Problem p00009 (C++) — buggy submission, verdict: Time Limit Exceeded

#include <algorithm>
#include <iostream>
#include <map>
#include <math.h>
#include <queue>
#include <string>
#include <vector>
using namespace std;
#define INF 1 << 20

struct edge {
  int to;
  int cost;
  edge(){};
  edge(int a, int b) {
    to = a;
    cost = b;
  }
};

struct Node {
  int pos;
  int cost;
  Node(){};
  Node(int a, int b) {
    pos = a;
    cost = b;
  }
  bool operator<(const Node &rhs) const { return cost > rhs.cost; }
};

void dijkistra(int st, int TO, int V, vector<vector<edge>> graph) {
  priority_queue<Node> q;
  vector<int> d(V);
  for (int i = 0; i < V; i++) {
    d[i] = INF;
  }
  d[st] = 0;
  q.push(Node(st, 0));
  while (!q.empty()) {
    Node s = q.top();
    q.pop();
    int pos = s.pos;
    for (int i = 0; i < graph[pos].size(); i++) {
      int to = graph[pos][i].to;
      int cost = graph[pos][i].cost;
      if (d[pos] + cost < d[to]) {
        d[to] = d[pos] + cost;
        q.push(Node(to, d[to]));
      }
    }
  }
  cout << d[TO] << endl;
}
int partition(int A[], int p, int r) {
  int x = A[r];
  int i = p - 1;
  for (int j = p; j < r; j++) {
    if (x >= A[j]) {
      i = i + 1;
      swap(A[i], A[j]);
    }
  }
  swap(A[i + 1], A[r]);
  return i + 1;
}
void quicksort(int A[], int p, int r) {
  if (p < r) {
    int q = partition(A, p, r);
    quicksort(A, p, q - 1);
    quicksort(A, q + 1, r);
  }
}

int getNumberOfCoin(int C[], int n, int m) {
  int T[50000];
  for (int j = 0; j <= n; j++) {
    T[j] = INF;
  }
  T[0] = 0;
  for (int i = 0; i < m; i++) {
    for (int j = 0; j <= n; j++) {
      if (j - C[i] >= 0) {
        T[j] = min(T[j], T[j - C[i]] + 1);
      }
    }
  }
  return T[n];
}
int knapsack(int w, int n, int *v, int *ww) {
  int W[101][10001];
  for (int j = 0; j <= w; j++) {
    W[0][j] = 0;
  }
  for (int i = 0; i < n; i++) {
    for (int k = 0; k < n; k++) {
      for (int j = 0; j <= w; j++) {
        if (j - ww[k] < 0) {
          W[i + 1][j] = W[i][j];
        }
        if (j - ww[i] >= 0) {
          W[i + 1][j] = max(W[i][j], W[i][j - ww[k]] + v[k]);
        }
      }
    }
  }
  return W[n][w];
}
int fib(int n) {
  vector<int> v(n + 1);
  v[0] = 1;
  v[1] = 1;
  for (int i = 2; i < n + 1; i++) {
    v[i] = v[i - 2] + v[i - 1];
  }
  return v[n];
}
int isPrime(int x) {
  if (x <= 1)
    return false;
  if (x > 2 && x % 2 == 0)
    return false;
  for (int i = 2; i <= sqrt(x); i++) {
    if (x % i == 0) {
      return false;
    }
  }
  if (x == 2)
    return true;
  return true;
}
int Prime(int x) {
  vector<int> prime(x + 1, 1);
  int num = 2;
  while (num <= sqrt(x)) {
    for (int i = 2; i <= x / num; i++) {
      prime[num * i] = 0;
    }
    num++;
    while (prime[num] == 0)
      num++;
  }
  int count = 0;
  for (int i = 2; i < x + 1; i++) {
    if (prime[i] == 1)
      count++;
  }
  return count;
}
int main() {
  int n;
  vector<int> a;
  while (cin >> n) {
    a.push_back(n);
  }
  for (int j = 0; j < a.size(); j++) {
    cout << Prime(a[j]) << endl;
  }
}

Error: TLE

Fixed submission:

#include <algorithm>
#include <iostream>
#include <map>
#include <math.h>
#include <queue>
#include <string>
#include <vector>
using namespace std;
#define INF 1 << 20

struct edge {
  int to;
  int cost;
  edge(){};
  edge(int a, int b) {
    to = a;
    cost = b;
  }
};

struct Node {
  int pos;
  int cost;
  Node(){};
  Node(int a, int b) {
    pos = a;
    cost = b;
  }
  bool operator<(const Node &rhs) const { return cost > rhs.cost; }
};

void dijkistra(int st, int TO, int V, vector<vector<edge>> graph) {
  priority_queue<Node> q;
  vector<int> d(V);
  for (int i = 0; i < V; i++) {
    d[i] = INF;
  }
  d[st] = 0;
  q.push(Node(st, 0));
  while (!q.empty()) {
    Node s = q.top();
    q.pop();
    int pos = s.pos;
    for (int i = 0; i < graph[pos].size(); i++) {
      int to = graph[pos][i].to;
      int cost = graph[pos][i].cost;
      if (d[pos] + cost < d[to]) {
        d[to] = d[pos] + cost;
        q.push(Node(to, d[to]));
      }
    }
  }
  cout << d[TO] << endl;
}
int partition(int A[], int p, int r) {
  int x = A[r];
  int i = p - 1;
  for (int j = p; j < r; j++) {
    if (x >= A[j]) {
      i = i + 1;
      swap(A[i], A[j]);
    }
  }
  swap(A[i + 1], A[r]);
  return i + 1;
}
void quicksort(int A[], int p, int r) {
  if (p < r) {
    int q = partition(A, p, r);
    quicksort(A, p, q - 1);
    quicksort(A, q + 1, r);
  }
}

int getNumberOfCoin(int C[], int n, int m) {
  int T[50000];
  for (int j = 0; j <= n; j++) {
    T[j] = INF;
  }
  T[0] = 0;
  for (int i = 0; i < m; i++) {
    for (int j = 0; j <= n; j++) {
      if (j - C[i] >= 0) {
        T[j] = min(T[j], T[j - C[i]] + 1);
      }
    }
  }
  return T[n];
}
int knapsack(int w, int n, int *v, int *ww) {
  int W[101][10001];
  for (int j = 0; j <= w; j++) {
    W[0][j] = 0;
  }
  for (int i = 0; i < n; i++) {
    for (int k = 0; k < n; k++) {
      for (int j = 0; j <= w; j++) {
        if (j - ww[k] < 0) {
          W[i + 1][j] = W[i][j];
        }
        if (j - ww[i] >= 0) {
          W[i + 1][j] = max(W[i][j], W[i][j - ww[k]] + v[k]);
        }
      }
    }
  }
  return W[n][w];
}
int fib(int n) {
  vector<int> v(n + 1);
  v[0] = 1;
  v[1] = 1;
  for (int i = 2; i < n + 1; i++) {
    v[i] = v[i - 2] + v[i - 1];
  }
  return v[n];
}
int isPrime(int x) {
  if (x <= 1)
    return false;
  if (x > 2 && x % 2 == 0)
    return false;
  for (int i = 2; i <= sqrt(x); i++) {
    if (x % i == 0) {
      return false;
    }
  }
  if (x == 2)
    return true;
  return true;
}
int main() {
  int n;
  vector<int> a;
  while (cin >> n) {
    a.push_back(n);
  }
  for (int j = 0; j < a.size(); j++) {
    int count = 0;
    for (int i = 2; i <= a[j]; i++) {

      if (isPrime(i)) {
        count++;
      }
    }
    cout << count << endl;
  }
}