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| 1 | + |
| 2 | +/** |
| 3 | +
|
| 4 | +Range Prime Factorization (OPTIMIZED) |
| 5 | +=============================== |
| 6 | +
|
| 7 | +Bruteforce Range Factorization |
| 8 | +------------------------------ |
| 9 | +Segmented Sieve + Bruteforce prime factorization |
| 10 | +O( (R-L+1) log(logR) ) + O( (R-L) * (sqrt(N)/log(sqrt(N)) * log(N)) ) |
| 11 | +
|
| 12 | +For larger N value (like 1e12) it will get TLE |
| 13 | +
|
| 14 | +Optimization Idea 1 (will eventually fail) : |
| 15 | +-------------------------------------------- |
| 16 | +using SPF (smallest prime factors) we could do prime factorization in logN , but we can't find all spf in [1,1e12] in memory limit |
| 17 | +will optimize for lower values of N |
| 18 | +
|
| 19 | +Optimization Idea 2 : |
| 20 | +-------------------- |
| 21 | +modifying segmented sieve to get all the prime factors of a number without the powers |
| 22 | +and later use this information to Factorize in logN |
| 23 | +As we don't need to iterate upto logN for every Number , the factorization complexity will reduce to logN |
| 24 | +
|
| 25 | +**/ |
| 26 | + |
| 27 | + |
| 28 | +/** Which of the favors of your Lord will you deny ? **/ |
| 29 | + |
| 30 | +#include<bits/stdc++.h> |
| 31 | +using namespace std; |
| 32 | + |
| 33 | +#define LL long long |
| 34 | +#define PII pair<int,int> |
| 35 | +#define PLL pair<LL,LL> |
| 36 | +#define F first |
| 37 | +#define S second |
| 38 | + |
| 39 | +#define ALL(x) (x).begin(), (x).end() |
| 40 | +#define READ freopen("alu.txt", "r", stdin) |
| 41 | +#define WRITE freopen("vorta.txt", "w", stdout) |
| 42 | + |
| 43 | +#ifndef ONLINE_JUDGE |
| 44 | +#define DBG(x) cout << __LINE__ << " says: " << #x << " = " << (x) << endl |
| 45 | +#else |
| 46 | +#define DBG(x) |
| 47 | +#define endl "\n" |
| 48 | +#endif |
| 49 | + |
| 50 | +template<class T1, class T2> |
| 51 | +ostream &operator <<(ostream &os, pair<T1,T2>&p); |
| 52 | +template <class T> |
| 53 | +ostream &operator <<(ostream &os, vector<T>&v); |
| 54 | +template <class T> |
| 55 | +ostream &operator <<(ostream &os, set<T>&v); |
| 56 | + |
| 57 | +inline void optimizeIO() |
| 58 | +{ |
| 59 | + ios_base::sync_with_stdio(false); |
| 60 | + cin.tie(NULL); |
| 61 | +} |
| 62 | + |
| 63 | +const int nmax = 2e5+7; |
| 64 | + |
| 65 | +#define int long long |
| 66 | + |
| 67 | +/** |
| 68 | +Segmented Sieve |
| 69 | +Range of [L,R] ~ 1e7 and R ~ 1e12 |
| 70 | +**/ |
| 71 | + |
| 72 | +const int pnmax = 1e6+10; /** ~ sqrt(R) **/ |
| 73 | +const int pnmax2 = 5e5+10; /** diff of R and L **/ |
| 74 | + |
| 75 | +LL LIM = 1e6+5; /** ~ sqrt(R) **/ |
| 76 | +vector<LL>primes; |
| 77 | +bool isP[pnmax]; |
| 78 | +bool isPFinal[pnmax2]; |
| 79 | + |
| 80 | +void sieve() |
| 81 | +{ |
| 82 | + for(LL i=2; i<=LIM; i++) |
| 83 | + isP[i] = true; |
| 84 | + |
| 85 | + for(LL i=2; i<=LIM; i++) /** If I don't want to know the primes , it is enough to loop upto sqrt(LIM) here **/ |
| 86 | + { |
| 87 | + if(isP[i]==true) |
| 88 | + { |
| 89 | + primes.push_back(i); |
| 90 | + for(LL j=i*i; j<=LIM; j+=i) |
| 91 | + isP[j]=false; |
| 92 | + } |
| 93 | + } |
| 94 | +} |
| 95 | + |
| 96 | +/** Prime Factorization **/ |
| 97 | +vector<vector<LL>>factorization(pnmax2); |
| 98 | + |
| 99 | +vector<LL> primeFactors(LL N,vector<LL>&factors_init) |
| 100 | +{ |
| 101 | + vector<LL>factors; |
| 102 | + for(auto PF:factors_init) /// stop at sqrt(N), but N can get smaller |
| 103 | + { |
| 104 | + while (N % PF == 0) |
| 105 | + { |
| 106 | + N /= PF; /// remove this PF |
| 107 | + factors.push_back(PF); |
| 108 | + } |
| 109 | + } |
| 110 | + if (N != 1) factors.push_back(N); /// special case if N is actually a prime |
| 111 | + |
| 112 | + return factors; |
| 113 | +} |
| 114 | + |
| 115 | +/** Number of Divisors **/ |
| 116 | +LL NOD[pnmax2]; |
| 117 | + |
| 118 | +LL number_of_Divisors(LL N,vector<LL>&factors_init) |
| 119 | +{ |
| 120 | + LL ans = 1; |
| 121 | + |
| 122 | + for(auto PF:factors_init) |
| 123 | + { |
| 124 | + LL power = 0; /// count the power |
| 125 | + while (N % PF == 0) |
| 126 | + { |
| 127 | + N /= PF; |
| 128 | + power++; |
| 129 | + } |
| 130 | + ans *= (power + 1); /// according to the formula |
| 131 | + } |
| 132 | + |
| 133 | + if (N != 1) ans *= 2; /// (last factor has pow = 1, we add 1 to it) |
| 134 | + |
| 135 | + return ans; |
| 136 | +} |
| 137 | + |
| 138 | +void segmented_sieve(LL L,LL R) |
| 139 | +{ |
| 140 | + vector<vector<int>>factors_init(pnmax2); |
| 141 | + |
| 142 | + /// Step 1: we will identify all the prime factors of Numbers in the range [l,r] . But we will not know the powers . That we will do in Step 2 |
| 143 | + for(auto p:primes) |
| 144 | + { |
| 145 | + if(p*p > R) |
| 146 | + break; |
| 147 | + |
| 148 | + LL j=p*p; |
| 149 | + |
| 150 | + if(j<L) |
| 151 | + j=((L+p-1)/p)*p; |
| 152 | + |
| 153 | + for(; j<=R; j+=p) |
| 154 | + factors_init[j-L].push_back(p); |
| 155 | + |
| 156 | + } |
| 157 | + |
| 158 | + /// Step 2 : Find the powers |
| 159 | + for(LL i=L; i<=R; i++) |
| 160 | + { |
| 161 | + NOD[i-L] = number_of_Divisors(i,factors_init[i-L]); |
| 162 | + factorization[i-L] = primeFactors(i,factors_init[i-L]); |
| 163 | + } |
| 164 | +} |
| 165 | + |
| 166 | +int32_t main() |
| 167 | +{ |
| 168 | + optimizeIO(); |
| 169 | + |
| 170 | + sieve(); |
| 171 | + |
| 172 | + int tc; |
| 173 | + cin>>tc; |
| 174 | + |
| 175 | + while(tc--) |
| 176 | + { |
| 177 | + LL l,r; |
| 178 | + cin>>l>>r; |
| 179 | + |
| 180 | + segmented_sieve(l,r); |
| 181 | + |
| 182 | + for(int i=l;i<=r;i++) |
| 183 | + { |
| 184 | + cout<<"Number : "<<i<<endl; |
| 185 | + cout<<"NOD : "<<NOD[i-l]<<endl; |
| 186 | + cout<<"Factorization : "<<endl; |
| 187 | + DBG(factorization[i-l]); |
| 188 | + } |
| 189 | + } |
| 190 | + |
| 191 | + return 0; |
| 192 | +} |
| 193 | + |
| 194 | +/** |
| 195 | +
|
| 196 | +**/ |
| 197 | + |
| 198 | +template<class T1, class T2> |
| 199 | +ostream &operator <<(ostream &os, pair<T1,T2>&p) |
| 200 | +{ |
| 201 | + os<< "{"<<p. first<< ", "<<p. second<< "} "; |
| 202 | + return os; |
| 203 | +} |
| 204 | +template <class T> |
| 205 | +ostream &operator <<(ostream &os, vector<T>&v) |
| 206 | +{ |
| 207 | + os<<"[ "; |
| 208 | + for(T i:v) |
| 209 | + { |
| 210 | + os<<i<<" " ; |
| 211 | + } |
| 212 | + os<<" ]"; |
| 213 | + return os; |
| 214 | +} |
| 215 | + |
| 216 | +template <class T> |
| 217 | +ostream &operator <<(ostream &os, set<T>&v) |
| 218 | +{ |
| 219 | + os<<"[ "; |
| 220 | + for(T i:v) |
| 221 | + { |
| 222 | + os<<i<<" "; |
| 223 | + } |
| 224 | + os<<" ]"; |
| 225 | + return os; |
| 226 | +} |
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