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I am a bit confused. Somehow I have a problem connecting two problems together. The Closest String problem and the problem of matching with mismatches. They seam to be related but, I fail to see the connection. The Closest String problem is defined as :

Instance: Strings $S_{1},S_{2}...S_{n}$ over alphabet $\Sigma$ of length $L$ each and a non-negative integers $d$ and $n$.

Parameters: $n,d$

 

Question: Is there a string $s$ of length $L$ such that $\delta(s,S_{i})\leq d$ for all $i=1..n$?

Note: $\delta(x,y)\leq d$ is the Hamming distance between $x$ and $y$.

This problem is proven to be NP-complete.

On the other hand we have a problem of matching with mismatches, which is described as:

The problem of string matching with $d$ mismatches consists of finding all occurrences of a pattern of length $m$ in a text of length $n$ such that in at most $d$ positions the text and the pattern have different symbols. In the following, we assume that $0 < d < m$ and $m\leq n$.

Landau and Vishkin gave the first (to my knowledge) efficient algorithm to solve this problem in $O(kn)$ time.

Now my question is:

Is matching with mismatches, or can it be seen, as a special parametrized case of the Closest String problem and how is this connection made?

I am a bit confused. Somehow I have a problem connecting two problems together. The Closest String problem and the problem of matching with mismatches. They seam to be related but, I fail to see the connection. The Closest String problem is defined as :

Instance: Strings $S_{1},S_{2}...S_{n}$ over alphabet $\Sigma$ of length $L$ each and a non-negative integers $d$ and $n$.

Parameters: $n,d$

Question: Is there a string $s$ of length $L$ such that $\delta(s,S_{i})\leq d$ for all $i=1..n$?

Note: $\delta(x,y)\leq d$ is the Hamming distance between $x$ and $y$.

This problem is proven to be NP-complete.

On the other hand we have a problem of matching with mismatches, which is described as:

The problem of string matching with $d$ mismatches consists of finding all occurrences of a pattern of length $m$ in a text of length $n$ such that in at most $d$ positions the text and the pattern have different symbols. In the following, we assume that $0 < d < m$ and $m\leq n$.

Landau and Vishkin gave the first (to my knowledge) efficient algorithm to solve this problem in $O(kn)$ time.

Now my question is:

Is matching with mismatches, or can it be seen, as a special parametrized case of the Closest String problem and how is this connection made?

I am a bit confused. Somehow I have a problem connecting two problems together. The Closest String problem and the problem of matching with mismatches. They seam to be related but, I fail to see the connection. The Closest String problem is defined as :

Instance: Strings $S_{1},S_{2}...S_{n}$ over alphabet $\Sigma$ of length $L$ each and a non-negative integers $d$ and $n$.

Parameters:

1. $n$
2. $d$

Question: Is there a string $s$ of length $L$ such that $\delta(s,S_{i})\leq d$ for all $i=1..n$?

Note: $\delta(x,y)\leq d$ is the Hamming distance between $x$ and $y$;

This problem is proven to be NP-complete.

On the other hand we have a problem of matching with mismatches, which is described as:

The problem of string matching with $d$ mismatches consists of finding all occurrences of a pattern of length $m$ in a text of length $n$ such that in at most $d$ positions the text and the pattern have different symbols. In the following, we assume that $0 < d < m$ and $m\leq n$.

Landau and Vishkin gave the first (to my knowledge) efficient algorithm to solve this problem in O(kn) time.

Now my question is:

Is matching with mismatches, or can it be seen, as a special parametrized case of the Closest String problem and how is this connection made?

Thank you

I am a bit confused. Somehow I have a problem connecting two problems together. The Closest String problem and the problem of matching with mismatches. They seam to be related but, I fail to see the connection. The Closest String problem is defined as :

Instance: Strings $S_{1},S_{2}...S_{n}$ over alphabet $\Sigma$ of length $L$ each and a non-negative integers $d$ and $n$.

Parameters: $n,d$

Question: Is there a string $s$ of length $L$ such that $\delta(s,S_{i})\leq d$ for all $i=1..n$?

Note: $\delta(x,y)\leq d$ is the Hamming distance between $x$ and $y$.

This problem is proven to be NP-complete.

On the other hand we have a problem of matching with mismatches, which is described as:

The problem of string matching with $d$ mismatches consists of finding all occurrences of a pattern of length $m$ in a text of length $n$ such that in at most $d$ positions the text and the pattern have different symbols. In the following, we assume that $0 < d < m$ and $m\leq n$.

Landau and Vishkin gave the first (to my knowledge) efficient algorithm to solve this problem in $O(kn)$ time.

Now my question is:

Is matching with mismatches, or can it be seen, as a special parametrized case of the Closest String problem and how is this connection made?

I am a bit confused. Somehow I have a problem connecting two problems together. The Closest String problem and the problem of matching with mismatches. They seam to be related but, I fail to see the connection. The Closest String problem is defined as :

Instance: Strings $S_{1},S_{2}...S_{n}$ over alphabet $\Sigma$ of length $L$ each and a non-negative integers $d$ and $n$.

Parameters:

1. $n$
2. $d$

Question: Is there a string $s$ of length $L$ such that $\delta(s,S_{i})\leq d$ for all $i=1..n$?

Note: $\delta(x,y)\leq d$ is the Hamming distance between $x$ and $y$;

This problem is proven to be NP-complete.

On the other hand we have a problem of matching with mismatches, which is described as:

The problem of string matching with $d$ mismatches consists of finding all occurrences of a pattern of length $m$ in a text of length $n$ such that in at most $d$ positions the text and the pattern have different symbols. In the following, we assume that $0 < d < m$ and $m\leq n$.

Landau and Vishk

The problem of string matching with d mismatches consists of finding all occurrences of a pattern of length m in a text of length n such that in at most d positions the text and the pattern have different symbols. In the following, we assume that 0<d<m and m≤n.

Landau and Vishkin gave the first (to my knowledge) efficient algorithm to solve this problem in O(kn) time.

Now my question is:

Is matching with mismatches, or can it be seen, as a special parametrized case of the Closest String problem and how is this connection made?

Thank you in gave the first (to my knowledge) efficient algorithm to solve this problem in $O(kn)$ time.

Now my question is:

Is matching with mismatches, or can it be seen, as a special parametrized case of the Closest String problem and how is this connection made?

Thank you

I am a bit confused. Somehow I have a problem connecting two problems together. The Closest String problem and the problem of matching with mismatches. They seam to be related but, I fail to see the connection. The Closest String problem is defined as :

Instance: Strings $S_{1},S_{2}...S_{n}$ over alphabet $\Sigma$ of length $L$ each and a non-negative integers $d$ and $n$.

Parameters:

1. $n$
2. $d$

Question: Is there a string $s$ of length $L$ such that $\delta(s,S_{i})\leq d$ for all $i=1..n$?

Note: $\delta(x,y)\leq d$ is the Hamming distance between $x$ and $y$;

This problem is proven to be NP-complete.

On the other hand we have a problem of matching with mismatches, which is described as:

The problem of string matching with $d$ mismatches consists of finding all occurrences of a pattern of length $m$ in a text of length $n$ such that in at most $d$ positions the text and the pattern have different symbols. In the following, we assume that $0 < d < m$ and $m\leq n$.

Landau and Vishkin gave the first (to my knowledge) efficient algorithm to solve this problem in O(kn) time.

Now my question is:

Is matching with mismatches, or can it be seen, as a special parametrized case of the Closest String problem and how is this connection made?

Thank you