Nghiên cứu về độ phức tạp của thuật toán (phần 6)

· Thuật toán, Tin học, Uncategorized

Trích từ trang web :

Using recursion to generate combinatorial objects

One common use of recursion is to implement a backtracking algorithm to generate all possible solutions of a problem. The general idea is to generate the solution incrementally and to step back and try another way once all solutions for the current branch have been exhausted.

This approach is not absolutely universal, there may be problems where it is impossible to generate the solution incrementally. However, very often the set of all possible solutions of a problem corresponds to the set of all combinatorial objects of some kind. Most often it is the set of all permutations (of a given size), but other objects (combinations, partitions, etc.) can be seen from time to time.

As a side note, it is always possible to generate all strings of zeroes and ones, check each of them (i.e. check whether it corresponds to a valid solution) and keep the best found so far. If we can find an upper bound on the size of the best solution, this approach is finite. However, this approach is everything but fast. Don’t use it if there is any other way.

Example 2. A trivial algorithm to generate all permutations of numbers 0 to N – 1.


vector<int> permutation(N);
vector<int> used(N,0);

void try(int which, int what) {
  // try taking the number "what" as the "which"-th element
  permutation[which] = what;
  used[what] = 1;

  if (which == N-1)
    // try all possibilities for the next element
    for (int next=0; next<N; next++)
      if (!used[next])
        try(which+1, next);

  used[what] = 0;

int main() {
  // try all possibilities for the first element
  for (int first=0; first<N; first++)

In this case a trivial lower bound on the time complexity is the number of possible solutions. Backtracking algorithms are usually used to solve hard problems – i.e. such that we don’t know whether a significantly more efficient solution exists. Usually the solution space is quite large and uniform and the algorithm can be implemented so that its time complexity is close to the theoretical lower bound. To get an upper bound it should be enough to check how much additional (i.e. unnecessary) work the algorithm does.

The number of possible solutions, and thus the time complexity of such algorithms, is usually exponential – or worse.

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