Sufixové stromy a polia

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Intro to suffix trees

• on slides
  • example of generalized suffix tree for S1=aba$, S2=bba$
• at the end: uses of suffix trees (assume we can build in O(n))
  • string matching - build tree, then each pattern in O(m+k) - search through a subtree - its size?
  • find longest word that occurs twice in S - node with highest string depth O(n)
  • find longest word that is at least two strings in input set - node with highest string depth that has two different string represented - how to do details O(n)
  • next all maximal repeats

Maximal pairs

• Definition, motivation on a slide
• example: S=xabxabyabz, max pairs xab-xab, ab-ab (2x)
• Each maximal repeat is an internal node (cannot end in the middle of an edge). Why?
• Let v is a leaf for suffix S[i..n-1], let l(v)=S[i-1] - left character
• Def. Node v is diverse (roznorody) if its subtree contains leaves labeled by at least 2 different values of l(v)
• Theorem. Node v corresponds to a maximal repeat <=> v is diverse
• Proof: => obvious.
• <= Let v corresponds to string A
• v is diverse => xA, yA are substrings in S for some x!=y
• v is a node => Ap, Aq are substrings in S for some p!=q
• If xA is always followed by the same character, e.g. p, then Aq must be preceded by some z!=x, we have pairs xAp, zAq
• Otherwise we have xA is followed by at least 2 different characters, say p and q. If yAp in S, use maximal pairs yAp, xAq, otherwise use maximal pairs xAp, yA?

Summary

• create suffix tree for #S$
• compute l(v) for each leaf
• compute diversity of nodes in a bottom-up fashion
• print all substrings (their indices) corresponding to diverse nodes

Further extensions: find only repeats which are not parts of other repeats, finding all maximal pairs,...

LCA v sufixovych stromoch

• def na slajde
• majme listy zodpovedajuce sufixom S[i..n-1] a S[j..n-1]
• lca(u,v) = najdlhsi spolocny prefix S[i..n-1] a S[j..n-1]
• funguje aj v zovseobecnenych sufixovych stromoch, vtedy sufixy mozu byt z roznych retazcov

Aproximate string matching

• on slides, do running time analysis O(nk)
• wildcards

Counting documents

• list of leaves in DFS order
• each subtree an interval of the list
• separate to sublists: L_i = list of suffixes of S_i in DFS order
• compute lca for each two consecutive members of each L_i
• in each node counter h(v): how many times found as lca
• compute in each node l(v): number of leaves in subtree, s(v): sum of h(v) in subtree
• C(v) = l(v)-s(v)

Proof:

• consider node v and string Si
• if Si occurs k>0 times in subtree of v, it should be counted once in c(v)
  • counted k-times in l(v), k-1 times in s(v)
• if Si not in subtree of v, it should be counted 0 times
  • counted 0 times in both l(v) and s(v)

Printing documents

Muthukrishnan, S. "Efficient algorithms for document retrieval problems." SODA 2002. Strany 657-666.

• Preprocess texts $\{S_{1},\dots ,S_{z}\}$
• Query: which documents contain pattern P?
• We can do O(m+k) where k=number of occurrences of P by simple search in suffix tree
• Using RMQ and print+small, we can achieve O(m+p) where p is the number of documents containing P
• see slides

Searching in suffix arrays

Algorithm 1

//find max i such that T[SA[i]..n] < X 
L = 0; R = n;
while(L < R){
  k = (L + R + 1) / 2;
  h = 0;
  while(T[SA[k] + h] == X[h]) h++;
  if(T[SA[k]+h] < X[h]) L = k;
  else R = k - 1;
}
return L;

• O(log n) iterations
• String comparison O(m)
• Run twice: once for X=P¢, once for X=P#, where ¢ < $ < a < # for all a in the alphabet
• If the first search returns i, the second search j, patter occurrences will be at SA[i+1],...,SA[j] in T
• Counting occurrences O(m log n), printing their positions O(m log n + k), where k is the number of occurrences

Algorithm 2

Keep invariant:

• XL = lcp(T[SA[L]..n], X)
• XR = lcp(T[SA[R]..n], X)

//find max i such that T[SA[i]..n-1]< X 
L = 0; R = n;
XL = lcp(X, T[SA[L]..n]); XR = lcp(X, T[SA[R]..n]);
while(R - L > 1){
  k = (L + R + 1) / 2;
  h = min(XL, XR);
  while(T[SA[k]+h]==X[h]) { h++; }
  if(T[SA[k]+h]<X[h]){ L = k; XL = h; }
  else {  R = k; XR = h; }
}
sequential search in SA[L..R];

Faster on some inputs, but still O(m log n) - see exercise

Algorithm 3

• similar to alg 2, but want to start from max(XL, XR), not min(XL, XR)
• recall that LCP(i,j) = lcp(T[SA[i]..n],T[SA[j]..n])
• assume we know LCP(i,j) for any i,j (later)

Assume XL>=XR in a given iteration (XL<XR is solved similarly. Consider three cases:

• if LCP(L,k) > XL, this imples T[SA[k]..n] < X and therefore set L=k, XL stays the same
• if LCP(L,k) < XL, this imples T[SA[k]..n] >X and therefore set R=k, XR=LCP(L,k)
• if LCP(L,k) = XL, compare X and T[SA[k]..n] starting at XL+1 (which is maximum of XL, XR)

Running time

• Overall O(log n) iterations
• each O(1) + character comparisons
• max(XL,XR) does not decrease
• overall O(log n) character comparisons ends with inequality
• character equality increases max(XL, XR) by 1
• max(XL, XR) <=m
• therefore at most m such comparisons with equality
• running time O(m+\log n)

Which LCP values are needed?

Literatúra

• Dan Gusfield (1997) Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. Cambridge University Press. Prezenčne v knižnici so signatúrou I-INF-G-8.
• Poznámky z predmetu Vyhľadávanie v texte [1] (zčasti písané študentami a nedokončené)