Vyhľadávanie kľúčových slov

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Keyword search, inverted index, tries, set intersection

• uvod: slides

Full-text keyword search (Plnotextove vyhladavanie klucovych slov)

• definicia problemu: slides
• zacat diskusiu o tom, co treba indexovat
• Inverted index: for each word list the documents in which it appears.

Ema: 0,2
Emu: 1
ma: 0,1,2
Mama: 1,2
mamu: 0
sa: 2

• What data structures do we need:
  • Map from words to lists (e.g. sorted array, binary search tree, hash)
  • List of documents for each word (array or linked list)

• Budeme navrhovat riesenia a analyzovat zlozitost:
  • cas predspracovania, cas na jeden dotaz
  • Parametre: n pocet roznych slov, N celkovy pocet vyskytov (sucet dlzok zoznamov), m - max. dlzka slova, p - pocet najdenych vyskytov v danom dotaze, sigma - velkost abecedy

 
Utriedené pole            O(m\log n + p)     O(mN\log N)  
Vyhľ. strom               O(m\log n + p)     O(mN\log n) 
Hašovanie - najh. prípad  O(mn + p)          O(mNn)
Hašovanie - priem. prípad O(m + p)           O(mN)
Lex. strom                O(m\log\sigma + p) O(mN\log\sigma)  (neskor)

Tries (lexikograficke stromy)

• A different structure for storing a collection of words
• Edges from each node labeled with characters of the alphabet (0 to |Sigma| edges)
• Each node represents a string obtained by concatenation of characters on the edges
  • If this string is in the collection, the node stores pointer to the list of occurrences (or other data)
• Small alphabet 0..sigma-1: in each node array of pointers child of length sigma-1, list of occurrences (possibly empty)
• Search for a word w of length m in the trie:

node = root;
for(i=0; i<m; i++) {
   node = node->child[w[i]];
   if(! node) return empty_list;
}
return node->list;

Time: O(m)

• Adding an occurrence of word w in document d to the trie:

node = root;
for(i=0; i<m; i++) {
   if(! node->child[w[i]]) {
     node->child[w[i]] = new node;
   }
   node = node->child[w[i]];
}
node->list.add(d)

Time: O(m) (assuming list addition in O(1))

Deletion of a word: what needs to be done (when do we delete a vertex), possibly a problem with deletion from list of occurrences

Summary and implementations

• several words of lengths n1..nz, total length N, alphabet size sigma, query size m
• full array: memory O(N sigma), search O(m), insert all words O(N sigma)
• sorted array of variable size memory O(N), search O(m log sigma), add one word O(n_i log sigma + sigma), all words O(N log sigma + z sigma)
• binary tree with letters as keywords: as before, but add only O(n_i log sigma)
• hashing of alphabet: on average get rid of log sigma but many practical issues (resizing...)

Multiple keywords

• Given several keywords, find documents that contain all of them (intersection, AND)
• Let us consider 2 keywords, one with m documents, other with n, m<n
• Assume that for each keyword we have a sorted array of occurrences (sorted e.g. by document ID, pagerank,...)
• Solution 1: Similar to merge in mergesort O(m+n) = O(n):

i=0; j=0;
while(i<m && j<n) {
  if(a[i]==b[j]) { print a[i]; i++; j++; }
  else if(a[i]<b[j]) { i++; } 
  else { j++; }
}

What if m<<n?

• do m times binary search

for(i=0; i<m; i++) {
   k = binary_search(b, a[i])
   if(a[i]==b[k]) print k;
}

• Rewrite a little bit to get an algorithm that generalizes the two (and possibly works faster)

j = 0; // current position in b
for(i=0; i<m; i++) {
    find smallest k>=j s.t. b[k]>=a[i]; (*)
    if(k==n) { break; }
    j=k;
    if(a[i]==b[j]) {
      print a[i]; 
      j++;
    }
}

How to do (*):

• linear search k=j; while(k<n && b[k]<a[i]) { k++; } the same algorithm as before
• binary search in b[j..n] O(m\log n), faster for very small m
• doubling search/galloping search Bentley, Yao 1976 (general idea), Demaine, Lopez-Ortiz, Munro 2000 (application to intersection, more complex alg.), see also Baeza-Yates, Salinger (nice overview):

step = 1; k = j; k'=k;
while(k<n && b[k]<a[i]) {
   k'=k; k=j+step; step*=2; 
}
binary search in b[k'..k];

• If k=j+x, doubling search needs O(log x) steps
• Overall running time of the algorithm: doubling search at most m times, each time eliminates some number of elements x_i, time log(x1)+log(x2)+...log(xm) where x1+x2+...+xm<=n For what value of x_i is the sum is maximized? Happens for equally sized xi's, each n/m elements, O(m\log(n/m)) time. For constant m logarithmic, for large m linear. Also better than simple merge if different clusters of similar values in each array.
• If more than 2 keywords, iterate (possibly starting with smallest lists - why better?), or extend the algorithm