Let’s solve the problem of finding the length of the longest increasing subsequence in a given array. An increasing subsequence is a subarray of elements from the array that have different indices and are sorted in ascending order. For example, for the array Write a method that returns the length of the longest increasing subsequence in a given integer array Print the length of the longest increasing subsequence.<\/p>\n Among the various algorithms for finding the longest increasing subsequence, the most commonly used method is Dynamic Programming. This approach breaks the problem into smaller subproblems, solving each one and using the results to solve the overall problem.<\/p>\n[10, 22, 9, 33, 21, 50, 41, 60, 80]<\/code>, the longest increasing subsequence is [10, 22, 33, 50, 60, 80]<\/code>, which has a length of 6.<\/p>\nProblem Definition<\/h2>\n
arr<\/code>.<\/p>\nInput<\/h3>\n
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n<\/code>. (1 \u2264 n \u2264 1000)<\/li>\nn<\/code> integers. Each integer is -10^6 \u2264 arr[i] \u2264 10^6<\/code>.<\/li>\n<\/ul>\nOutput<\/h3>\n
Example<\/h2>\n
Input<\/h4>\n
9\n10 22 9 33 21 50 41 60 80<\/code><\/pre>\nOutput<\/h4>\n
6<\/code><\/pre>\nSolution Method<\/h2>\n
Step-by-Step Explanation<\/h3>\n
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arr<\/code> is n<\/code>. We initialize a dynamic programming array dp<\/code> to store the length of the longest increasing subsequence up to each index i<\/code>. Initially, we set the value to 1 for each element, as each element can form a subsequence by itself.<\/li>\ni<\/code> and j<\/code>. Here, i<\/code> should be greater than j<\/code>, and if arr[i]<\/code> is greater than arr[j]<\/code> (i.e., satisfying the increasing condition), we update the value of dp[i]<\/code>. Specifically, we set dp[i] = max(dp[i], dp[j] + 1)<\/code>. This means adding 1 to the length of the sequence that includes j<\/code>.<\/li>\ndp<\/code> array and return it.<\/li>\n<\/ol>\nC# Implementation<\/h2>\n
using System;\n\nclass Program\n{\n static void Main()\n {\n int n = int.Parse(Console.ReadLine());\n int[] arr = Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);\n\n int[] dp = new int[n];\n for (int i = 0; i < n; i++)\n {\n dp[i] = 1; \/\/ Each element can form at least 1 subsequence\n }\n\n for (int i = 1; i < n; i++)\n {\n for (int j = 0; j < i; j++)\n {\n if (arr[i] > arr[j] && dp[i] < dp[j] + 1)\n {\n dp[i] = dp[j] + 1;\n }\n }\n }\n\n int maxLength = dp[0];\n for (int i = 1; i < n; i++)\n {\n if (dp[i] > maxLength)\n {\n maxLength = dp[i];\n }\n }\n\n Console.WriteLine(maxLength);\n }\n}\n<\/code><\/pre>\nTime Complexity<\/h2>\n