I am learning suffix arrays. I understood the O(nlogn) implementation of suffix array. But I am not being able to understand LCP calculation. Could someone explain how to calculate LCP from suffix arrays? Thanks in advance.

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I am learning suffix arrays. I understood the O(nlogn) implementation of suffix array. But I am not being able to understand LCP calculation. Could someone explain how to calculate LCP from suffix arrays? Thanks in advance.

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Kasai's algorithm is pretty easy and works in

O(n).Let's look at the two continuous suffixes in the suffix array. Let their indexes in suffix array be

i_{1}andi_{1}+ 1. If theirlcp> 0, then if we delete first letter from both of them. We can easily see that new strings will have the same relative order. Also we can see that lcp of new strings will be exactlylcp- 1.Let's now look at the string wich we have got from the

isuffix by deleting its first character. Obviously it is some suffix of the string too. Let its index bei_{2}. Let's look at the lcp of suffixesi_{2}andi_{2}+ 1. We can see that it'slcpwill be at least already mentionedlcp- 1. This is associated with certain properties of lcp array, in particular, thatlcp(i,j) =min(lcp_{i},lcp_{i + 1}, ...,lcp_{j - 1}).And finally let's make the algorithm based on the mentioned above. We will need an additional array

rank[n], wich will contain the index in the suffix array of the suffix starting in indexi. Firstly we should calculate the lcp of the suffix with indexrank[0]. Then let's iterate through all suffixes in order in which we meet them in the string and calculatelcp[rank[i]] in naive way, BUT starting it fromlcp[rank[i- 1]] - 1. Easy to see that now we haveO(n) algorithm because on the each step ourlcpdecreasing not more than by 1 (except the case whenrank[i] =n- 1).Implementation:

There is also a way to build it in

O(nlogn) with a segment tree described on the e-maxx, but in my opinion it is much harder and slower.And there's another

O(NlogN) algorithm which is much more intuitive: find LCP of each pair of consecutive suffixes using binary search and hashes. However, I'm not really sure what's easier and faster to implement: this method or Kasai's (and several other guys')Yes, but hashes are evil, we don't want use them :)

Why exactly? Due to anti-hash tests? Try hashing mod 2^64 and a randomly chosen reasonably small prime.

I just dislike hashes and trying to avoid them almost always when I have such opportunity. Also, double hashing is quite slow :(

That's why 2^64 — what makes double hashing slow is especially the modulo operation, if you use just long long, then it's fast, but it's easy to make anti-hash tests, which is what the other part (mod smaller prime) takes care of while retaining decent runtime.

Interesting trick. But I still dislike hashes :)

Relevant username

where can I learn hashes ? Any tutorial or blog ?

That was helpful.I got the idea. Thanks a lot.But Could you explain why lcp(i, j) = min(lcpi, lcpi + 1, ..., lcpj-1). this property is true?

write a suffix array + lcp in a paper, you will notice that property.

For example, we know

lcp(i,j- 1). Obviously iflcp[j- 1] <lcp(i,j- 1) thenlcp(i,j) =lcp[j- 1], otherwiselcp(i,j) =lcp(i,j- 1), i.e.lcp(i,j) =min(lcp(i,j- 1),lcp[j- 1]). Now we could rewritelcp(i,j- 1) in this formula in the same way and get what we get.It is clear now. Thanks :)

Do you mean lcp(1,4) in abcabcd = min(lcp(1,2),lcp(2,3),lcp(3,4)) = min(0,0,0) = 0 ?

here lcp(i,j) means lcp(suffix from sa[i],suffix from sa[j]) ,where sa=>suffix array

what is rank array....please explain a little more !!!!!

rank array is just a reverse function for suffix array

what if we used j=sa[i]+1;

??/

The idea is following thing. Let s='abcdefghi' Then LCP(0)=lcp(abcdefghi, bcdefghi) =|bcdefghi|=8

And then we cut one character from left from each string and move one position forward and calculate LCP(1). It would be LCP(1)=|cdefghi|=7=LCP(0)-1

So if j=sa[i] +1 we can't say that LCP(i) =k-1, we should check prefix fully.

lcp(abcdefghi, bcdefghi) =|bcdefghi|=8

wat

lcp(abcdefghi, bcdefghi) =|bcdefghi|=0

right??

Sorry, I did a huge mistake.

I thought actually about the following.

We have string suff[i] ='abcdefghi'. Let LCP(i) =X.

Then for suff[j] ='bcdefghi' LCP(j) >=X-1.

I thought about something like example of cutting It should be something like LPC(0)=lcp(abcdhhh, abcdiii) = |abcd|=4 => lcp(bcdhhh,bcdiii)=|bcd|=LCP(0)-1

Maybe that is some sort if mild brain damage

haha

Adamant mentioned that lcp(i, j) =min(LCP(k)) {k=[i:j)} That is because we can take lcp between cur and next suffixes in the string, not in the suffix array, as a lower bound.

I spent some time trying to run algorithm in my imagination.

Despite algorithm is very confusing and it seems that it can't work it works actually. I tested it on my machine today morning.

We consider i-th suffix of original string.

Find where in suffix array i-th suffix lies and then find where lexicographically next suffix lies in original string. Then compare characters one by one and count of characters successfully matched is LCP.

And for some reason it is legal to start comparing not from first character but from k-th.

Why? What is k? k is LCP of rank[i-1]. And i-th suffix is also a suffix of (i-1)-suffix.

Moreover, suff(i-1)=c+suff(i).

So if k >1 then x=y+z, where |y|=k-1.

Postfix increment is bad!

You're bad!

I saw postfix increment implementation in GCC C++.

It was like this. [] [int &x] {int y=++x; return y;} So, postfix uses prefix form as a subroutine and therefore is slower.

UPD: Postfix increment does unnecessarily job. So it consumes additional energy without a real purpose. Energy is not infinite you know. You are bringing us closer to the

heat death of the Universe.

UPD2: I understood. You are talking about compiler optimization. It is almost certain that compiler will remove postfix and put prefix form.

Nonetheless, it requires additional time, additional energy and thus additional heating.

Kill yourself, then you can slow down heat death of universe.

It might help or might not. Now if kill myself then everybody would think I have killed myself because of codeforces.com user advised me to do so. It may trigger a huge response. Or may not. The question is pretty complex. We should discuss that question very hard.

Can you please explain the notation used above. I am having doubt in lcp(i, j) and lcp_i

https://www.hackerrank.com/challenges/find-strings/topics

http://www.mi.fu-berlin.de/wiki/pub/ABI/RnaSeqP4/suffix-array.pdf

Very well explained. Thanks!