u0644 \u0628\u0627\u06cc\u062f \u0645\u0631\u0627\u062d\u0644 \u0632\u06cc\u0631 \u0631\u0627 \u0637\u06cc \u06a9\u0646\u06cc\u0645:<\/strong><\/em><\/p>\n<p><strong>1.<\/strong> \u0639\u0628\u0648\u0631 \u0627\u0632 <em>\u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc<\/em> \u0648 \u062a\u0639\u062f\u0627\u062f \u0645\u0631\u0628\u0648\u0637 \u0628\u0647 \u0647\u0631 \u0639\u0646\u0635\u0631 \u0631\u0627 \u0627\u0641\u0632\u0627\u06cc\u0634 \u062f\u0647\u06cc\u062f <code>1<\/code><\/p>\n<p>\u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u062b\u0627\u0644\u060c \u0627\u06af\u0631 \u0628\u0647 \u0639\u0646\u0635\u0631\u06cc \u0628\u0627 \u0645\u0642\u062f\u0627\u0631 \u0622\u0646 \u0628\u0631\u062e\u0648\u0631\u062f \u06a9\u0646\u06cc\u0645 <code>2<\/code> \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc<\/em> (<code>I<\/code>\u060c 1 \u0631\u0627 \u0628\u0647 \u0639\u0646\u0635\u0631 \u0628\u0627 \u0634\u0627\u062e\u0635 \u0627\u0636\u0627\u0641\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645 <code>2<\/code> \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0634\u0645\u0627\u0631\u0634<\/em>:<\/p>\n<pre><code class=\"hljs\">I = (<span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">6<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">9<\/span>, <span class=\"hljs-number\">9<\/span>, <span class=\"hljs-number\">7<\/span>) \n     ^\n        \nC = (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>) \n\n<\/code><\/pre>\n<p>\u067e\u0633 \u0627\u0632 \u0627\u06cc\u0646 \u0645\u0631\u062d\u0644\u0647\u060c <em>\u0622\u0631\u0627\u06cc\u0647 \u0634\u0645\u0627\u0631\u0634<\/em> \u062a\u0639\u062f\u0627\u062f \u062f\u0641\u0639\u0627\u062a \u0647\u0631 \u0639\u0646\u0635\u0631 \u0631\u0627 \u062f\u0631 \u0642\u0633\u0645\u062a \u0630\u062e\u06cc\u0631\u0647 \u0645\u06cc \u06a9\u0646\u062f <em>\u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc<\/em>:<\/p>\n<pre><code class=\"hljs\">C = (<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">2<\/span>) \n\n   \n\n\n\n\n<\/code><\/pre>\n<p><strong>2.<\/strong> \u0628\u0631\u0627\u06cc \u0647\u0631 \u0639\u0646\u0635\u0631 \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0634\u0645\u0627\u0631\u0634<\/em>\u060c \u0645\u0642\u062f\u0627\u0631 \u0622\u0646 \u0631\u0627 \u0628\u0627 \u0645\u0642\u062f\u0627\u0631 \u062a\u0645\u0627\u0645 \u0639\u0646\u0627\u0635\u0631 \u0642\u0628\u0644\u06cc \u0622\u0646 \u062e\u0644\u0627\u0635\u0647 \u06a9\u0646\u06cc\u062f \u0648 \u0633\u067e\u0633 \u0622\u0646 \u0645\u0642\u062f\u0627\u0631 \u0631\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u062f\u0627\u0631 \u0639\u0646\u0635\u0631 \u0641\u0639\u0644\u06cc \u0630\u062e\u06cc\u0631\u0647 \u06a9\u0646\u06cc\u062f:<\/p>\n<pre><code class=\"hljs\">C = (<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">2<\/span>, <span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">4<\/span>, <span class=\"hljs-number\">5<\/span>, <span class=\"hljs-number\">6<\/span>, <span class=\"hljs-number\">6<\/span>, <span class=\"hljs-number\">8<\/span>) \n\n\n\n\n\n\n<\/code><\/pre>\n<p>\u0628\u0647 \u0627\u06cc\u0646 \u062a\u0631\u062a\u06cc\u0628\u060c \u0645\u0627 \u062c\u0645\u0639 \u062a\u062c\u0645\u0639\u06cc \u0639\u0646\u0627\u0635\u0631 \u0631\u0627 \u0630\u062e\u06cc\u0631\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645 <em>\u0622\u0631\u0627\u06cc\u0647 \u0634\u0645\u0627\u0631\u0634<\/em>\u060c \u0631\u0648\u06cc \u0647\u0631 \u0645\u0631\u062d\u0644\u0647<\/p>\n<p><strong>3.<\/strong> \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u0648\u0642\u0639\u06cc\u062a \u0639\u0646\u0635\u0631 \u0628\u0631 \u0627\u0633\u0627\u0633 \u0631\u0648\u06cc \u0631\u0627 <em>\u0622\u0631\u0627\u06cc\u0647 \u0634\u0645\u0627\u0631\u0634<\/em> \u0627\u0631\u0632\u0634 \u0647\u0627\u06cc<\/p>\n<p>\u0628\u0631\u0627\u06cc \u0630\u062e\u06cc\u0631\u0647 \u0627\u06cc\u0646 \u062a\u0648\u0627\u0644\u06cc \u0645\u0631\u062a\u0628 \u0634\u062f\u0647\u060c \u0628\u0627\u06cc\u062f \u06cc\u06a9 \u0622\u0631\u0627\u06cc\u0647 \u062c\u062f\u06cc\u062f \u0627\u06cc\u062c\u0627\u062f \u06a9\u0646\u06cc\u0645.  \u0628\u06cc\u0627\u06cc\u06cc\u062f \u0627\u0633\u0645\u0634 \u0631\u0627 \u0628\u06af\u0630\u0627\u0631\u06cc\u0645 <em>\u0622\u0631\u0627\u06cc\u0647 \u062e\u0631\u0648\u062c\u06cc<\/em> (<code>O<\/code>) \u0648 \u0622\u0646 \u0631\u0627 \u0628\u0627 \u0645\u0642\u062f\u0627\u0631\u062f\u0647\u06cc \u0627\u0648\u0644\u06cc\u0647 \u06a9\u0646\u06cc\u062f <code>k<\/code> \u0635\u0641\u0631\u060c \u06a9\u062c\u0627 <code>k<\/code> \u062a\u0639\u062f\u0627\u062f \u0639\u0646\u0627\u0635\u0631 \u0645\u0648\u062c\u0648\u062f \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc<\/em>:<\/p>\n<pre><code class=\"hljs\">O = (<span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>, <span class=\"hljs-number\">0<\/span>) \/\/ Initialized output array\n\n<\/code><\/pre>\n<p><em><strong>\u0628\u0631\u0627\u06cc \u0647\u0631 \u0639\u0646\u0635\u0631 <code>I(i)<\/code> (\u0634\u0631\u0648\u0639 \u0627\u0632 \u0627\u0646\u062a\u0647\u0627) \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc<\/em>:<\/strong><\/em><\/p>\n<ol>\n<li>\u0634\u0627\u062e\u0635 \u0631\u0627 \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0634\u0645\u0627\u0631\u0634<\/em> \u06a9\u0647 \u0628\u0631\u0627\u0628\u0631 \u0628\u0627 \u0645\u0642\u062f\u0627\u0631 \u0639\u0646\u0635\u0631 \u0641\u0639\u0644\u06cc \u0627\u0633\u062a <code>I(i)<\/code>\n<ul>\n<li>\u0627\u06cc\u0646 \u0639\u0646\u0635\u0631 \u0627\u0633\u062a <code>C(j)<\/code> \u062c\u0627\u06cc\u06cc \u06a9\u0647 <code>j=I(i)<\/code><\/li>\n<\/ul>\n<\/li>\n<li>\u062a\u0641\u0631\u06cc\u0642 \u06a9\u0631\u062f\u0646 <code>1<\/code> \u0627\u0632 \u0627\u0631\u0632\u0634 <code>C(i)<\/code>\n<ul>\n<li>\u0627\u06a9\u0646\u0648\u0646 \u062f\u0627\u0631\u06cc\u0645 <code>newValue = C(i)-1<\/code><\/li>\n<\/ul>\n<\/li>\n<li>\u0630\u062e\u06cc\u0631\u0647 \u06a9\u0646\u06cc\u062f <code>I(i)<\/code> \u0628\u0647 <code>O(newValue)<\/code><\/li>\n<li>\u0631\u0627 \u0628\u0647 \u0631\u0648\u0632 \u06a9\u0646\u06cc\u062f <code>C(i)<\/code> \u0628\u0627 <code>newValue<\/code><\/li>\n<\/ol>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/radix-sort-in-python-02.png\" alt=\"\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647 \u062f\u0631-python-02.png\" title=\"\"><\/p>\n<p>\u062f\u0631 \u067e\u0627\u06cc\u0627\u0646\u060c <em>\u0622\u0631\u0627\u06cc\u0647 \u062e\u0631\u0648\u062c\u06cc<\/em> \u0634\u0627\u0645\u0644 \u0639\u0646\u0627\u0635\u0631 \u0645\u0631\u062a\u0628 \u0634\u062f\u0647 \u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc \u0627\u0633\u062a!<\/p>\n<h4 id=\"implementingcountingsortinpython\">\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<\/h4>\n<p>\u062d\u0627\u0644\u0627\u060c \u0628\u0627 \u0648\u062c\u0648\u062f \u0647\u0645\u0647 \u0686\u06cc\u0632\u0647\u0627\u06cc\u06cc \u06a9\u0647 \u062f\u0631 \u0631\u0627\u0647 \u0646\u06cc\u0633\u062a &#8211; \u0628\u06cc\u0627\u06cc\u06cc\u062f \u067e\u06cc\u0634 \u0628\u0631\u0648\u06cc\u0645 \u0648 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0631\u0627 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">countingSort<\/span>(<span class=\"hljs-params\">inputArray<\/span>):<\/span>\n    \n    maxEl = <span class=\"hljs-built_in\">max<\/span>(inputArray)\n\n    countArrayLength = maxEl+<span class=\"hljs-number\">1<\/span>\n\n    \n    countArray = (<span class=\"hljs-number\">0<\/span>) * countArrayLength\n\n    \n    \n    <span class=\"hljs-keyword\">for<\/span> el <span class=\"hljs-keyword\">in<\/span> inputArray: \n        countArray(el) += <span class=\"hljs-number\">1<\/span>\n\n    \n    \n    \n    \n    <span class=\"hljs-keyword\">for<\/span> i <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">range<\/span>(<span class=\"hljs-number\">1<\/span>, countArrayLength):\n        countArray(i) += countArray(i-<span class=\"hljs-number\">1<\/span>) \n\n    \n    \n    outputArray = (<span class=\"hljs-number\">0<\/span>) * <span class=\"hljs-built_in\">len<\/span>(inputArray)\n    i = <span class=\"hljs-built_in\">len<\/span>(inputArray) - <span class=\"hljs-number\">1<\/span>\n    <span class=\"hljs-keyword\">while<\/span> i &gt;= <span class=\"hljs-number\">0<\/span>:\n        currentEl = inputArray(i)\n        countArray(currentEl) -= <span class=\"hljs-number\">1<\/span>\n        newPosition = countArray(currentEl)\n        outputArray(newPosition) = currentEl\n        i -= <span class=\"hljs-number\">1<\/span>\n\n    <span class=\"hljs-keyword\">return<\/span> outputArray\n\ninputArray = (<span class=\"hljs-number\">2<\/span>,<span class=\"hljs-number\">2<\/span>,<span class=\"hljs-number\">0<\/span>,<span class=\"hljs-number\">6<\/span>,<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">9<\/span>,<span class=\"hljs-number\">9<\/span>,<span class=\"hljs-number\">7<\/span>)\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-string\">\"Input array = \"<\/span>, inputArray)\n\nsortedArray = countingSort(inputArray)\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-string\">\"Counting sort result = \"<\/span>, sortedArray)\n<\/code><\/pre>\n<p>\u0628\u0627 \u0627\u062c\u0631\u0627\u06cc \u06a9\u062f \u0628\u0627\u0644\u0627 \u062e\u0631\u0648\u062c\u06cc \u0632\u06cc\u0631 \u0628\u0647 \u062f\u0633\u062a \u0645\u06cc \u0622\u06cc\u062f:<\/p>\n<pre><code class=\"hljs\">Input array =  (2, 2, 0, 6, 1, 9, 9, 7)\nCounting sort result =  (0, 1, 2, 2, 6, 7, 9, 9)\n<\/code><\/pre>\n<h4 id=\"countingsortcomplexity\">\u0634\u0645\u0627\u0631\u0634 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc<\/h4>\n<p>\u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0632\u0645\u0627\u0646\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0627\u0633\u062a <code>O(n+k)<\/code>\u060c \u062c\u0627\u06cc\u06cc \u06a9\u0647 <code>n<\/code> \u062a\u0639\u062f\u0627\u062f \u0639\u0646\u0627\u0635\u0631 \u0645\u0648\u062c\u0648\u062f \u062f\u0631 <em>\u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc<\/em>\u060c \u0648 <code>k<\/code> \u0627\u0631\u0632\u0634 \u0622\u0646 \u0627\u0633\u062a <code>max<\/code> \u0639\u0646\u0635\u0631 \u062f\u0631 \u0622\u0631\u0627\u06cc\u0647.<\/p>\n<p>\u0645\u0634\u06a9\u0644 \u0632\u0645\u0627\u0646\u06cc \u0631\u062e \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0645\u0642\u062f\u0627\u0631 \u0628\u0632\u0631\u06af\u062a\u0631\u06cc\u0646 \u0639\u0646\u0635\u0631 \u0628\u0647 \u0634\u062f\u062a \u0627\u0632 \u062a\u0639\u062f\u0627\u062f \u0639\u0646\u0627\u0635\u0631 \u0645\u0648\u062c\u0648\u062f \u062f\u0631 \u0622\u0631\u0627\u06cc\u0647 \u0628\u06cc\u0634\u062a\u0631 \u0634\u0648\u062f.  \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 <code>k<\/code> \u0646\u0632\u062f\u06cc\u06a9 \u0645\u06cc \u0634\u0648\u062f <code>n\u00b2<\/code>\u060c \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0632\u0645\u0627\u0646\u06cc \u0646\u0632\u062f\u06cc\u06a9\u062a\u0631 \u0645\u06cc \u0634\u0648\u062f <code>O(n\u00b2)<\/code>\u060c \u06a9\u0647 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0632\u0645\u0627\u0646\u06cc \u0648\u062d\u0634\u062a\u0646\u0627\u06a9\u06cc \u0628\u0631\u0627\u06cc \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0627\u0633\u062a.<\/p>\n<p>\u0627\u06cc\u0646\u062c\u0627 \u062c\u0627\u06cc\u06cc \u0627\u0633\u062a \u06a9\u0647 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc Radix \u0634\u0631\u0648\u0639 \u0628\u0647 \u06a9\u0627\u0631 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<h3 id=\"radixsortalgorithm\"><span class=\"ez-toc-section\" id=\"%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_radix\"><\/span>\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc Radix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0628\u0647 \u062c\u0627\u06cc \u0634\u0645\u0627\u0631\u0634 \u0639\u0646\u0627\u0635\u0631 \u0628\u0627 \u0645\u0642\u062f\u0627\u0631 \u06a9\u0644\u06cc\u062f \u0645\u062a\u0645\u0627\u06cc\u0632\u0634\u0627\u0646 &#8211; \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u0627\u0631\u0642\u0627\u0645 \u0631\u0627 \u0628\u0631 \u0627\u0633\u0627\u0633 \u0622\u0646\u0647\u0627 \u06af\u0631\u0648\u0647\u200c\u0628\u0646\u062f\u06cc \u0645\u06cc\u200c\u06a9\u0646\u062f. <em>\u0627\u0631\u0632\u0634 \u0645\u0648\u0642\u0639\u06cc\u062a<\/em> \u0648 \u0627\u062c\u0631\u0627\u06cc \u0634\u0645\u0627\u0631\u0634 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u062f\u0631 \u0647\u0631 \u06af\u0631\u0648\u0647.  \u0645\u0648\u0642\u0639\u06cc\u062a \u0634\u0631\u0648\u0639 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u0645\u062a\u0641\u0627\u0648\u062a \u0628\u0627\u0634\u062f &#8211; <em>LSD (\u06a9\u0645\u062a\u0631\u06cc\u0646 \u0627\u0631\u0642\u0627\u0645 \u0645\u0647\u0645)<\/em> \u06cc\u0627 <em>MSD (\u0645\u0647\u0645\u062a\u0631\u06cc\u0646 \u0627\u0631\u0642\u0627\u0645)<\/em> \u062f\u0648 \u0646\u0648\u0639 \u0645\u062a\u062f\u0627\u0648\u0644 \u0647\u0633\u062a\u0646\u062f \u0648 \u0628\u0631 \u0627\u06cc\u0646 \u0627\u0633\u0627\u0633\u060c \u0627\u06cc\u0646 \u062a\u063a\u06cc\u06cc\u0631\u0627\u062a Radix Sort LSD Radix Sort \u0648 MSD Radix Sort \u0646\u0627\u0645\u06cc\u062f\u0647 \u0645\u06cc \u0634\u0648\u0646\u062f.<\/p>\n<p>\u0627\u062c\u0627\u0632\u0647 \u062f\u0647\u06cc\u062f <code>I = (2, 20, 61, 997, 1, 619)<\/code> \u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc \u0628\u0627\u0634\u062f \u06a9\u0647 \u0645\u06cc \u062e\u0648\u0627\u0647\u06cc\u0645 \u0645\u0631\u062a\u0628 \u06a9\u0646\u06cc\u0645:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/radix-sort-in-python-03.png\" alt=\"\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647 \u062f\u0631-python-03.png\" title=\"\"><\/p>\n<p>\u0645\u0627 \u062a\u0645\u0631\u06a9\u0632 \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u0631\u0648\u06cc <em>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u0627\u0644 \u0627\u0633 \u062f\u06cc<\/em>.<\/p>\n<h4 id=\"radixsortalgorithm\">\u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633<\/h4>\n<p>\u0645\u0631\u0627\u062d\u0644 \u0627\u0646\u062c\u0627\u0645 \u0634\u062f\u0647 \u062a\u0648\u0633\u0637 Radix Sort \u0646\u0633\u0628\u062a\u0627\u064b \u0633\u0627\u062f\u0647 \u0627\u0633\u062a:<\/p>\n<ol>\n<li>\u062d\u062f\u0627\u06a9\u062b\u0631 \u0639\u0646\u0635\u0631 \u0631\u0627 \u062f\u0631 \u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc \u067e\u06cc\u062f\u0627 \u06a9\u0646\u06cc\u062f &#8211; <code>max = 997<\/code><\/li>\n<li>\u062a\u0639\u062f\u0627\u062f \u0627\u0631\u0642\u0627\u0645 \u0645\u0648\u062c\u0648\u062f \u062f\u0631 <code>max<\/code> \u0639\u0646\u0635\u0631 &#8211; <code>D = 3<\/code><\/li>\n<li>\u0627\u0631\u0632\u0634 \u0645\u06a9\u0627\u0646\u06cc \u0631\u0627 \u0628\u0647 \u06a9\u0645\u062a\u0631\u06cc\u0646 \u0645\u06a9\u0627\u0646 \u0646\u0634\u0627\u0646 \u062f\u0647\u06cc\u062f &#8211; <code>placeVal = 1<\/code><\/li>\n<li>\u0628\u0631\u0627\u06cc <code>D<\/code> \u0632\u0645\u0627\u0646 \u0627\u0646\u062c\u0627\u0645:\n<ol>\n<li>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0631\u0627 \u0628\u0631 \u0627\u0633\u0627\u0633 \u0627\u0631\u0632\u0634 \u0645\u06a9\u0627\u0646\u06cc \u0641\u0639\u0644\u06cc \u0627\u0646\u062c\u0627\u0645 \u062f\u0647\u06cc\u062f<\/li>\n<li>\u0628\u0627 \u0636\u0631\u0628 \u0628\u0647 \u0627\u0631\u0632\u0634 \u0645\u06a9\u0627\u0646\u06cc \u0628\u0639\u062f\u06cc \u0628\u0631\u0648\u06cc\u062f <code>placeVal<\/code> \u062a\u0648\u0633\u0637 10<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/radix-sort-in-python-04.png\" alt=\"\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647 \u062f\u0631-python-04.png\" title=\"\"><\/p>\n<h3 id=\"howtoimplementradixsortinpython\"><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_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3_%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 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0648 \u062f\u0631 \u0646\u0647\u0627\u06cc\u062a\u060c \u0628\u0627 \u0627\u06cc\u0646 \u06a9\u0627\u0631\u060c \u0628\u06cc\u0627\u06cc\u06cc\u062f \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc Radix \u0631\u0627 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646 \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">countingSortForRadix<\/span>(<span class=\"hljs-params\">inputArray, placeValue<\/span>):<\/span>\n    \n    \n    countArray = (<span class=\"hljs-number\">0<\/span>) * <span class=\"hljs-number\">10<\/span>\n    inputSize = <span class=\"hljs-built_in\">len<\/span>(inputArray)\n\n    \n    \n    \n    \n    <span class=\"hljs-keyword\">for<\/span> i <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">range<\/span>(inputSize): \n        placeElement = (inputArray(i) \/\/ placeValue) % <span class=\"hljs-number\">10<\/span>\n        countArray(placeElement) += <span class=\"hljs-number\">1<\/span>\n\n    <span class=\"hljs-keyword\">for<\/span> i <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">range<\/span>(<span class=\"hljs-number\">1<\/span>, <span class=\"hljs-number\">10<\/span>):\n        countArray(i) += countArray(i-<span class=\"hljs-number\">1<\/span>)\n\n    \n    outputArray = (<span class=\"hljs-number\">0<\/span>) * inputSize\n    i = inputSize - <span class=\"hljs-number\">1<\/span>\n    <span class=\"hljs-keyword\">while<\/span> i &gt;= <span class=\"hljs-number\">0<\/span>:\n        currentEl = inputArray(i)\n        placeElement = (inputArray(i) \/\/ placeValue) % <span class=\"hljs-number\">10<\/span>\n        countArray(placeElement) -= <span class=\"hljs-number\">1<\/span>\n        newPosition = countArray(placeElement)\n        outputArray(newPosition) = currentEl\n        i -= <span class=\"hljs-number\">1<\/span>\n        \n    <span class=\"hljs-keyword\">return<\/span> outputArray\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">radixSort<\/span>(<span class=\"hljs-params\">inputArray<\/span>):<\/span>\n    \n    maxEl = <span class=\"hljs-built_in\">max<\/span>(inputArray)\n\n    \n    D = <span class=\"hljs-number\">1<\/span>\n    <span class=\"hljs-keyword\">while<\/span> maxEl &gt; <span class=\"hljs-number\">0<\/span>:\n        maxEl \/= <span class=\"hljs-number\">10<\/span>\n        D += <span class=\"hljs-number\">1<\/span>\n    \n    \n    placeVal = <span class=\"hljs-number\">1<\/span>\n\n    \n    outputArray = inputArray\n    <span class=\"hljs-keyword\">while<\/span> D &gt; <span class=\"hljs-number\">0<\/span>:\n        outputArray = countingSortForRadix(outputArray, placeVal)\n        placeVal *= <span class=\"hljs-number\">10<\/span>  \n        D -= <span class=\"hljs-number\">1<\/span>\n\n    <span class=\"hljs-keyword\">return<\/span> outputArray\n    \n<span class=\"hljs-built_in\">input<\/span> = (<span class=\"hljs-number\">2<\/span>,<span class=\"hljs-number\">20<\/span>,<span class=\"hljs-number\">61<\/span>,<span class=\"hljs-number\">997<\/span>,<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">619<\/span>)\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-built_in\">input<\/span>)\n<span class=\"hljs-built_in\">sorted<\/span> = radixSort(<span class=\"hljs-built_in\">input<\/span>)\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-built_in\">sorted<\/span>)\n<\/code><\/pre>\n<p>\u0628\u0627 \u0627\u062c\u0631\u0627\u06cc \u06a9\u062f \u0628\u0627\u0644\u0627 \u062e\u0631\u0648\u062c\u06cc \u0632\u06cc\u0631 \u0628\u0647 \u062f\u0633\u062a \u0645\u06cc \u0622\u06cc\u062f:<\/p>\n<pre><code class=\"hljs\">(2, 20, 61, 997, 1, 619)\n(1, 2, 20, 61, 619, 997)\n<\/code><\/pre>\n<h3 id=\"radixsortcomplexity\"><span class=\"ez-toc-section\" id=\"%d9%be%db%8c%da%86%db%8c%d8%af%da%af%db%8c_%d9%85%d8%b1%d8%aa%d8%a8_%d8%b3%d8%a7%d8%b2%db%8c_%d8%b1%d8%a7%d8%af%db%8c%da%a9%d8%b3\"><\/span>\u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0642\u0628\u0644\u0627\u064b \u0628\u06cc\u0627\u0646 \u06a9\u0631\u062f\u06cc\u0645\u060c Radix Sort \u062f\u0627\u0631\u062f <strong>\u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0632\u0645\u0627\u0646\u06cc \u062e\u0637\u06cc<\/strong>.  \u0627\u06af\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u06cc\u0645 <em>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634<\/em> \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0632\u06cc\u0631\u0628\u0631\u0646\u0627\u0645\u0647 \u0627\u0635\u0644\u06cc\u060c \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647 \u0627\u0633\u062a <code>O(d(n+k))<\/code>.  \u062f\u0644\u06cc\u0644\u0634 \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0645\u0627 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634 \u0631\u0627 \u0627\u062c\u0631\u0627 \u0645\u06cc \u06a9\u0646\u06cc\u0645 <code>d<\/code> \u0632\u0645\u0627\u0646\u060c \u0648 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc <em>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0634\u0645\u0627\u0631\u0634<\/em> \u062e\u0648\u062f\u0634 \u0627\u0633\u062a <code>O(n+k)<\/code>.<\/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>\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc Radix \u06cc\u06a9 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0639\u0627\u0644\u06cc \u0628\u0631\u0627\u06cc \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u062f\u0631 \u0628\u0631\u062e\u06cc \u0645\u0648\u0627\u0631\u062f \u062e\u0627\u0635 \u0627\u0633\u062a.  \u0628\u0631\u062e\u06cc \u0627\u0632 \u0645\u0639\u06cc\u0627\u0631\u0647\u0627 \u062d\u062a\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0647\u200c\u0627\u0646\u062f \u06a9\u0647 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647 \u0645\u06cc\u200c\u062a\u0648\u0627\u0646\u062f \u062a\u0627 3 \u0628\u0631\u0627\u0628\u0631 \u0633\u0631\u06cc\u0639\u200c\u062a\u0631 \u0627\u0632 \u0633\u0627\u06cc\u0631 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u200c\u0647\u0627\u06cc \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0639\u0645\u0648\u0645\u06cc\u200c\u062a\u0631 \u0627\u062c\u0631\u0627 \u0634\u0648\u062f.<\/p>\n<p>\u0647\u0646\u06af\u0627\u0645\u06cc \u06a9\u0647 \u0622\u0631\u0627\u06cc\u0647 \u0648\u0631\u0648\u062f\u06cc \u06a9\u0644\u06cc\u062f\u0647\u0627\u06cc \u06a9\u0648\u062a\u0627\u0647 \u062a\u0631\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f \u06cc\u0627 \u0645\u062d\u062f\u0648\u062f\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0639\u0646\u0627\u0635\u0631 \u06a9\u0648\u0686\u06a9\u062a\u0631 \u0628\u0627\u0634\u062f\u060c \u0645\u06cc \u062f\u0631\u062e\u0634\u062f.  \u0627\u0645\u0627 \u062f\u0631 \u0645\u0648\u0627\u0631\u062f \u062f\u06cc\u06af\u0631\u060c \u0632\u0645\u0627\u0646\u06cc \u06a9\u0647 \u062f\u0627\u0645\u0646\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0639\u0646\u0627\u0635\u0631 \u0628\u0633\u06cc\u0627\u0631 \u0632\u06cc\u0627\u062f \u0627\u0633\u062a \u0648 \u0639\u0646\u0627\u0635\u0631 \u062f\u0627\u0631\u0627\u06cc \u0627\u0631\u0642\u0627\u0645 \u0628\u06cc\u0634 \u0627\u0632 \u062d\u062f \u062f\u0631 \u0646\u0645\u0627\u06cc\u0634 \u062e\u0648\u062f \u0647\u0633\u062a\u0646\u062f\u060c \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0641\u0636\u0627\u06cc \u0636\u0639\u06cc\u0641\u06cc \u062f\u0627\u0631\u062f.<\/p>\n<p>\u0627\u06cc\u0646 \u062f\u0644\u06cc\u0644 \u0627\u0635\u0644\u06cc \u0627\u06cc\u0646 \u0627\u0633\u062a \u06a9\u0647 \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0631\u06cc\u0634\u0647 \u0628\u0647 \u0627\u0646\u062f\u0627\u0632\u0647 \u0628\u0631\u062e\u06cc \u062f\u06cc\u06af\u0631 \u0627\u0632 \u0627\u0646\u0648\u0627\u0639 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u200c\u0647\u0627\u06cc \u0645\u0631\u062a\u0628\u200c\u0633\u0627\u0632\u06cc \u0628\u0647 \u0637\u0648\u0631 \u06af\u0633\u062a\u0631\u062f\u0647 \u0645\u0648\u0631\u062f \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0642\u0631\u0627\u0631 \u0646\u0645\u06cc\u200c\u06af\u06cc\u0631\u062f\u060c \u062d\u062a\u06cc \u0627\u06af\u0631 \u067e\u06cc\u0686\u06cc\u062f\u06af\u06cc \u0632\u0645\u0627\u0646\u06cc \u062e\u0637\u06cc \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\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               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data-payload='{&quot;align&quot;:&quot;center&quot;,&quot;id&quot;:&quot;15345&quot;,&quot;slug&quot;:&quot;default&quot;,&quot;valign&quot;:&quot;bottom&quot;,&quot;ignore&quot;:&quot;&quot;,&quot;reference&quot;:&quot;auto&quot;,&quot;class&quot;:&quot;&quot;,&quot;count&quot;:&quot;0&quot;,&quot;legendonly&quot;:&quot;&quot;,&quot;readonly&quot;:&quot;&quot;,&quot;score&quot;:&quot;0&quot;,&quot;starsonly&quot;:&quot;&quot;,&quot;best&quot;:&quot;5&quot;,&quot;gap&quot;:&quot;5&quot;,&quot;greet&quot;:&quot;\u0627\u0645\u062a\u06cc\u0627\u0632 \u0634\u0645\u0627 \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0637\u0644\u0628&quot;,&quot;legend&quot;:&quot;0\\\/5 (0 \u0631\u0627\u06cc)&quot;,&quot;size&quot;:&quot;30&quot;,&quot;title&quot;:&quot;\u0645\u0631\u062a\u0628 \u0633\u0627\u0632\u06cc \u0631\u0627\u062f\u06cc\u06a9\u0633 \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646&quot;,&quot;width&quot;:&quot;0&quot;,&quot;_legend&quot;:&quot;{score}\\\/{best} ({count} \u0631\u0627\u06cc)&quot;,&quot;font_factor&quot;:&quot;1.25&quot;}'>\n            \n<div class=\"kksr-stars\">\n    \n<div class=\"kksr-stars-inactive\">\n            <div class=\"kksr-star\" data-star=\"1\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"2\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"3\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"4\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"5\" style=\"padding-left: 5px\">\n            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30px;\"><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n                \n\n<div class=\"kksr-legend\" style=\"font-size: 24px;\">\n            <span class=\"kksr-muted\">\u0627\u0645\u062a\u06cc\u0627\u0632 \u0634\u0645\u0627 \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0637\u0644\u0628<\/span>\n    <\/div>\n    <\/div>\n","protected":false},"excerpt":{"rendered":"<p><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\"> 6<\/span> <span class=\"rt-label rt-postfix\">\u062f\u0642\u06cc\u0642\u0647<\/span><\/span>\u0645\u0639\u0631\u0641\u06cc \u0631\u06cc\u0634\u0647 (\u06cc\u0627 \u067e\u0627\u06cc\u0647) \u0627\u0633\u062a \u062a\u0639\u062f\u0627\u062f \u0627\u0631\u0642\u0627\u0645 \u0628\u0631\u0627\u06cc \u0646\u0645\u0627\u06cc\u0634 \u0627\u0639\u062f\u0627\u062f \u062f\u0631 a 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[&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":15346,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1743,620],"tags":[1749,1748,1746,1754,1753,1744,2750,3861,1805,3258,1752],"class_list":["post-15345","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-python","category-programming","tag-python","tag-python-hosting","tag-vps-","tag-1754","tag-1753","tag-1744","tag-2750","tag-3861","tag-1805","tag-3258","tag-1752"],"acf":[],"_links":{"self":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts\/15345","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/comments?post=15345"}],"version-history":[{"count":0,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts\/15345\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/media\/15346"}],"wp:attachment":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/media?parent=15345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/categories?post=15345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/tags?post=15345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}