• Assignment Overview
For this project you will implement a dynamic program that solves the longest increasing subsequence problem. A sequence S is a subsequence of another sequence T if you can produce S by removing some items from T (without reordering them). For instance, [0, 2, 5] is a subsequence of [0, 1, 2, 3, 4, 5,], but [5, 2, 0] is not. Your objective is to nd the longest possible subsequence whose items are strictly increasing (that is if i < j then si < sj ).
• Assignment Deliverables
You must turn in completed versions of the following les:
Lis.py
Be sure to use the speci ed le name and to submit your les for grading via Mimir before the project deadline.
• Assignment Speci cations
Your task will be to complete the methods listed below:
verify subseq: Determines whether one sequence is a subsequence of another.
verify increasing: Determines whether a sequence is in increasing order.
find lis: Finds the longest increasing subsequence of the given sequence.
If multiple subsequences each have the maximum possible size you may return any of them.
You may use any of the standard collections discussed in class as well as the built-in functions that act upon them. You may add any additional helper functions that you need.
You may assume that each sequence contains mutually comparable items (they have a < operator), but you cannot make other assumptions about their types. You are not allowed to modify the input sequence in any way. The sequence is indexable (but it might not be a list).
Both of the veri cation methods should run in O(n) time and require only a constant amount of extra space. While a O(n2) solution to the find lis method is readily apparent, your solution should take O(n log n) time and require at most O(n) extra space.
You should include comments where appropriate. In particular, you should describe the method used for computing the LIS. You must also add documentation for each of the above methods (as well as for any helper functions created).
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• Assignment Notes
Points will be deducted if your solution has warnings of any type.
You have been supplied with stubs for the required methods. You must complete these methods to complete the project. You may add more functions than what was provided, but you may not modify the signatures of the provided methods.
You have been provided with some unit tests that can be used to assess your solution. There are additional test cases that will be kept hidden.
It is your responsibility to check for potential edge cases. The supplied unit tests do not account for all possible valid inputs
You may use a wide variety of collections classes. You may want to use list, set, dict, heapq, deque, or defaultdict.
The items in the subsequence must be strictly increasing. Equal items are not allowed. You may not modify the input sequence.
Some of the tests give you a string (character sequence) instead of a list. You may still return a list rather than a string.
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