#include "bits/stdc++.h" using namespace std; #define all(x) x.begin(),x.end() template<typename A, typename B> ostream& operator<<(ostream &os, const pair<A, B> &p) { return os << p.first << " " << p.second; } template<typename T_container, typename T = typename enable_if<!is_same<T_container, string>::value, typename T_container::value_type>::type> ostream& operator<<(ostream &os, const T_container &v) { string sep; for (const T &x : v) os << sep << x, sep = " "; return os; } #ifdef LOCAL #include "debug.h" #else #define debug(...) 42 #define ASSERT(...) 42 #endif typedef long long ll; typedef vector<int> vi; typedef vector<vi> vvi; typedef pair<int,int> pi; const int oo = 1e9; /* for A with some monotonic sequences. A, B look at bigger of the two? Can always split to the optimum plus 1. So best is <= (N/M)+1 Assume N>=M small edge cases are trashy 2 is doable usually? 1 | 2 | 3 ? 1 3 2 4 If string lengths are roughly equal? a a a b a b (if you know max s is something?) You can keep it the same. if becomes too much: a a a bad? a<a<b bad? a<b>a>a bad? a<b>a>b bad? So the best you can do a a a a a a a a a > b < a? currently have: binary search on answer s. then do dp[i][j][flag] = min current length of monotonic in a direction. Ok, observation, given lengths N>=M can add some < or > sign inside N for 1 cost, max cost<=M What is minimum number of adjacent <<<< signs or >>>> signs? For a given string: can binary search on s-1 = number of signs, and check it by greedily adding the signs? If <<<<<< (too many signs), can add > at regular intervals and not at the ends, you need ceil(B/s) signs. So total number of cost needed is sum of ceil((BLOCK_i)/s)<=M. So for some s to be optimal you have: sum ceil((BLOCK_i)/s)<=M < sum ceil((BLOCK_i)/(s-1)). For some s to be >= optimum? You need: sum ceil((BLOCK_i)/s)<=M And you need: sum from (m-M+1) to N This is easy to work with? If you do divide and conquer: substrings that go over the splitting line: s < sqrt(n) ? Can loop over them s > sqrt(n) Only O(sqrt(n)) segments are important. I would just like to now all floor((Block_i-1)/s) as a frequency array. only have O(n/s) important segments, res of the segments contribute 0!. rest of the segments just contribute something flat. Want to know contributions of weight1* p_j - p_i*weight2, but only when p_j>p_i Can do it with a single FFT! ok, so have important and stupid segments. stupid segments: can group them together. can do some FFT? want to know frequencies of those Afterwards: For given s: now the amount of those? Can just calc that easily, and subtract with <=(s+1) inside block need to be careful. need to subtract those again... in total <= 3e5*3e5, so can easily fit with regular FFT. */ const long long MD = 1e9+7; template<long long MOD=MD> struct mdint { int d; mdint () {d=0;} mdint (long long _d) : d(_d%MOD){ if(d<0) d+=MOD; }; friend mdint& operator+=(mdint& a, const mdint& o) { a.d+=o.d; if(a.d>=MOD) a.d-=MOD; return a; } friend mdint& operator-=(mdint& a, const mdint& o) { a.d-=o.d; if(a.d<0) a.d+=MOD; return a; } friend mdint& operator*=(mdint& a, const mdint& o) { return a = mdint((ll)a.d*o.d); } mdint operator*(const mdint& o) const { mdint res = *this; res*=o; return res; } mdint operator+(const mdint& o) const { mdint res = *this; res+=o; return res; } mdint operator-(const mdint& o) const { mdint res = *this; res-=o; return res; } mdint operator^(long long b) const { mdint tmp = 1; mdint power = *this; while(b) { if(b&1) { tmp = tmp*power; } power = power*power; b/=2; } return tmp; } friend mdint operator/=(mdint& a, const mdint& o) { a *= (o^(MOD-2)); return a; } mdint operator/(const mdint& o) { mdint res = *this; res/=o; return res; } bool operator==(const mdint& o) { return d==o.d;} bool operator!=(const mdint& o) { return d!=o.d;} friend istream& operator>>(istream& c, mdint& a) {return c >> a.d;} friend ostream& operator<<(ostream& c, const mdint& a) {return c << a.d;} }; using mint = mdint<MD>; vector<bool> read(int n) { vi a(n); for(auto& i : a) cin >> i; vector<bool> b(n-1); for(int i=0;i<n-1;++i) b[i] = a[i]<a[i+1]; return b; } vector<mint> solve(vector<bool> a, int m) { int n = a.size(); // first write n^4! vector<mint> ans(n+1); // s<=k, for all k for(int len=2;len<=min(n+1,m);++len) { for(int olen=len;olen<=m;++olen) { ans[0]+=(n+1-len+1)*ll(m-olen+1); } } for(int s=1;s<=n;++s) { // answer is <=s mint res=0; for(int i=0;i<n;++i) { int need=0; int cur=1; for(int j=i;j<n;++j) { if(j>i and a[j]==a[j-1]) cur++; else { need+=(cur-1)/s; cur=1; } int need2 = need + (cur-1)/s; // assume substring ends here! for(int len=max(1,need2);len<=m;++len) { res+=(m-len+1); } } } ans[s]=res; } // s=k --> s<=k and s>k-1 for(int i=ans.size()-1;i>0;--i) ans[i]-=ans[i-1]; ans.erase(ans.begin()); return ans; } mint arith(ll a, ll b) { if(a>b) return 0; return (a+b)*(b-a+1)/2; } const mint inv2 = mint(1)/2; vector<mint> solve2(vector<bool> a, int m) { int n = a.size(); // first write n^4! vector<mint> ans(n+1); // s<=k, for all k for(int len=2;len<=min(n+1,m);++len) { ans[0]+=arith(1,m-len+1)*(n+1-len+1); } struct B { int l; int mid; int r; }; vector<B> blocks; for(int i=0;i<n;) { int j = i; while(j<n and a[i]==a[j]) ++j; blocks.push_back({0,j-i,0}); i=j; } for(int s=1;s<=n;++s) { // answer is <=s if(!blocks.empty()) { vector<B> nb; nb.reserve(blocks.size()); int freebefore = blocks[0].l; for(auto b : blocks) { if(b.mid<=s) { freebefore+=b.mid; freebefore+=b.r; } else { b.l = freebefore; nb.push_back(b); freebefore = b.r; } } if(!nb.empty()) nb.back().r=freebefore; for(int i=1;i<nb.size();++i) { nb[i-1].r=nb[i].l; } swap(nb,blocks); } mint res=0; if(blocks.empty()) { // all subarrays are good? res = arith(1,n)*arith(1,m); } else { // do it faster! mint zeros=0; typedef array<mint,3> W; vector<W> prefsums = {{}}; int lastpref=0; auto addL = [&](mint num, int mypref) { assert(mypref>=lastpref); while(lastpref<mypref) { prefsums.push_back(prefsums.back()); ++lastpref; } W nls = {num,num*mypref,num*mypref*mypref}; auto& ls = prefsums.back(); for(int i=0;i<3;++i) ls[i]+=nls[i]; }; auto addC = [&](mint num, int sum) { if(sum==0) zeros+=num; res+=num*arith(1,m+1-sum); }; auto addR = [&](mint num, int mypref) { W rs = {num,num*mypref,num*mypref*mypref}; assert(mypref>=lastpref); if(mypref==lastpref) { mint cur = prefsums[mypref][0]; if(mypref) cur-=prefsums[mypref-1][0]; zeros+=cur*num; } auto ls = prefsums.back(); int bef = mypref - m-1; if(bef>lastpref) return; if(bef>=0) { for(int j=0;j<3;++j) ls[j]-=prefsums[bef][j]; } auto contribution = ls[2]*rs[0]*inv2 - ls[1]*rs[1] + ls[1]*rs[0]*(inv2*3+m); contribution+=ls[0]*rs[2]*inv2 - rs[1]*ls[0]*(inv2*3+m) + ls[0]*rs[0]*(mint(m)*(m+3)*inv2 + 1); res+=contribution; }; /* do just arith(1,m+1-(pb-pa)) = Problem when pb-pa> a (a/2 - b + m + 3/2) + b (b/2 - m - 3/2) + (m/2 + 3/2) m + 1 and for all 0's do something special? */ int pref=0; // in all blocks, inside: // loop over the sum // for this sum do the addition manually for(auto b : blocks) { // only strictly inside // add the ls. addC(arith(1,b.l),0); addL(b.l, pref); for(int sum=0;sum<=(b.mid-1)/s;++sum) { // strictly inside: int lo = max(1,(sum*s+1)), hi = min(b.mid,(sum+1)*s); mint num = hi-lo+1; if(lo<=hi) addR(num,pref+sum); } for(int sum=0;sum<=(b.mid-1)/s;++sum) { int lo = max(1,(sum*s+1)), hi = min(b.mid,(sum+1)*s); if(lo<=hi) { addC(arith(b.mid+1-hi,b.mid+1-lo),sum); } } pref+=(b.mid-1)/s; for(int sum=(b.mid-1)/s;sum>=0;--sum) { int lo = max(1,sum*s+1), hi = min(b.mid,(sum+1)*s); mint num = hi-lo+1; if(lo<=hi) addL(num,pref-sum); } addR(b.r,pref); } addC(arith(1,blocks.back().r), 0); res+=zeros*(arith(1,m)-arith(1,m+1)); // mint brute=0; // for(int i=0;i<n;++i) { // int need=0; // int cur=1; // for(int j=i;j<n;++j) { // if(j>i and a[j]==a[j-1]) cur++; // else { // need+=(cur-1)/s; // cur=1; // } // int need2 = need + (cur-1)/s; // // assume substring ends here! // // need2,1 ? // brute+=arith(1, m-max(need2,1)+1); // } // } // debug(brute,res); // if(brute!=res) exit(0); } ans[s]=res; } // s=k --> s<=k and s>k-1 for(int i=ans.size()-1;i>0;--i) ans[i]-=ans[i-1]; ans.erase(ans.begin()); return ans; } int main() { ios_base::sync_with_stdio(false); cin.tie(NULL); int n,m; cin >> n >> m; auto a = read(n), b = read(m); vector<mint> ans(n+m); auto res = solve2(a,m); for(int i=0;i<res.size();++i) { ans[i]+=res[i]; } res = solve2(b,n); for(int i=0;i<res.size();++i) { ans[i]+=res[i]; } for(int len=1;len<=min(n,m);++len) { // for all with this length, answer is 2: ans[0]+=mint(n-len+1)*(m-len+1); } ans.insert(ans.begin(),0); // ans.push_back(0); ans.pop_back(); // assert(ans.size()==n+m); cout << ans << ' ' << '\n'; }
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 | #include "bits/stdc++.h" using namespace std; #define all(x) x.begin(),x.end() template<typename A, typename B> ostream& operator<<(ostream &os, const pair<A, B> &p) { return os << p.first << " " << p.second; } template<typename T_container, typename T = typename enable_if<!is_same<T_container, string>::value, typename T_container::value_type>::type> ostream& operator<<(ostream &os, const T_container &v) { string sep; for (const T &x : v) os << sep << x, sep = " "; return os; } #ifdef LOCAL #include "debug.h" #else #define debug(...) 42 #define ASSERT(...) 42 #endif typedef long long ll; typedef vector<int> vi; typedef vector<vi> vvi; typedef pair<int,int> pi; const int oo = 1e9; /* for A with some monotonic sequences. A, B look at bigger of the two? Can always split to the optimum plus 1. So best is <= (N/M)+1 Assume N>=M small edge cases are trashy 2 is doable usually? 1 | 2 | 3 ? 1 3 2 4 If string lengths are roughly equal? a a a b a b (if you know max s is something?) You can keep it the same. if becomes too much: a a a bad? a<a<b bad? a<b>a>a bad? a<b>a>b bad? So the best you can do a a a a a a a a a > b < a? currently have: binary search on answer s. then do dp[i][j][flag] = min current length of monotonic in a direction. Ok, observation, given lengths N>=M can add some < or > sign inside N for 1 cost, max cost<=M What is minimum number of adjacent <<<< signs or >>>> signs? For a given string: can binary search on s-1 = number of signs, and check it by greedily adding the signs? If <<<<<< (too many signs), can add > at regular intervals and not at the ends, you need ceil(B/s) signs. So total number of cost needed is sum of ceil((BLOCK_i)/s)<=M. So for some s to be optimal you have: sum ceil((BLOCK_i)/s)<=M < sum ceil((BLOCK_i)/(s-1)). For some s to be >= optimum? You need: sum ceil((BLOCK_i)/s)<=M And you need: sum from (m-M+1) to N This is easy to work with? If you do divide and conquer: substrings that go over the splitting line: s < sqrt(n) ? Can loop over them s > sqrt(n) Only O(sqrt(n)) segments are important. I would just like to now all floor((Block_i-1)/s) as a frequency array. only have O(n/s) important segments, res of the segments contribute 0!. rest of the segments just contribute something flat. Want to know contributions of weight1* p_j - p_i*weight2, but only when p_j>p_i Can do it with a single FFT! ok, so have important and stupid segments. stupid segments: can group them together. can do some FFT? want to know frequencies of those Afterwards: For given s: now the amount of those? Can just calc that easily, and subtract with <=(s+1) inside block need to be careful. need to subtract those again... in total <= 3e5*3e5, so can easily fit with regular FFT. */ const long long MD = 1e9+7; template<long long MOD=MD> struct mdint { int d; mdint () {d=0;} mdint (long long _d) : d(_d%MOD){ if(d<0) d+=MOD; }; friend mdint& operator+=(mdint& a, const mdint& o) { a.d+=o.d; if(a.d>=MOD) a.d-=MOD; return a; } friend mdint& operator-=(mdint& a, const mdint& o) { a.d-=o.d; if(a.d<0) a.d+=MOD; return a; } friend mdint& operator*=(mdint& a, const mdint& o) { return a = mdint((ll)a.d*o.d); } mdint operator*(const mdint& o) const { mdint res = *this; res*=o; return res; } mdint operator+(const mdint& o) const { mdint res = *this; res+=o; return res; } mdint operator-(const mdint& o) const { mdint res = *this; res-=o; return res; } mdint operator^(long long b) const { mdint tmp = 1; mdint power = *this; while(b) { if(b&1) { tmp = tmp*power; } power = power*power; b/=2; } return tmp; } friend mdint operator/=(mdint& a, const mdint& o) { a *= (o^(MOD-2)); return a; } mdint operator/(const mdint& o) { mdint res = *this; res/=o; return res; } bool operator==(const mdint& o) { return d==o.d;} bool operator!=(const mdint& o) { return d!=o.d;} friend istream& operator>>(istream& c, mdint& a) {return c >> a.d;} friend ostream& operator<<(ostream& c, const mdint& a) {return c << a.d;} }; using mint = mdint<MD>; vector<bool> read(int n) { vi a(n); for(auto& i : a) cin >> i; vector<bool> b(n-1); for(int i=0;i<n-1;++i) b[i] = a[i]<a[i+1]; return b; } vector<mint> solve(vector<bool> a, int m) { int n = a.size(); // first write n^4! vector<mint> ans(n+1); // s<=k, for all k for(int len=2;len<=min(n+1,m);++len) { for(int olen=len;olen<=m;++olen) { ans[0]+=(n+1-len+1)*ll(m-olen+1); } } for(int s=1;s<=n;++s) { // answer is <=s mint res=0; for(int i=0;i<n;++i) { int need=0; int cur=1; for(int j=i;j<n;++j) { if(j>i and a[j]==a[j-1]) cur++; else { need+=(cur-1)/s; cur=1; } int need2 = need + (cur-1)/s; // assume substring ends here! for(int len=max(1,need2);len<=m;++len) { res+=(m-len+1); } } } ans[s]=res; } // s=k --> s<=k and s>k-1 for(int i=ans.size()-1;i>0;--i) ans[i]-=ans[i-1]; ans.erase(ans.begin()); return ans; } mint arith(ll a, ll b) { if(a>b) return 0; return (a+b)*(b-a+1)/2; } const mint inv2 = mint(1)/2; vector<mint> solve2(vector<bool> a, int m) { int n = a.size(); // first write n^4! vector<mint> ans(n+1); // s<=k, for all k for(int len=2;len<=min(n+1,m);++len) { ans[0]+=arith(1,m-len+1)*(n+1-len+1); } struct B { int l; int mid; int r; }; vector<B> blocks; for(int i=0;i<n;) { int j = i; while(j<n and a[i]==a[j]) ++j; blocks.push_back({0,j-i,0}); i=j; } for(int s=1;s<=n;++s) { // answer is <=s if(!blocks.empty()) { vector<B> nb; nb.reserve(blocks.size()); int freebefore = blocks[0].l; for(auto b : blocks) { if(b.mid<=s) { freebefore+=b.mid; freebefore+=b.r; } else { b.l = freebefore; nb.push_back(b); freebefore = b.r; } } if(!nb.empty()) nb.back().r=freebefore; for(int i=1;i<nb.size();++i) { nb[i-1].r=nb[i].l; } swap(nb,blocks); } mint res=0; if(blocks.empty()) { // all subarrays are good? res = arith(1,n)*arith(1,m); } else { // do it faster! mint zeros=0; typedef array<mint,3> W; vector<W> prefsums = {{}}; int lastpref=0; auto addL = [&](mint num, int mypref) { assert(mypref>=lastpref); while(lastpref<mypref) { prefsums.push_back(prefsums.back()); ++lastpref; } W nls = {num,num*mypref,num*mypref*mypref}; auto& ls = prefsums.back(); for(int i=0;i<3;++i) ls[i]+=nls[i]; }; auto addC = [&](mint num, int sum) { if(sum==0) zeros+=num; res+=num*arith(1,m+1-sum); }; auto addR = [&](mint num, int mypref) { W rs = {num,num*mypref,num*mypref*mypref}; assert(mypref>=lastpref); if(mypref==lastpref) { mint cur = prefsums[mypref][0]; if(mypref) cur-=prefsums[mypref-1][0]; zeros+=cur*num; } auto ls = prefsums.back(); int bef = mypref - m-1; if(bef>lastpref) return; if(bef>=0) { for(int j=0;j<3;++j) ls[j]-=prefsums[bef][j]; } auto contribution = ls[2]*rs[0]*inv2 - ls[1]*rs[1] + ls[1]*rs[0]*(inv2*3+m); contribution+=ls[0]*rs[2]*inv2 - rs[1]*ls[0]*(inv2*3+m) + ls[0]*rs[0]*(mint(m)*(m+3)*inv2 + 1); res+=contribution; }; /* do just arith(1,m+1-(pb-pa)) = Problem when pb-pa> a (a/2 - b + m + 3/2) + b (b/2 - m - 3/2) + (m/2 + 3/2) m + 1 and for all 0's do something special? */ int pref=0; // in all blocks, inside: // loop over the sum // for this sum do the addition manually for(auto b : blocks) { // only strictly inside // add the ls. addC(arith(1,b.l),0); addL(b.l, pref); for(int sum=0;sum<=(b.mid-1)/s;++sum) { // strictly inside: int lo = max(1,(sum*s+1)), hi = min(b.mid,(sum+1)*s); mint num = hi-lo+1; if(lo<=hi) addR(num,pref+sum); } for(int sum=0;sum<=(b.mid-1)/s;++sum) { int lo = max(1,(sum*s+1)), hi = min(b.mid,(sum+1)*s); if(lo<=hi) { addC(arith(b.mid+1-hi,b.mid+1-lo),sum); } } pref+=(b.mid-1)/s; for(int sum=(b.mid-1)/s;sum>=0;--sum) { int lo = max(1,sum*s+1), hi = min(b.mid,(sum+1)*s); mint num = hi-lo+1; if(lo<=hi) addL(num,pref-sum); } addR(b.r,pref); } addC(arith(1,blocks.back().r), 0); res+=zeros*(arith(1,m)-arith(1,m+1)); // mint brute=0; // for(int i=0;i<n;++i) { // int need=0; // int cur=1; // for(int j=i;j<n;++j) { // if(j>i and a[j]==a[j-1]) cur++; // else { // need+=(cur-1)/s; // cur=1; // } // int need2 = need + (cur-1)/s; // // assume substring ends here! // // need2,1 ? // brute+=arith(1, m-max(need2,1)+1); // } // } // debug(brute,res); // if(brute!=res) exit(0); } ans[s]=res; } // s=k --> s<=k and s>k-1 for(int i=ans.size()-1;i>0;--i) ans[i]-=ans[i-1]; ans.erase(ans.begin()); return ans; } int main() { ios_base::sync_with_stdio(false); cin.tie(NULL); int n,m; cin >> n >> m; auto a = read(n), b = read(m); vector<mint> ans(n+m); auto res = solve2(a,m); for(int i=0;i<res.size();++i) { ans[i]+=res[i]; } res = solve2(b,n); for(int i=0;i<res.size();++i) { ans[i]+=res[i]; } for(int len=1;len<=min(n,m);++len) { // for all with this length, answer is 2: ans[0]+=mint(n-len+1)*(m-len+1); } ans.insert(ans.begin(),0); // ans.push_back(0); ans.pop_back(); // assert(ans.size()==n+m); cout << ans << ' ' << '\n'; } |