定点的二进制搜索

Question

I need to write a program that being

S = x ₁ , …, x _n

A sequence of integer numbers such that

x ₁ < … < x _n

For every integer number a and every index

1 ≤ i ≤ n

Define

f _a(i) = x _i + a.

Given S and a , tells whether there is some i such that

f _a(i) = i.

I have implemented the following program:

#include <iostream>
#include <vector>
using namespace std;

int fixpoint(const vector<int>& v, int a, int esq, int dre) {
  if(esq > dre) return -1;
  int m = (esq+dre)/2;
  if(v[m]+a == m+1) return m+1;
  if(v[m]+a > m+1) return fixpoint(v, a, m+1, dre);
  if(v[m]+a < m+1) return fixpoint(v, a, esq, m-1);


}


int main() {
  int s;
  while(cin >> s) {
    vector<int> v(s);
    for(int i = 0; i<s; ++i) cin >> v[i];
    int n;
    cin >> n;
    for(int i = 0; i<n; ++i) {
      int a;
      cin >> a;
      int fix = fixpoint(v, a, 0, v.size()-1);
      if(fix == -1) cout << "no fixed point for " << a << endl;
      else cout << "fixed point for " << a << ": " << fix << endl;
    }
    cout << endl;
  }
}

The program must write the first i that satisfies the condition, but if i use the following input:

5
-7 -2 0 4 8
1
0

5
0 1 2 3 4
3
0 -1 1

The output is:

Sequence #1
no fixed point for 0

Sequence #2
no fixed point for 0
no fixed point for -1
fixed point for 1: 3

And the fixed point for 1 in the second sequence should be 1 , because it is the first i that satisfies the condition.

Could you help me?

Answer 1

I see two problems.

First in your function you should invert the branches of the binary search, as being f monotone increasing (as i) if f > i you have to search in the intervall with smaller i.

int fixpoint(const vector<int>& v, int a, int esq, int dre) {
    if ( esq > dre ) return -1;
    int m = (esq + dre)/2;
    int m1 = m + 1;
    int f = v[m] + a;
    if( f == m1)
        return m1;
    else if( f < m1)
        return fixpoint(v, a, m1, dre);
    else
        return fixpoint(v, a, esq, m-1);
}

Second, your second example is ill formed, as f == i for every i. Your function returns 3 only because it's the first index it checks and 3 is as correct an answer as 1.