{"id":2804,"date":"2025-06-07T13:48:45","date_gmt":"2025-06-07T13:48:45","guid":{"rendered":"https:\/\/diznr.com\/?p=2804"},"modified":"2025-06-07T13:48:45","modified_gmt":"2025-06-07T13:48:45","slug":"aad-dynamic-programming-in-hindi-introduction-and-general-method-multistage-graphs-1-part","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/aad-dynamic-programming-in-hindi-introduction-and-general-method-multistage-graphs-1-part\/","title":{"rendered":"AAD- Dynamic Programming In Hindi Introduction and General Method multistage graphs Part 1"},"content":{"rendered":"<p>AAD- Dynamic Programming In Hindi%20%20 Introduction and%20 General Method multistage graphs Part 1<\/p>\n<p>[fvplayer id=&#8221;121&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"79\"><strong data-start=\"4\" data-end=\"79\">\u0921\u0940ynamic Programming (\u0921\u0940\u092a\u0940) &#8211; \u092a\u0930\u093f\u091a\u092f \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0927\u093f (Multistage Graphs)<\/strong><\/h3>\n<p data-start=\"81\" data-end=\"516\"><strong data-start=\"81\" data-end=\"106\">\u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915 \u092a\u094d\u0930\u094b\u0917\u094d\u0930\u093e\u092e\u093f\u0902\u0917<\/strong> (Dynamic Programming, DP) \u090f\u0915 \u0936\u0915\u094d\u0924\u093f\u0936\u093e\u0932\u0940 \u0924\u0915\u0928\u0940\u0915 \u0939\u0948 \u091c\u093f\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u091b\u094b\u091f\u0947-\u091b\u094b\u091f\u0947 \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0924\u0915\u0928\u0940\u0915 \u0915\u093e \u092e\u0941\u0916\u094d\u092f \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0938\u092e\u093e\u0928 \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u094b \u092a\u0941\u0928\u0903 \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0928\u093e \u0939\u0948 \u0924\u093e\u0915\u093f \u0938\u092e\u092f \u0914\u0930 \u0938\u0902\u0938\u093e\u0927\u0928\u094b\u0902 \u0915\u0940 \u092c\u091a\u0924 \u0915\u0940 \u091c\u093e \u0938\u0915\u0947\u0964 \u091c\u092c \u0939\u092e \u090f\u0915 \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u094b \u0939\u0932 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e \u092a\u0939\u0932\u0947 \u0909\u0928 \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u092e\u093e\u0927\u093e\u0928 \u0922\u0942\u0902\u0922\u0924\u0947 \u0939\u0948\u0902, \u092b\u093f\u0930 \u0909\u0928\u094d\u0939\u0947\u0902 \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u092a\u0942\u0930\u0947 \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u093e \u0938\u092e\u093e\u0927\u093e\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<h3 data-start=\"518\" data-end=\"555\"><strong data-start=\"522\" data-end=\"554\">Dynamic Programming \u0915\u093e \u092a\u0930\u093f\u091a\u092f<\/strong>:<\/h3>\n<p data-start=\"557\" data-end=\"727\">\u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915 \u092a\u094d\u0930\u094b\u0917\u094d\u0930\u093e\u092e\u093f\u0902\u0917 (DP) \u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0924\u092c \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u091c\u092c \u0915\u094b\u0908 \u0938\u092e\u0938\u094d\u092f\u093e <strong data-start=\"624\" data-end=\"651\">\u0913\u0935\u0930\u0932\u0948\u092a\u093f\u0902\u0917 \u0938\u092c\u092a\u094d\u0930\u094b\u092c\u094d\u0932\u0947\u092e\u094d\u0938<\/strong> \u0914\u0930 <strong data-start=\"655\" data-end=\"678\">\u0911\u092a\u094d\u091f\u093f\u092e\u0932 \u0938\u092c\u0938\u094d\u091f\u094d\u0930\u0915\u094d\u091a\u0930<\/strong> \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u090f\u0901 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938\u0915\u093e \u0905\u0930\u094d\u0925 \u0939\u0948 \u0915\u093f:<\/p>\n<ol data-start=\"728\" data-end=\"1045\">\n<li data-start=\"728\" data-end=\"850\"><strong data-start=\"731\" data-end=\"784\">\u0913\u0935\u0930\u0932\u0948\u092a\u093f\u0902\u0917 \u0938\u092c\u092a\u094d\u0930\u094b\u092c\u094d\u0932\u0947\u092e\u094d\u0938 (Overlapping Subproblems)<\/strong>: \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u094b \u0939\u0932 \u0915\u0930\u0924\u0947 \u0938\u092e\u092f \u0939\u092e \u090f\u0915 \u0939\u0940 \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e \u0915\u094b \u092c\u093e\u0930-\u092c\u093e\u0930 \u0939\u0932 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/li>\n<li data-start=\"851\" data-end=\"1045\"><strong data-start=\"854\" data-end=\"900\">\u0911\u092a\u094d\u091f\u093f\u092e\u0932 \u0938\u092c\u0938\u094d\u091f\u094d\u0930\u0915\u094d\u091a\u0930 (Optimal Substructure)<\/strong>: \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u0947 \u0938\u092e\u093e\u0927\u093e\u0928 \u0915\u094b \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u092e\u093e\u0927\u093e\u0928 \u0938\u0947 \u0928\u093f\u0930\u094d\u092e\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u0938\u0947 \u0939\u092e\u0947\u0902 \u0915\u0947\u0935\u0932 \u090f\u0915 \u092c\u093e\u0930 \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u092e\u093e\u0927\u093e\u0928 \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0939\u094b\u0924\u0940 \u0939\u0948\u0964<\/li>\n<\/ol>\n<p data-start=\"1047\" data-end=\"1195\">\u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915 \u092a\u094d\u0930\u094b\u0917\u094d\u0930\u093e\u092e\u093f\u0902\u0917 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 <strong data-start=\"1078\" data-end=\"1093\">\u092a\u094d\u0930\u094b\u092c\u094d\u0932\u0947\u092e\u094d\u0938<\/strong> \u091c\u0948\u0938\u0947 \u0915\u093f <strong data-start=\"1102\" data-end=\"1122\">\u092b\u093f\u092c\u094b\u0928\u093e\u091a\u094d\u091a\u0940 \u0938\u0940\u0930\u0940\u091c<\/strong>, <strong data-start=\"1124\" data-end=\"1153\">\u0936\u0949\u0930\u094d\u091f\u0947\u0938\u094d\u091f \u092a\u093e\u0925 \u092a\u094d\u0930\u0949\u092c\u094d\u0932\u0947\u092e\u094d\u0938<\/strong>, \u0914\u0930 <strong data-start=\"1158\" data-end=\"1177\">\u0938\u094d\u091f\u094d\u0930\u093f\u0902\u0917 \u092e\u0948\u091a\u093f\u0902\u0917<\/strong> \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<h3 data-start=\"1202\" data-end=\"1267\"><strong data-start=\"1206\" data-end=\"1227\">Multistage Graphs<\/strong> (\u092e\u0932\u094d\u091f\u0940\u0938\u094d\u091f\u0947\u091c \u0917\u094d\u0930\u093e\u092b\u093c) &#8211; \u090f\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0909\u0926\u093e\u0939\u0930\u0923<\/h3>\n<p data-start=\"1269\" data-end=\"1768\"><strong data-start=\"1269\" data-end=\"1290\">\u092e\u0932\u094d\u091f\u0940\u0938\u094d\u091f\u0947\u091c \u0917\u094d\u0930\u093e\u092b\u093c<\/strong> \u0935\u0939 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0917\u094d\u0930\u093e\u092b \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0938\u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0928\u094b\u0921 (\u0936\u093f\u0916\u0930) \u0915\u094b \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u091a\u0930\u0923\u094b\u0902 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0928\u092e\u0947\u0902 \u090f\u0915 \u0938\u094d\u0930\u094b\u0924 \u0936\u093f\u0916\u0930 (Source Node) \u0914\u0930 \u090f\u0915 \u0932\u0915\u094d\u0937\u094d\u092f \u0936\u093f\u0916\u0930 (Target Node) \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 \u0917\u094d\u0930\u093e\u092b \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u091a\u0930\u0923\u094b\u0902 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948, \u0914\u0930 \u090f\u0915 \u091a\u0930\u0923 \u0938\u0947 \u0926\u0942\u0938\u0930\u0947 \u091a\u0930\u0923 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u0941\u091b \u090f\u091c (Edge) \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u090f\u091c \u092e\u0947\u0902 \u090f\u0915 \u0935\u091c\u0928 (Weight) \u0939\u094b\u0924\u093e \u0939\u0948, \u0914\u0930 \u0939\u092e\u0947\u0902 \u0907\u0938 \u0917\u094d\u0930\u093e\u092b \u092e\u0947\u0902 \u0938\u094d\u0930\u094b\u0924 \u0938\u0947 \u0932\u0915\u094d\u0937\u094d\u092f \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u093e \u0938\u092c\u0938\u0947 \u0915\u092e \u0935\u091c\u0928 (minimum cost) \u092f\u093e \u0938\u092c\u0938\u0947 \u0905\u091a\u094d\u091b\u093e \u0930\u093e\u0938\u094d\u0924\u093e (optimal path) \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0939\u094b\u0924\u0940 \u0939\u0948\u0964<\/p>\n<p data-start=\"1770\" data-end=\"1874\"><strong data-start=\"1770\" data-end=\"1874\">Multistage Graphs \u0915\u0947 \u0938\u093e\u0925 \u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u092e\u093e\u0927\u093e\u0928 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915 \u092a\u094d\u0930\u094b\u0917\u094d\u0930\u093e\u092e\u093f\u0902\u0917 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/strong><\/p>\n<h3 data-start=\"1881\" data-end=\"2021\"><strong data-start=\"1885\" data-end=\"1967\">General Method for Solving Multistage Graph Problems Using Dynamic Programming<\/strong> (Multistage Graph \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0940 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0927\u093f):<\/h3>\n<h4 data-start=\"2023\" data-end=\"2060\"><strong data-start=\"2028\" data-end=\"2060\">Step 1: Graph Representation<\/strong><\/h4>\n<p data-start=\"2061\" data-end=\"2252\">\u0917\u094d\u0930\u093e\u092b \u0915\u094b \u0938\u0939\u0940 \u0924\u0930\u0940\u0915\u0947 \u0938\u0947 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0930\u0947\u0902\u0964 \u092e\u0932\u094d\u091f\u0940\u0938\u094d\u091f\u0947\u091c \u0917\u094d\u0930\u093e\u092b \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0928\u094b\u0921 \u0915\u094b \u090f\u0915 \u091a\u0930\u0923 (Stage) \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u091a\u093f\u0924\u094d\u0930\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u0914\u0930 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u090f\u091c (Edge) \u090f\u0915 \u091a\u0930\u0923 \u0938\u0947 \u0926\u0942\u0938\u0930\u0947 \u091a\u0930\u0923 \u092e\u0947\u0902 \u091f\u094d\u0930\u093e\u0902\u0938\u092b\u0930 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<h4 data-start=\"2254\" data-end=\"2285\"><strong data-start=\"2259\" data-end=\"2285\">Step 2: Initialization<\/strong><\/h4>\n<p data-start=\"2286\" data-end=\"2425\">\u0938\u092c\u0938\u0947 \u092a\u0939\u0932\u0947, \u0905\u0902\u0924\u093f\u092e \u091a\u0930\u0923 (Stage) \u092e\u0947\u0902 \u0932\u0915\u094d\u0937\u094d\u092f \u0928\u094b\u0921 (Target Node) \u0938\u0947 \u091c\u0941\u0921\u093c\u0940 \u091c\u093e\u0928\u0915\u093e\u0930\u0940 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0947\u0902\u0964 \u0907\u0938\u0947 <strong data-start=\"2379\" data-end=\"2403\">\u0935\u0930\u094d\u0924\u092e\u093e\u0928 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924<\/strong> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0936\u0941\u0930\u0942 \u0915\u0930\u0947\u0902\u0964<\/p>\n<h4 data-start=\"2427\" data-end=\"2478\"><strong data-start=\"2432\" data-end=\"2478\">Step 3: Compute Minimum Cost for Each Node<\/strong><\/h4>\n<p data-start=\"2479\" data-end=\"2726\">\u0905\u092c \u0939\u092e \u0909\u0932\u094d\u091f\u0947 \u0915\u094d\u0930\u092e (Reverse Order) \u092e\u0947\u0902 \u0917\u094d\u0930\u093e\u092b \u0915\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0928\u094b\u0921 \u0915\u0947 \u0932\u093f\u090f \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924 (Minimum Cost) \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902\u0917\u0947\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0928\u094b\u0921 \u0915\u0947 \u0932\u093f\u090f, \u0939\u092e \u0909\u0928 \u0928\u094b\u0921\u094d\u0938 \u0915\u0947 \u092e\u093e\u0930\u094d\u0917 \u0915\u093e \u092e\u0942\u0932\u094d\u092f\u093e\u0902\u0915\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u0907\u0938 \u0928\u094b\u0921 \u0938\u0947 \u0938\u0940\u0927\u0947 \u091c\u0941\u0921\u093c\u0947 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u0938\u092c\u0938\u0947 \u0915\u092e \u0932\u093e\u0917\u0924 \u0935\u093e\u0932\u0947 \u092e\u093e\u0930\u094d\u0917 \u0915\u094b \u091a\u0941\u0928\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<h4 data-start=\"2728\" data-end=\"2754\"><strong data-start=\"2733\" data-end=\"2754\">Step 4: Recursion<\/strong><\/h4>\n<p data-start=\"2755\" data-end=\"2987\">\u0907\u0938 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0915\u094b \u0905\u0917\u0932\u0947 \u0928\u094b\u0921 \u092a\u0930 \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902 \u0914\u0930 \u092b\u093f\u0930 \u0907\u0938\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092a\u0930 \u0932\u093e\u0917\u0942 \u0915\u0930\u0947\u0902 \u091c\u092c \u0924\u0915 \u0939\u092e \u0938\u094d\u0930\u094b\u0924 \u0928\u094b\u0921 \u0924\u0915 \u0928\u0939\u0940\u0902 \u092a\u0939\u0941\u0901\u091a \u091c\u093e\u0924\u0947\u0964 \u0939\u092e \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0928\u094b\u0921 \u0915\u0947 \u0932\u093f\u090f \u0907\u0938\u0915\u093e \u0938\u092e\u093e\u0927\u093e\u0928 \u0926\u0930\u094d\u091c \u0915\u0930\u0947\u0902\u0917\u0947 \u0914\u0930 \u0905\u0902\u0924 \u092e\u0947\u0902 \u0938\u094d\u0930\u094b\u0924 \u0928\u094b\u0921 \u0938\u0947 \u0932\u0915\u094d\u0937\u094d\u092f \u0928\u094b\u0921 \u0924\u0915 \u0915\u093e \u0938\u092c\u0938\u0947 \u0915\u092e \u0930\u093e\u0938\u094d\u0924\u093e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0947\u0902\u0917\u0947\u0964<\/p>\n<h4 data-start=\"2989\" data-end=\"3040\"><strong data-start=\"2994\" data-end=\"3040\">Step 5: Backtrack to Find the Optimal Path<\/strong><\/h4>\n<p data-start=\"3041\" data-end=\"3248\">\u090f\u0915 \u092c\u093e\u0930 \u091c\u092c \u0939\u092e \u0938\u092c\u0938\u0947 \u0915\u092e \u0932\u093e\u0917\u0924 (minimum cost) \u0915\u093e \u092e\u093e\u0930\u094d\u0917 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930 \u0932\u0947\u0902, \u0924\u094b \u0939\u092e \u0935\u093e\u092a\u0938\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e (Backtracking) \u0926\u094d\u0935\u093e\u0930\u093e \u0909\u0938 \u092e\u093e\u0930\u094d\u0917 \u0915\u094b \u091f\u094d\u0930\u0948\u0915 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u0907\u0938 \u0924\u0930\u0939 \u0938\u0947 \u0939\u092e \u0938\u092c\u0938\u0947 \u0905\u091a\u094d\u091b\u0947 \u092e\u093e\u0930\u094d\u0917 (optimal path) \u0915\u093e \u091a\u092f\u0928 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<h3 data-start=\"3255\" data-end=\"3301\"><strong data-start=\"3259\" data-end=\"3300\">Multistage Graph Problem \u0915\u093e \u090f\u0915 \u0909\u0926\u093e\u0939\u0930\u0923<\/strong>:<\/h3>\n<p data-start=\"3303\" data-end=\"3530\">\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0939\u092e\u0947\u0902 \u090f\u0915 \u092e\u0932\u094d\u091f\u0940\u0938\u094d\u091f\u0947\u091c \u0917\u094d\u0930\u093e\u092b \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0928\u094b\u0921 \u090f\u0915 \u091a\u0930\u0923 (Stage) \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 \u090f\u091c (Edge) \u0915\u0947 \u0935\u091c\u0928 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964 \u0939\u092e\u093e\u0930\u093e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0939\u0948 \u0915\u093f \u0939\u092e <strong data-start=\"3459\" data-end=\"3470\">Stage 1<\/strong> \u0938\u0947 <strong data-start=\"3474\" data-end=\"3485\">Stage 5<\/strong> \u0924\u0915 \u0938\u092c\u0938\u0947 \u0915\u092e \u0932\u093e\u0917\u0924 \u0935\u093e\u0932\u0947 \u0930\u093e\u0938\u094d\u0924\u0947 (path) \u0915\u094b \u0916\u094b\u091c\u0947\u0902\u0964<\/p>\n<h4 data-start=\"3532\" data-end=\"3558\">Graph Representation:<\/h4>\n<ul data-start=\"3560\" data-end=\"3803\">\n<li data-start=\"3560\" data-end=\"3629\"><strong data-start=\"3562\" data-end=\"3573\">Stage 1<\/strong> \u2192 <strong data-start=\"3576\" data-end=\"3587\">Stage 2<\/strong> \u2192 <strong data-start=\"3590\" data-end=\"3601\">Stage 3<\/strong> \u2192 <strong data-start=\"3604\" data-end=\"3615\">Stage 4<\/strong> \u2192 <strong data-start=\"3618\" data-end=\"3629\">Stage 5<\/strong><\/li>\n<li data-start=\"3630\" data-end=\"3803\">\u090f\u091c\u094b\u0902 \u092e\u0947\u0902 \u0935\u091c\u0928 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0939\u0948:\n<ul data-start=\"3662\" data-end=\"3803\">\n<li data-start=\"3662\" data-end=\"3695\">Stage 1 \u0938\u0947 Stage 2 (weight = 2)<\/li>\n<li data-start=\"3698\" data-end=\"3731\">Stage 2 \u0938\u0947 Stage 3 (weight = 3)<\/li>\n<li data-start=\"3734\" data-end=\"3767\">Stage 3 \u0938\u0947 Stage 4 (weight = 1)<\/li>\n<li data-start=\"3770\" data-end=\"3803\">Stage 4 \u0938\u0947 Stage 5 (weight = 4)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h4 data-start=\"3805\" data-end=\"3831\">Step-by-step Process:<\/h4>\n<ol data-start=\"3833\" data-end=\"4553\">\n<li data-start=\"3833\" data-end=\"3944\">\n<p data-start=\"3836\" data-end=\"3944\"><strong data-start=\"3836\" data-end=\"3864\">Stage 5 \u092e\u0947\u0902 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924<\/strong>: \u0932\u0915\u094d\u0937\u094d\u092f \u0928\u094b\u0921 (Stage 5) \u0938\u0947 \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0930\u0947\u0902, \u091c\u0939\u093e\u0902 \u0932\u093e\u0917\u0924 0 \u0939\u094b\u0917\u0940 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u0905\u0902\u0924\u093f\u092e \u0928\u094b\u0921 \u0939\u0948\u0964<\/p>\n<\/li>\n<li data-start=\"3946\" data-end=\"4057\">\n<p data-start=\"3949\" data-end=\"4057\"><strong data-start=\"3949\" data-end=\"3977\">Stage 4 \u092e\u0947\u0902 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924<\/strong>: Stage 4 \u0938\u0947 Stage 5 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0940 \u0932\u093e\u0917\u0924 4 \u0939\u0948, \u0907\u0938\u0932\u093f\u090f Stage 4 \u0915\u0940 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924 4 \u0939\u094b\u0917\u0940\u0964<\/p>\n<\/li>\n<li data-start=\"4059\" data-end=\"4222\">\n<p data-start=\"4062\" data-end=\"4222\"><strong data-start=\"4062\" data-end=\"4090\">Stage 3 \u092e\u0947\u0902 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924<\/strong>: Stage 3 \u0938\u0947 Stage 4 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0940 \u0932\u093e\u0917\u0924 1 \u0939\u0948, \u0914\u0930 Stage 4 \u0938\u0947 Stage 5 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0940 \u0932\u093e\u0917\u0924 4 \u0939\u0948, \u0907\u0938\u0932\u093f\u090f Stage 3 \u0915\u0940 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924 1 + 4 = 5 \u0939\u094b\u0917\u0940\u0964<\/p>\n<\/li>\n<li data-start=\"4224\" data-end=\"4387\">\n<p data-start=\"4227\" data-end=\"4387\"><strong data-start=\"4227\" data-end=\"4255\">Stage 2 \u092e\u0947\u0902 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924<\/strong>: Stage 2 \u0938\u0947 Stage 3 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0940 \u0932\u093e\u0917\u0924 3 \u0939\u0948, \u0914\u0930 Stage 3 \u0938\u0947 Stage 4 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0940 \u0932\u093e\u0917\u0924 5 \u0939\u0948, \u0907\u0938\u0932\u093f\u090f Stage 2 \u0915\u0940 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924 3 + 5 = 8 \u0939\u094b\u0917\u0940\u0964<\/p>\n<\/li>\n<li data-start=\"4389\" data-end=\"4553\">\n<p data-start=\"4392\" data-end=\"4553\"><strong data-start=\"4392\" data-end=\"4420\">Stage 1 \u092e\u0947\u0902 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924<\/strong>: Stage 1 \u0938\u0947 Stage 2 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0940 \u0932\u093e\u0917\u0924 2 \u0939\u0948, \u0914\u0930 Stage 2 \u0938\u0947 Stage 3 \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u0940 \u0932\u093e\u0917\u0924 8 \u0939\u0948, \u0907\u0938\u0932\u093f\u090f Stage 1 \u0915\u0940 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u0932\u093e\u0917\u0924 2 + 8 = 10 \u0939\u094b\u0917\u0940\u0964<\/p>\n<\/li>\n<\/ol>\n<h4 data-start=\"4555\" data-end=\"4577\"><strong data-start=\"4560\" data-end=\"4576\">Final Answer<\/strong>:<\/h4>\n<p data-start=\"4578\" data-end=\"4640\">\u0938\u094d\u0930\u094b\u0924 \u0938\u0947 \u0932\u0915\u094d\u0937\u094d\u092f \u0924\u0915 \u091c\u093e\u0928\u0947 \u0915\u093e \u0938\u092c\u0938\u0947 \u0915\u092e \u0932\u093e\u0917\u0924 \u0935\u093e\u0932\u093e \u0930\u093e\u0938\u094d\u0924\u093e <strong data-start=\"4630\" data-end=\"4636\">10<\/strong> \u0939\u0948\u0964<\/p>\n<h3 data-start=\"4647\" data-end=\"4664\"><strong data-start=\"4651\" data-end=\"4663\">\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937<\/strong>:<\/h3>\n<p data-start=\"4666\" data-end=\"4983\">\u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915 \u092a\u094d\u0930\u094b\u0917\u094d\u0930\u093e\u092e\u093f\u0902\u0917 (DP) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 <strong data-start=\"4702\" data-end=\"4723\">Multistage Graphs<\/strong> \u091c\u0948\u0938\u0940 \u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u092c\u0939\u0941\u0924 \u092a\u094d\u0930\u092d\u093e\u0935\u0940 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0907\u0938\u092e\u0947\u0902 \u0939\u092e \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0939\u0932 \u0915\u093e \u092a\u0941\u0928\u0903 \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0938\u092e\u092f \u0915\u0940 \u092c\u091a\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u0914\u0930 \u0915\u093f\u0938\u0940 \u092d\u0940 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0913\u0935\u0930\u0932\u0948\u092a\u093f\u0902\u0917 \u0915\u0940 \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u094b \u0939\u0932 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938\u0915\u0947 \u0926\u094d\u0935\u093e\u0930\u093e \u0939\u092e \u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0915\u094d\u0930\u092e\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0939\u0932 \u0915\u0930\u0915\u0947 \u0905\u0902\u0924\u093f\u092e \u0938\u092e\u093e\u0927\u093e\u0928 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p data-start=\"4985\" data-end=\"5060\" data-is-last-node=\"\" data-is-only-node=\"\">\u092f\u0926\u093f \u0906\u092a\u0915\u094b \u0907\u0938 \u0935\u093f\u0937\u092f \u092f\u093e \u0915\u093f\u0938\u0940 \u0905\u0928\u094d\u092f \u0939\u093f\u0938\u094d\u0938\u0947 \u092a\u0930 \u0905\u0927\u093f\u0915 \u091c\u093e\u0928\u0915\u093e\u0930\u0940 \u091a\u093e\u0939\u093f\u090f, \u0924\u094b \u0915\u0943\u092a\u092f\u093e \u092c\u0924\u093e\u090f\u0902!<\/p>\n<h3 data-start=\"4985\" data-end=\"5060\"><a href=\"https:\/\/www.bbau.ac.in\/dept\/UIET\/EMER-601%20Operation%20Research%20Queuing%20theory.pdf\" target=\"_blank\" rel=\"noopener\">AAD- Dynamic Programming In Hindi Introduction and General Method multistage graphs Part 1<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.aicte-india.org\/sites\/default\/files\/Vol.%20I_UG.pdf\" target=\"_blank\" rel=\"noopener\">Vol. I_UG.pdf<\/a><\/h3>\n<p data-start=\"0\" data-end=\"290\">Here&#8217;s a clear and simplified explanation of <strong data-start=\"45\" data-end=\"73\">Dynamic Programming (DP)<\/strong> with an introduction to <strong data-start=\"98\" data-end=\"119\">Multistage Graphs<\/strong>, explained in <strong data-start=\"134\" data-end=\"162\">Hindi-English (Hinglish)<\/strong> for easy understanding \u2013 suitable for students preparing for competitive exams or learning <strong data-start=\"254\" data-end=\"289\">Advanced Algorithm Design (AAD)<\/strong>.<\/p>\n<hr data-start=\"292\" data-end=\"295\" \/>\n<h2 data-start=\"297\" data-end=\"354\">\ud83d\udcd8 AAD \u2013 <strong data-start=\"309\" data-end=\"354\">Dynamic Programming Introduction in Hindi<\/strong><\/h2>\n<h3 data-start=\"355\" data-end=\"412\">\ud83e\udde0 Part 1: General Method + Multistage Graph Approach<\/h3>\n<hr data-start=\"414\" data-end=\"417\" \/>\n<h2 data-start=\"419\" data-end=\"458\">\ud83d\udd37 What is Dynamic Programming (DP)?<\/h2>\n<p data-start=\"459\" data-end=\"671\"><strong data-start=\"459\" data-end=\"482\">Dynamic Programming<\/strong> ek technique hai jo kisi <strong data-start=\"508\" data-end=\"527\">complex problem<\/strong> ko chhote-chhote subproblems mein tod kar solve karti hai, <strong data-start=\"587\" data-end=\"637\">aur har subproblem ka solution store karti hai<\/strong> taaki dobara solve na karna pade.<\/p>\n<blockquote data-start=\"673\" data-end=\"736\">\n<p data-start=\"675\" data-end=\"736\"><strong data-start=\"675\" data-end=\"687\">Keyword:<\/strong> <em data-start=\"688\" data-end=\"736\">Overlapping Subproblems + Optimal Substructure<\/em><\/p>\n<\/blockquote>\n<hr data-start=\"738\" data-end=\"741\" \/>\n<h2 data-start=\"743\" data-end=\"790\">\ud83d\udccc Dynamic Programming ke 2 Important Steps:<\/h2>\n<ol data-start=\"792\" data-end=\"911\">\n<li data-start=\"792\" data-end=\"845\">\n<p data-start=\"795\" data-end=\"845\"><strong data-start=\"795\" data-end=\"823\">Recursive Formula Banana<\/strong> (Recurrence relation)<\/p>\n<\/li>\n<li data-start=\"846\" data-end=\"911\">\n<p data-start=\"849\" data-end=\"911\"><strong data-start=\"849\" data-end=\"878\">Memoization ya Tabulation<\/strong> ka use karke problem solve karna<\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"913\" data-end=\"916\" \/>\n<h2 data-start=\"918\" data-end=\"944\">\ud83d\udcca DP Use Hota Hai Jab:<\/h2>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"945\" data-end=\"1143\">\n<thead data-start=\"945\" data-end=\"968\">\n<tr data-start=\"945\" data-end=\"968\">\n<th data-start=\"945\" data-end=\"957\" data-col-size=\"sm\">Condition<\/th>\n<th data-start=\"957\" data-end=\"968\" data-col-size=\"md\">Meaning<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"993\" data-end=\"1143\">\n<tr data-start=\"993\" data-end=\"1064\">\n<td data-start=\"993\" data-end=\"1026\" data-col-size=\"sm\">\ud83d\udd01 <strong data-start=\"998\" data-end=\"1025\">Overlapping Subproblems<\/strong><\/td>\n<td data-col-size=\"md\" data-start=\"1026\" data-end=\"1064\">Same subproblem baar-baar aata hai<\/td>\n<\/tr>\n<tr data-start=\"1065\" data-end=\"1143\">\n<td data-start=\"1065\" data-end=\"1094\" data-col-size=\"sm\">\u2705 <strong data-start=\"1069\" data-end=\"1093\">Optimal Substructure<\/strong><\/td>\n<td data-start=\"1094\" data-end=\"1143\" data-col-size=\"md\">Final solution sub-solutions se ban sakta hai<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"1145\" data-end=\"1148\" \/>\n<h2 data-start=\"1150\" data-end=\"1200\">\ud83c\udfd7\ufe0f <strong data-start=\"1157\" data-end=\"1185\">Multistage Graph Problem<\/strong> \u2013 Introduction<\/h2>\n<h3 data-start=\"1202\" data-end=\"1216\">\u2705 Problem:<\/h3>\n<p data-start=\"1217\" data-end=\"1332\">Multistage graph ek <strong data-start=\"1237\" data-end=\"1269\">directed acyclic graph (DAG)<\/strong> hota hai jisme nodes ko stages mein divide kiya gaya hota hai.<\/p>\n<h3 data-start=\"1334\" data-end=\"1351\">\ud83c\udfaf Objective:<\/h3>\n<p data-start=\"1352\" data-end=\"1421\"><strong data-start=\"1352\" data-end=\"1412\">Stage 1 se Stage N tak ka minimum (ya maximum) cost path<\/strong> nikalna.<\/p>\n<hr data-start=\"1423\" data-end=\"1426\" \/>\n<h2 data-start=\"1428\" data-end=\"1459\">\ud83d\udd22 Multistage Graph Example:<\/h2>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\">yaml<\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre!\"><span class=\"hljs-attr\">Stage 1:<\/span>    <span class=\"hljs-string\">A<\/span><br \/>\n<span class=\"hljs-string\">|<\/span><br \/>\n<span class=\"hljs-attr\">Stage 2:<\/span>   <span class=\"hljs-string\">B<\/span>  <span class=\"hljs-string\">C<\/span><br \/>\n<span class=\"hljs-string\">\/<\/span>    <span class=\"hljs-string\">\\<\/span><br \/>\n<span class=\"hljs-attr\">Stage 3:<\/span>  <span class=\"hljs-string\">D<\/span>      <span class=\"hljs-string\">E<\/span><br \/>\n<span class=\"hljs-string\">\\<\/span>    <span class=\"hljs-string\">\/<\/span><br \/>\n<span class=\"hljs-attr\">Stage 4:<\/span>     <span class=\"hljs-string\">F<\/span> <span class=\"hljs-string\">(Destination)<\/span><br \/>\n<\/code><\/div>\n<\/div>\n<p data-start=\"1598\" data-end=\"1632\">Har edge ke sath cost di hoti hai.<\/p>\n<hr data-start=\"1634\" data-end=\"1637\" \/>\n<h3 data-start=\"1639\" data-end=\"1664\">\ud83d\udcd0 DP Solution Steps:<\/h3>\n<ol data-start=\"1666\" data-end=\"1927\">\n<li data-start=\"1666\" data-end=\"1705\">\n<p data-start=\"1669\" data-end=\"1705\"><strong data-start=\"1669\" data-end=\"1705\">Graph ko stages mein divide karo<\/strong><\/p>\n<\/li>\n<li data-start=\"1706\" data-end=\"1762\">\n<p data-start=\"1709\" data-end=\"1762\"><strong data-start=\"1709\" data-end=\"1762\">Last stage se backward calculate karo (bottom-up)<\/strong><\/p>\n<\/li>\n<li data-start=\"1763\" data-end=\"1847\">\n<p data-start=\"1766\" data-end=\"1783\">Har node ke liye:<\/p>\n<div class=\"contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative bg-token-sidebar-surface-primary\">\n<div class=\"flex items-center text-token-text-secondary px-4 py-2 text-xs font-sans justify-between h-9 bg-token-sidebar-surface-primary dark:bg-token-main-surface-secondary select-none rounded-t-[5px]\">pgsql<\/div>\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-sidebar-surface-primary text-token-text-secondary dark:bg-token-main-surface-secondary flex items-center rounded-sm px-2 font-sans text-xs\"><button class=\"flex gap-1 items-center select-none px-4 py-1\" aria-label=\"Copy\">Copy<\/button><span class=\"\" data-state=\"closed\"><button class=\"flex items-center gap-1 px-4 py-1 select-none\">Edit<\/button><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre!\"><span class=\"hljs-keyword\">cost<\/span>[node] = min (<span class=\"hljs-keyword\">cost<\/span>[next_node] + edge_cost)<br \/>\n<\/code><\/div>\n<\/div>\n<\/li>\n<li data-start=\"1849\" data-end=\"1927\">\n<p data-start=\"1852\" data-end=\"1927\">Final result hoga:<br data-start=\"1870\" data-end=\"1873\" \/><code data-start=\"1876\" data-end=\"1927\">cost[start_node] = minimum cost from start to end<\/code><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"1929\" data-end=\"1932\" \/>\n<h2 data-start=\"1934\" data-end=\"1968\">\ud83d\udca1 General Method (Hindi Mein):<\/h2>\n<ol data-start=\"1970\" data-end=\"2216\">\n<li data-start=\"1970\" data-end=\"2024\">\n<p data-start=\"1973\" data-end=\"2024\"><strong data-start=\"1973\" data-end=\"2001\">Last stage (destination)<\/strong> ka cost <code data-start=\"2010\" data-end=\"2013\">0<\/code> set karo<\/p>\n<\/li>\n<li data-start=\"2025\" data-end=\"2158\">\n<p data-start=\"2028\" data-end=\"2068\"><strong data-start=\"2028\" data-end=\"2059\">Peeche ke stage ke har node<\/strong> ke liye:<\/p>\n<ul data-start=\"2072\" data-end=\"2158\">\n<li data-start=\"2072\" data-end=\"2123\">\n<p data-start=\"2074\" data-end=\"2123\">Check karo uske baad wale connected nodes ke cost<\/p>\n<\/li>\n<li data-start=\"2127\" data-end=\"2158\">\n<p data-start=\"2129\" data-end=\"2158\">Minimum cost path choose karo<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"2159\" data-end=\"2216\">\n<p data-start=\"2162\" data-end=\"2216\"><strong data-start=\"2162\" data-end=\"2176\">Start node<\/strong> tak aate-aatay minimum cost mil jaayegi<\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"2218\" data-end=\"2221\" \/>\n<h2 data-start=\"2223\" data-end=\"2248\">\ud83c\udf93 Real-Life Examples:<\/h2>\n<ul data-start=\"2249\" data-end=\"2312\">\n<li data-start=\"2249\" data-end=\"2273\">\n<p data-start=\"2251\" data-end=\"2273\">Shortest path problems<\/p>\n<\/li>\n<li data-start=\"2274\" data-end=\"2295\">\n<p data-start=\"2276\" data-end=\"2295\">Resource allocation<\/p>\n<\/li>\n<li data-start=\"2296\" data-end=\"2312\">\n<p data-start=\"2298\" data-end=\"2312\">Job scheduling<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2314\" data-end=\"2317\" \/>\n<h2 data-start=\"2319\" data-end=\"2333\">\ud83d\udcda Summary:<\/h2>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"2335\" data-end=\"2536\">\n<thead data-start=\"2335\" data-end=\"2358\">\n<tr data-start=\"2335\" data-end=\"2358\">\n<th data-start=\"2335\" data-end=\"2343\" data-col-size=\"sm\">Topic<\/th>\n<th data-start=\"2343\" data-end=\"2358\" data-col-size=\"md\">Description<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2383\" data-end=\"2536\">\n<tr data-start=\"2383\" data-end=\"2421\">\n<td data-start=\"2383\" data-end=\"2388\" data-col-size=\"sm\">DP<\/td>\n<td data-start=\"2388\" data-end=\"2421\" data-col-size=\"md\">Reuse of subproblem solutions<\/td>\n<\/tr>\n<tr data-start=\"2422\" data-end=\"2477\">\n<td data-start=\"2422\" data-end=\"2441\" data-col-size=\"sm\">Multistage Graph<\/td>\n<td data-col-size=\"md\" data-start=\"2441\" data-end=\"2477\">Staged graph with directed edges<\/td>\n<\/tr>\n<tr data-start=\"2478\" data-end=\"2536\">\n<td data-start=\"2478\" data-end=\"2485\" data-col-size=\"sm\">Goal<\/td>\n<td data-start=\"2485\" data-end=\"2536\" data-col-size=\"md\">Minimum\/maximum cost from source to destination<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"2538\" data-end=\"2541\" \/>\n<p data-start=\"2543\" data-end=\"2558\">Would you like:<\/p>\n<ul data-start=\"2559\" data-end=\"2700\">\n<li data-start=\"2559\" data-end=\"2609\">\n<p data-start=\"2561\" data-end=\"2609\">A <strong data-start=\"2563\" data-end=\"2608\">solved example with full diagram in Hindi<\/strong>?<\/p>\n<\/li>\n<li data-start=\"2610\" data-end=\"2650\">\n<p data-start=\"2612\" data-end=\"2650\"><strong data-start=\"2612\" data-end=\"2634\">Practice questions<\/strong> with solutions?<\/p>\n<\/li>\n<li data-start=\"2651\" data-end=\"2700\">\n<p data-start=\"2653\" data-end=\"2700\">A <strong data-start=\"2655\" data-end=\"2699\">video explanation link or animated chart<\/strong>?<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2702\" data-end=\"2739\" data-is-last-node=\"\" data-is-only-node=\"\">Bataiye, main turant help karta hoon!<\/p>\n<h3 data-start=\"2702\" data-end=\"2739\"><a href=\"https:\/\/www.jntua.ac.in\/gate-online-classes\/registration\/downloads\/material\/a159237887440.pdf\" target=\"_blank\" rel=\"noopener\">AAD- Dynamic Programming In Hindi Introduction and General Method multistage graphs Part 1<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/snscourseware.org\/snsrcas\/files\/CW_5c4d67ab82c50\/Dynamic%20Programming%20-%20Unit%20IV.pdf\" target=\"_blank\" rel=\"noopener\">DYNAMIC PROGRAMING MULTISTAGE GRAPH<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.recw.ac.in\/v1.8\/wp-content\/uploads\/2021\/10\/DAA-UNIT-II.pdf\" target=\"_blank\" rel=\"noopener\">UNIT-2 Part \u2013 I : Greedy Method<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>AAD- Dynamic Programming In Hindi%20%20 Introduction and%20 General Method multistage graphs Part 1 [fvplayer id=&#8221;121&#8243;] \u0921\u0940ynamic Programming (\u0921\u0940\u092a\u0940) &#8211; \u092a\u0930\u093f\u091a\u092f \u0914\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0927\u093f (Multistage Graphs) \u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915 \u092a\u094d\u0930\u094b\u0917\u094d\u0930\u093e\u092e\u093f\u0902\u0917 (Dynamic Programming, DP) \u090f\u0915 \u0936\u0915\u094d\u0924\u093f\u0936\u093e\u0932\u0940 \u0924\u0915\u0928\u0940\u0915 \u0939\u0948 \u091c\u093f\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u091b\u094b\u091f\u0947-\u091b\u094b\u091f\u0947 \u0909\u092a-\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-2804","post","type-post","status-publish","format-standard","hentry","category-algorithm-analysis-and-design"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2804","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2804"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2804\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2804"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2804"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2804"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}