Difference Between Time Complexities of Different Implementations - c++

Question

I have implemented the following algorithms for a competitive programming problem. But for the First method, it gives a TLE (Time Limit Exceeded).
In contrast, the second implementation was accepted as the correct answer even though its time complexity is higher than the first one since it uses the .erase method inside a loop.
May I know the reason why the second implementation was faster than the first
Please refer to the first if statement inside the while(true) loop
First Implementation (TLE)
bool isEmpty (vector<int> a) {
sort(a.begin(), a.end());
return a.size() == 0 || a[a.size() - 1] == 0;
}
int main() {
/* Enter your code here. Read input from STDIN. Print output to STDOUT */
int m, n, l;
cin>>n>>l;
vector<int> k;
vector<int> q;
for (int i=0; i<n; i++){
int ki;
cin>>ki;
k.push_back(ki);
}
cin>>m;
for (int i=0; i<m; i++){
int qi;
cin>>qi;
q.push_back(qi);
}
vector<int> bagsNeeded;
for (int qi:q){
if (qi % l == 0){
bagsNeeded.push_back(qi / l);
}
else {
bagsNeeded.push_back((qi / l) + 1);
}
}
sort(bagsNeeded.begin(), bagsNeeded.end());
int i = 0;
int c = 0;
while(true){
auto itr = lower_bound(bagsNeeded.begin(), bagsNeeded.end(), k[i]);
// Difference between the two implementations is inside this if statement
if (itr == bagsNeeded.end()){
if (isEmpty(bagsNeeded)) break;
else {
int bags = k[i];
int carriable = 0;
int j = bagsNeeded.size() - 1;
c++;
while(bags > 0 && j >= 0){
if (bagsNeeded[j] <= bags){
bags -= bagsNeeded[j];
bagsNeeded[j] = 0;
}
else {
bagsNeeded[j] -= bags;
bags = 0;
}
j--;
}
}
}
else if (itr == bagsNeeded.begin()){
bagsNeeded[0] -= k[i];
if (bagsNeeded[0] == 0){
bagsNeeded.erase(itr);
}
c++;
}
else {
bagsNeeded[itr - bagsNeeded.begin()] -= k[i];
if (bagsNeeded[itr - bagsNeeded.begin()] == 0){
bagsNeeded.erase(itr);
}
c++;
}
i++;
if (i == n){
i = 0;
}
}
cout<<c<<"\n";
return 0;
}
Second Implementation (Accepted)
int main() {
/* Enter your code here. Read input from STDIN. Print output to STDOUT */
int m, n, l;
cin>>n>>l;
vector<int> k;
vector<int> q;
for (int i=0; i<n; i++){
int ki;
cin>>ki;
k.push_back(ki);
}
cin>>m;
for (int i=0; i<m; i++){
int qi;
cin>>qi;
q.push_back(qi);
}
vector<int> bagsNeeded;
for (int qi:q){
if (qi % l == 0){
bagsNeeded.push_back(qi / l);
}
else {
bagsNeeded.push_back((qi / l) + 1);
}
}
sort(bagsNeeded.begin(), bagsNeeded.end());
int i = 0;
int c = 0;
while(true){
auto itr = lower_bound(bagsNeeded.begin(), bagsNeeded.end(), k[i]);
if (itr == bagsNeeded.end()){
if (bagsNeeded.size() == 0) break;
else {
int bags = k[i];
int carriable = 0;
int j = bagsNeeded.size() - 1;
c++;
while(bags > 0 && j >= 0){
if (bagsNeeded[j] <= bags){
bags -= bagsNeeded[j];
bagsNeeded.erase(bagsNeeded.begin() + j);
}
else {
bagsNeeded[j] -= bags;
bags = 0;
}
j--;
}
}
}
else if (itr == bagsNeeded.begin()){
bagsNeeded[0] -= k[i];
if (bagsNeeded[0] == 0){
bagsNeeded.erase(itr);
}
c++;
}
else {
bagsNeeded[itr - bagsNeeded.begin()] -= k[i];
if (bagsNeeded[itr - bagsNeeded.begin()] == 0){
bagsNeeded.erase(itr);
}
c++;
}
i++;
if (i == n){
i = 0;
}
}
cout<<c<<"\n";
return 0;
}