f.nasim's blog

Longest Increasing Subsequence
By f.nasim, 15 years ago, In English
Given an unsorted sequence A={a0,a1,a2,......an-1} of n integers. How do I find the longest increasing sub-sequence of the given sequence. Is there any dynamic programming solution.
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15 years ago, # |
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Just google for "longest increasing subsequence" and you'll find it...
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  • 15 years ago, # ^ |
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    Thanks for the suggestion. Can you suggest me some books, links or tutorials which can improve my understanding of dp or greedy algorithms.
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    • 15 years ago, # ^ |
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      The classical "Introduction to algorithms" has a very good chapter on these subjects; the number of examples is quite small, but they are discussed thoroughly. Other than that, I don't know... I can only suggest the tutorial at Topcoder (I haven't read it, but at least it contains a list of problems)
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15 years ago, # |
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It can be solved very easy with O(N^2) by DP.
Do you need this solution?
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  • 15 years ago, # ^ |
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    it can be solved easy with O(NlogN) by DP.
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    • 15 years ago, # ^ |
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      By sorting all values we had before and binary search on them?
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      • 15 years ago, # ^ |
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        No.
        A[i] - our sequence
        DP[i] = the length of the LIS ending with element numbered i.
        We'll keep track of vector V as follows:
        V[j] = A[k], such that dp[k] = j and k is the largest number < i, satisfying this condition.
        Suppose, we have found out all the dp[1..i-1], and V is up to date. How to find dp[i]? It's easy to find by binary search on V, isn't it? OK, find it, and don't forget to update V.
        Start with dp[1] = 1 and V[1] = A[1] (1-based sequence)
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        • 15 years ago, # ^ |
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          Condition "V[j] = A[k], such that dp[k] = j and k is the largest number < i, satisfying this condition" guaranties us than elements in V is already sorted.
          I mean that when saying "By sorting all values we had before and binary search on them?".
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          • 15 years ago, # ^ |
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            Yes, V remains sorted. But we do NOT sort V. All we do to V is
            1) override elements in V and
            2) append elements to V.
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            • 15 years ago, # ^ |
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              Yeah. I understood at once that we don't need to sort V . ^_^
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15 years ago, # |
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BTW, my favourite problem on LIS is problem J from NEERC 2004 "Joke with turtles".
You can try to solve it here: http://acm.tju.edu.cn/toj/showp2400.html
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