Question

Given a sequence such as S = {1,8,2,1,4,1,2,9,1,8,4}, I need to find the minimal-length subsequence that contains all element of S (no duplicates, order does not matter). How do find this subsequence in an efficient way?

Note: There are 5 distinct elements in S: {1,2,4,8,9}. The minimum-length subsequence must contain all these 5 elements.

Answer 1

Algorithm:

First, determine the quantity of different elements in the array – this can be easily done in linear time. Let there be k different elements.

Allocate an array cur of size 10^5, each showing how much of each element is used in current subsequence (see later).

Hold a cnt variable showing how many different elements are there currently in the considered sequence. Now, take two indexes, begin and end and iterate them through the array the following way:

1. initialize cnt and begin as 0, end as -1 (to get 0 after first increment). Then while possible perform follows:

2. If cnt != k:

   2.1. increment end. If end already is the end of array, then break. If cur[array[end]] is zero, increment cnt. Increment cur[array[end]].

   Else:

   2.2 {

   Try to increment the begin iterator: while cur[array[begin]] > 1, decrement it, and increment the begin (cur[array[begin]] > 1 means that we have another such element in our current subsequence). After all, compare the [begin, end] interval with current answer and store it if it is better.

   }

After the further process becomes impossible, you got the answer. The complexity is O(n) – just passing two interators through the array.

Implementation in C++:

    #include <iostream>

using namespace std;

const int MAXSIZE = 10000;

int arr[ MAXSIZE ];
int cur[ MAXSIZE ];

int main ()
{
   int n; // the size of array
   // read n and the array

   cin >> n;
   for( int i = 0; i < n; ++i )
      cin >> arr[ i ];

   int k = 0;
   for( int i = 0; i < n; ++i )
   {
      if( cur[ arr[ i ] ] == 0 )
         ++k;
      ++cur[ arr[ i ] ];
   }

   // now k is the number of distinct elements

   memset( cur, 0, sizeof( cur )); // we need this array anew
   int begin = 0, end = -1; // to make it 0 after first increment
   int best = -1; // best answer currently found
   int ansbegin, ansend; // interval of the best answer currently found
   int cnt = 0; // distinct elements in current subsequence

   while(1)
   {
      if( cnt < k )
      {
         ++end;
         if( end == n )
            break;
         if( cur[ arr[ end ]] == 0 )
            ++cnt; // this elements wasn't present in current subsequence;
         ++cur[ arr[ end ]];
         continue;
      }
      // if we're here it means that [begin, end] interval contains all distinct elements
      // try to shrink it from behind
      while( cur[ arr[ begin ]] > 1 ) // we have another such element later in the subsequence
      {
         --cur[ arr[ begin ]];
         ++begin;
      }
      // now, compare [begin, end] with the best answer found yet
      if( best == -1 || end - begin < best )
      {
         best = end - begin;
         ansbegin = begin;
         ansend = end;
      }
      // now increment the begin iterator to make cur < k and begin increasing the end iterator again
      --cur[ arr[ begin]];
      ++begin;
      --cnt;
   }

   // output the [ansbegin, ansend] interval as it's the answer to the problem

   cout << ansbegin << ' ' << ansend << endl;
   for( int i = ansbegin; i <= ansend; ++i )
      cout << arr[ i ] << ' ';
   cout << endl;

   return 0;
}