PROBLEM LINK:
Div1
Div2
Practice
Setter-Kirill Gulin
Tester-Ivan Safonov
Editorialist-Abhishek Pandey
DIFFICULTY:
EASY
PRE-REQUISITES:
Sieve of Erastothenes, Binary Search
PROBLEM:
Given an array $A$ of $N$ integers, we have to maximize the GCD of all elements of the array. To do that, you can select any array element $A_i$ which is divisible by any number ${d}^{2}$, divide it by ${d}^{2}$ and multiply any other array element by $d$.
QUICK EXPLANATION:
Key to AC- Realize that manipulation of powers of one prime is independent of powers of other primes. Perceiving that this question is more about manipulating prime and their powers rather than some complex mathematics theorem was important.
Make a list of primes, and smallest prime divisors of all numbers in range using sieve. Now, for each prime $p_i$, keep a note of how many array elements are divisible by it, and what is the power of $p_i$ in their prime factorization form. Now whats left is finding what power of this prime can we maximum achieve in final answer using above operations - we can use BSH (Binary Search Heuristics) for this. Repeat for all primes and get an AC :).
EXPLANATION:
This editorial is divided into $2$ sections. The first section will highlight on the pre-processing required, and the second section will feature how to calculate answers using Binary Search Heuristics (BSH).
Note that, I assume you all realize that, using operation of same number (i.e. dividing the same element by ${d}^{2}$ and increasing it by $d$) is useless. It will just reduce the prime power of that prime, which we might want to maximize for GCD.
1. Pre-processing-
It seems kind of obvious that we might need factorization of elements. Also, usually such problems do involve manipulation of prime numbers. Hence the first step is to use sieve of erastothenes to pre-calculate-
- Prime numbers in range $[1,{10}^{6}]$
- The smallest prime factor of a given number. This ensures $O(LogN)$ factorization of elements.
The observation to make is that manipulation of exponents of prime factors of $A_i$ are independent. We can choose a prime $d$ every time for our operation so that only the chosen prime gets affected. If this doesnt ring a bell, you can refer to tab below.
View Content
Hide Content
Now, observe this thing by thinking in terms of prime numbers. You'd notice that, increasing or decreasing power of one prime by above operation has no effect on power of other primes. In other words, manipulation of primes is independent of one another. Changing exponent of one prime factor of $A_i$ will have no effect on exponent of other prime factors if $d$ is prime. Only the chosen prime power will get affected.
This is important as it helps in realizing that, increasing GCD prime by prime is the way to go. It gives an intuition that - "Ok, initialize GCD=1 for now. Lets pick a prime. Let me see what is the maximum possible power of it I can get for GCD of array using above operations. Multiply answer by it. Lets repeat this until we get answer."
What is the next thing we need if we are to do such manipulations? We'd need the prime numbers and their exponents for all elements of array. I specifically liked the tester's method of storing it (to use it later) and I think it can use some credit :) .Its in tab below for reference.
View Content
Hide Content
for (int i = 0; i < n; i++)
{
while (a[i] > 1)//Factorize a[i]
{
int p = mn[a[i]];
int k = 0;
while (a[i] % p == 0)
{
a[i] /= p;
k++;
}
id[p].push_back(k);//Some array element as an exponent k for prime p.
}
}
This is especially nice because now you can also find how many array elements are divisible by $p$ in $O(1)$ time by $idx[p].size()$.
2. Binary Search Heuristics-
I hope that the intuition so far, especially that of manipulating prime by prime, is clear to you.
Now, for each prime, we will Binary Search on the highest possible power achievable (of that prime) in the final GCD. It is actually quite simple.
Lets say we have to check if a power $k$ is achievable for some prime $p$. How do we do it? Recall that using our operation, we can reduce the power of $p$ in some $A_i$ by $2$ and increase power in another $A_i$ by $1$. In a way, its like "Sacrifice $2$ powers of some $A_i$ for $1$ power in another element." Since we aim to calculate Gcd, our aim is to maximize the minimum power $p$.
Now, we have to check if we can make exponent of $p$ equal to $k$ in GCD. Hence, it makes sense to calculate how much "excess exponent" of $p$ can we take from "rich $A_i$", and how much exponent do we need to "poor $A_i$" to make exponent of $p$ equal to $k$ in GCD.
It can be elegantly done if you have pre-processed the exponents and primes like tester did. The tester used a vector idx[100001] where $idx[p]$ had "Exponents of $p$ which are $>0$ among all elements of array." Hence, he can simply loop over $idx[p]$ and calculate the excess and deficit exponents. (Do not forget that we are currently checking if we can achieve an exponent of $k$ in GCD).
Now, things are quite simple. If we have gained sufficient exponent from "Rich $A_i$" to give to "Poor $A_i$", then a power of at least $k$ is achievable in the final GCD. We check for the highest power achievable using binary search (proof in tab below).
View Content
Hide Content
Say, we are checking for power $k$. Either we can achieve it, or we cannot achieve it. If we achieve it, we can discard checking for ALL powers in range $[0,k-1]$ as if exponent $k$ is achievable, then exponents $<k$ are obviously achievable as well. If we cannot achieve $k$, then any exponent greater than $k$ is also not achievable. Hence, the condition of Binary Search holds which allows its use.
The code to implement this is given below :)
View Content
Hide Content
for (int x : prime)//For all primes
if (!id[x].empty()) // we can do operations for all primes independently, x - current prime
{
int l = 0;
int r = 0;
for (int k : id[x]) r = max(r, k + 1);//Find upper bound. r= Max exponent+1
int num0 = n - id[x].size();//num0 = number of array elements with 0 exponent of x
while (r - l > 1) // binary search for maximum possible power of x in answer
{
int h = (l + r) >> 1; // check x^h
int now = 0;
for (int k : id[x])
if (k >= h)//Find excess power
now += ((k - h) >> 1); // sum, that we can use
for (int k : id[x])
if (k < h)//Find deficit power.
now -= (h - k); // sum, that we need
now -= num0 * h; // sum, that we need
//-num0 *h accounts for deficit exponent for Ai where exponent of x is 0.
if (now >= 0)
l = h;
else
r = h;
}
for (int i = 0; i < l; i++) ans *= x;//multiply answer by x^l
}
SOLUTION:
The solutions are also pasted in tab below for you guys to see. Please refer to them in case links dont work. @admin may need time to link solutions up :)
Setter
View Content
Hide Content
#include <bits/stdc++.h>
using namespace std;
const int maxl = (int)1e6 + 10;
int lp[maxl];
int cnt[maxl];
vector <int> d[maxl];
bool can(int n, int md, const vector <int> &d) {
int need = md * (n - (int)d.size());
for(int x: d) {
if (x < md) need += md - x;
else need -= (x - md) / 2;
}
return need <= 0;
}
int main() {
lp[1] = 1;
for(int i = 2; i < maxl; ++i) {
if (lp[i] > 0) continue;
for(int j = i; j < maxl; j += i) {
if (lp[j] == 0) lp[j] = i;
}
}
ios_base::sync_with_stdio(0);
int t;
cin >> t;
while(t--) {
int n;
cin >> n;
memset(cnt, 0, sizeof cnt);
for(int i = 1; i < maxl; ++i) {
d[i].clear();
}
for(int i = 0; i < n; ++i) {
int x;
cin >> x;
while(x > 1) {
int p = lp[x];
int c = 0;
while(x > 1 && x % p == 0) x /= p, c++;
d[p].push_back(c);
cnt[p] += c;
}
}
int gcd = 1;
for(int p = 2; p < maxl; ++p) {
if (cnt[p] < n) continue;
int lf = -1, rg = 20;
while(rg - lf > 1) {
int md = (lf + rg) / 2;
if (md == -1 || can(n, md, d[p])) {
lf = md;
} else rg = md;
}
//lf now is the maximum power of p in gcd
while(lf--) gcd *= p;
}
cout << gcd << '\n';
}
return 0;
}
Tester
View Content
Hide Content
#include <bits/stdc++.h>
#include <cassert>
using namespace std;
typedef long long ll;
typedef long double ld;
// read template
long long readInt(long long l,long long r,char endd){
long long x=0;
int cnt=0;
int fi=-1;
bool is_neg=false;
while(true){
char g=getchar();
if(g=='-'){
assert(fi==-1);
is_neg=true;
continue;
}
if('0'<=g && g<='9'){
x*=10;
x+=g-'0';
if(cnt==0){
fi=g-'0';
}
cnt++;
assert(fi!=0 || cnt==1);
assert(fi!=0 || is_neg==false);
assert(!(cnt>19 || ( cnt==19 && fi>1) ));
} else if(g==endd){
assert(cnt>0);
if(is_neg){
x= -x;
}
assert(l<=x && x<=r);
return x;
} else {
assert(false);
}
}
}
string readString(int l,int r,char endd){
string ret="";
int cnt=0;
while(true){
char g=getchar();
assert(g!=-1);
if(g==endd){
break;
}
cnt++;
ret+=g;
}
assert(l<=cnt && cnt<=r);
return ret;
}
long long readIntSp(long long l,long long r){
return readInt(l,r,' ');
}
long long readIntLn(long long l,long long r){
return readInt(l,r,'\n');
}
string readStringLn(int l,int r){
return readString(l,r,'\n');
}
string readStringSp(int l,int r){
return readString(l,r,' ');
}
// end
const int M = 1e5 + 239;
const int X = 1e6 + 239;
int n, a[M], mn[X];
vector<int> id[X];
vector<int> prime;
void solve()
{
n = readIntLn(1, (int)(1e5));
for (int i = 0; i < n; i++)
{
if (i < n - 1)
a[i] = readIntSp(1, (int)(1e6));
else
a[i] = readIntLn(1, (int)(1e6));
}
for (int x : prime) id[x].clear();
for (int i = 0; i < n; i++)
{
while (a[i] > 1)
{
int p = mn[a[i]];
int k = 0;
while (a[i] % p == 0)
{
a[i] /= p;
k++;
}
id[p].push_back(k);
}
}
int ans = 1;
for (int x : prime)
if (!id[x].empty()) // we can do operations for all primes independently, x - current prime
{
int l = 0;
int r = 0;
for (int k : id[x]) r = max(r, k + 1);
int num0 = n - id[x].size();
while (r - l > 1) // binary search for maximum possible power of x in answer
{
int h = (l + r) >> 1; // check x^h
int now = 0;
for (int k : id[x])
if (k >= h)
now += ((k - h) >> 1); // sum, that we can use
for (int k : id[x])
if (k < h)
now -= (h - k); // sum, that we need
now -= num0 * h; // sum, that we need
if (now >= 0)
l = h;
else
r = h;
}
for (int i = 0; i < l; i++) ans *= x;
}
cout << ans << "\n";
}
int main()
{
memset(mn, -1, sizeof(mn));
for (int i = 2; i < X; i++)
if (mn[i] == -1)
{
prime.push_back(i);
for (int j = i; j < X; j += i)
if (mn[j] == -1)
mn[j] = i;
}
int T = readIntLn(1, 5);
while (T--) solve();
assert(getchar() == -1);
return 0;
}
$Time$ $Complexity=$ $O((K+N)logN)$ where $K$ is number of primes in given constraints of $A_i$
CHEF VIJJU'S CORNER :D
1. Is this problem solvable if we raise constraint of $A_i$ to $1 \le A_i \le {10}^{9}$?
2. Usually, many problems using GCD need knowledge of Mobius Function, Euler Totient functions etc. There is a very good blog describing them here
3.Linear Sieve deserves a mention here. :)
4. Prove that you can get an AC by this approach-
View Content
Hide Content
Another possible solution:
Let $g$ be gcd of initial numbers. Make $a[i] = a[i] / g$. Now, New gcd = 1.
Now consider only primes until $100$ to find the new solution. We will not be able to add any prime $> 100$ after the 2nd step. let $d$ be a prime $> 100$ such that ${d}^{2}$ is divisible by $A[i]$, then if we divide this $A[i]$ by ${d}^{2}$ then $A[i]$ will no longer be divisible by $d$. Let the answer for $A[i]/g$ be denoted by $ans$
Final solution: ans * g.
For above approach, solve the following-
- This approach considers only first $100$ prime number. Validate and comment on its correctness. "Proof by AC" not accepted.
- Can this approach be extended to $A_i \le {10}^{9}$. Why/Why not?
- What is the lower bound on number of primes to consider for making it useful for $A_i <{10}^{9}$ and $A_i <{10}^{18}$?
Credits for this question to - @codebreaker123