aditya1703's blog

Stuck in a problem
By aditya1703, history, 10 months ago, In English

Given a string Str, rearrange Str such that the resultant string T maximizes min (LCS(Str, T) and LCS(Str, reverse(T))).

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10 months ago, # |
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looks like min (LCS(Str, T) and LCS(Str, reverse(T))) cannot exceed longest palindromic subsequence of Str, hence T=Str should work, i may be wrong tho

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    10 months ago, # ^ |
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    Actually it can exceed that.

    Let $$$str = \text{aabb}$$$, $$$T = \text{abba}$$$. Now, $$$\min(\mathrm{LCS}(\text{aabb}, \text{abba}), \mathrm{LCS}(\text{aabb}, \text{abba})) = \min(3, 3) = 3$$$.

    If you choose $$$T = str$$$, you get $$$\min(\mathrm{LCS}(\text{aabb}, \text{aabb}), \mathrm{LCS}(\text{aabb}, \text{bbaa})) = \min(4, 2) = 2$$$.

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      10 months ago, # ^ |
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      yeah, i knew most probably my claim must be wrong

      anyways, please tell how to solve the above problem

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10 months ago, # |
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I think the answer is $$$\lfloor\frac{n + |P|}{2}\rfloor$$$, where $$$|P|$$$ is the maximum palindrome size.

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    10 months ago, # ^ |
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    why?

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    10 months ago, # ^ |
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    I will provide a construction as well as a proof of optimality. First, let's start with a proof of optimality. notice that $$$LCS(Str, reverse(T)) = LCS(reverse(Str), T)$$$. Now, let's say $$$s_1 = LCS(Str, T)$$$ and $$$s_2 = LCS(reverse(Str), T)$$$. Then, we must have that

    $$$|s_1| + |s_2| - |s_1 \cap s_2| \leq n = |T|$$$

    As $$$T$$$ must be an intertwining of the sub-sequences $$$s_1$$$ and $$$s_2$$$ with some positions that are common to both of them, and some positions that belong to neither of them. Here $$$s_1 \cap s_2$$$ denotes the positions in $$$T$$$ that are common to both $$$s_1$$$ and $$$s_2$$$. Now, notice that $$$s_1 \cap s_2 \leq |LCS(s_1, s_2)|$$$. Therefore, we must have that

    $$$|s_1| + |s_2| - |LCS(s_1, s_2)| \leq n$$$

    But $s_1$ is a sub-sequence of $$$Str$$$, and $$$s_2$$$ is a sub-sequence of $$$reverse(Str)$$$. Therefore,

    $$$|LCS(s_1, s_2)| \leq |LCS(Str, reverse(Str))| = |P|$$$

    which is the length of longest palindromic substring. Hence,

    $$$|s_1| + |s_2| - |P| \leq |s_1| + |s_2| - |LCS(s_1, s_2)| \leq n$$$

    Which implies that

    $$$min(|s_1|,|s_2|) \leq \frac{|s_1| + |s_2|}{2} \leq (N+|P|)/2$$$

    I will provide the construction in another comment.

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    10 months ago, # ^ |
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    Consider the longest palindromic substring of $$$Str$$$, call it $$$P = \{ p_1, p_2, ..., p_n \}$$$. Then \we must have that $$$p_i = p_{n+1-i} \forall \ 1\leq i\leq n$$$.

    Now, remove this substring from $$$Str$$$, and assign the left-half substring to $$$s_1$$$ and reverse of the right-half substring to $$$s_2$$$. So, the the assignment will look like

    $$$a_1 \ P_1 \ a_2 \ P_2 ... \ P_k \ a_{k+1} \ b_{n-k+1} \ P_{k+1} \ b_{n-k} \ P_{k+2} \ ... \ P_n \ b_1 $$$

    where $a$ denotes that this position belongs to the left-half substring and $$$b$$$ denotes that it belongs to the right half sub-string. Now, we're going to construct the string $$$T$$$ as follows

    $$$T = a_1 \ reverse(b_1) \ P_1 \ a_2 \ reverse(b_2) \ P_2 \cdots $$$

    Notice that for this sub-string,

    $$$LCS(T, Str) = \lfloor(\frac{n-|P|}{2})\rfloor + |P|$$$

    and

    $$$LCS( T, Reverse(Str)) = \lceil\frac{n-|P|}{2}\rceil + |P|$$$
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