The problem of finding the Longest Increasing Subsequence (LIS) involves finding the longest subsequence within a given sequence while maintaining the order of increasing values. A subsequence does not need to be contiguous, but the chosen numbers must follow an increasing order.<\/p>\n
Output the length of the longest increasing subsequence.<\/p>\n
6\n10 20 10 30 20 50<\/code><\/pre>\nOutput Example<\/h3>\n4<\/code><\/pre>\nProblem Solving Process<\/h2>\nStep 1: Understand the Problem<\/h3>\n
To understand the problem, let’s examine the sequence. For the given sequence [10, 20, 10, 30, 20, 50]<\/code>, there are several possible increasing subsequences. Among them, the longest increasing subsequence is [10, 20, 30, 50]<\/code>, which has a length of 4. Therefore, the answer is 4.<\/p>\nStep 2: Choose an Algorithm<\/h3>\n
There are various algorithms for finding the longest increasing subsequence, but the most efficient method is to use dynamic programming. This method has a time complexity of O(N^2)<\/code>. I will use this method to solve the problem.<\/p>\nStep 3: Dynamic Programming Solution<\/h3>\nfunction LIS(array) {\n const N = array.length;\n const dp = new Array(N).fill(1); \/\/ Initialize subsequence lengths to 1\n\n for (let i = 1; i < N; i++) {\n for (let j = 0; j < i; j++) {\n if (array[i] > array[j]) {\n dp[i] = Math.max(dp[i], dp[j] + 1);\n }\n }\n }\n\n return Math.max(...dp);\n }\n\n const sequence = [10, 20, 10, 30, 20, 50];\n console.log(LIS(sequence)); \/\/ Output: 4\n <\/code><\/pre>\nExplanation<\/h4>\n
1. Declare the dp<\/code> array to store the length of the longest increasing subsequence for each index. The initial value is set to 1 since each element can form a subsequence by itself.<\/p>\n2. Use two loops to compare indices i<\/code> and j<\/code>. If array[i]<\/code> is greater than array[j]<\/code>, the value of dp[i]<\/code> is updated to the maximum of dp[j] + 1<\/code> and the current value of dp[i]<\/code>. This considers the subsequence that includes array[j]<\/code> as well as array[i]<\/code>.<\/p>\n