Write in python programming language: The Longest Subsequence Problem is a well-studied problem in Computer Science, where given a sequence of distinct positive integers, the goal is to output the longest subsequence whose elements appear from smallest to largest, or from largest to smallest. For example, consider the sequence S= [9,7,4,10,6,8,2,1,3,5]. The longest increasing subsequence of S has length three ([4,6,8] or [2,3,5]), and the longest decreasing subsequence of S has length five([9,7,4,2,1] or [9,7,6,2,1]). And if we have the sequence S = [531,339,298,247,246,195,104,73,52,31], then the length of the longest increasing subsequence is 1 and the length of the longest decreasing subsequence is 10. Question: Find a sequence with nine distinct integers for which the length of the longest increasing subsequence is 3, and the length of the longest decreasing subsequence is 3. Briefly explain how youconstructed your sequence. Let S be a sequence with ten distinct integers. Prove by Contradiction that there must exist an increasing subsequence of length 4 (or more) or a decreasing subsequence of length 4 (or more). Hint: for each integer k in the sequence you found in the first part, define the ordered pair (x(k), y(k)), where x(k) is the length of the longest increasing subsequence beginning withk, and y(k) is the length of the longest decreasing subsequence beginning withk. You should notice that each of yourordered pairs is different. Explain why this is not a coincidence, i.e., why it is impossible for two different numbers in your sequence to be represented by the same ordered pair (x(k), y(k)).
Write in python programming language:
The Longest Subsequence Problem is a well-studied problem in Computer Science, where given a sequence of distinct positive integers, the goal is to output the longest subsequence whose elements appear from smallest to largest, or from largest to smallest. For example, consider the sequence S= [9,7,4,10,6,8,2,1,3,5]. The longest increasing subsequence of S has length three ([4,6,8] or [2,3,5]), and the longest decreasing subsequence of S has length five([9,7,4,2,1] or [9,7,6,2,1]). And if we have the sequence S = [531,339,298,247,246,195,104,73,52,31], then the length of the longest increasing subsequence is 1 and the length of the longest decreasing subsequence is 10.
Question:
- Find a sequence with nine distinct integers for which the length of the longest increasing subsequence is 3, and the length of the longest decreasing subsequence is 3. Briefly explain how youconstructed your sequence.
- Let S be a sequence with ten distinct integers. Prove by Contradiction that there must exist an increasing subsequence of length 4 (or more) or a decreasing subsequence of length 4 (or more).
Hint: for each integer k in the sequence you found in the first part, define the ordered pair (x(k), y(k)), where x(k) is the length of the longest increasing subsequence beginning withk, and y(k) is the length of the longest decreasing subsequence beginning withk. You should notice that each of yourordered pairs is different. Explain why this is not a coincidence, i.e., why it is impossible for two different numbers in your sequence to be represented by the same ordered pair (x(k), y(k)).
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