{"id":86,"date":"2025-01-11T23:20:47","date_gmt":"2025-01-11T15:20:47","guid":{"rendered":"http:\/\/150.230.63.147\/blog\/?p=86"},"modified":"2025-01-11T23:20:47","modified_gmt":"2025-01-11T15:20:47","slug":"d-concatenated-multiples-%e8%a7%a3%e6%9e%90%e6%80%9d%e7%b6%ad","status":"publish","type":"post","link":"https:\/\/blog.seekrealthing.net\/blog\/?p=86","title":{"rendered":"D. Concatenated Multiples \u89e3\u6790(\u601d\u7dad)"},"content":{"rendered":"\n

Codeforce 1029 D. Concatenated Multiples \u89e3\u6790(\u601d\u7dad)<\/h1>\n

\u4eca\u5929\u6211\u5011\u4f86\u770b\u770bCF1029D
\n\u984c\u76ee\u9023\u7d50<\/a><\/p>\n

\n

\u984c\u76ee<\/strong>
\n\u7d66\u4f60\u4e00\u500b\u5e8f\u5217$a$\u548c\u4e00\u500b\u6578\u5b57$k$\uff0c\u6c42\u6709\u5e7e\u7a2eindices pair\u53ef\u4ee5\u8b93\u5169\u500b\u6578\u5b57\u4e32\u63a5\u5728\u4e00\u8d77\u4e4b\u5f8c\u53ef\u4ee5\u88ab$k$\u6574\u9664\u3002<\/p>\n<\/blockquote>\n

\u524d\u8a00<\/h3>\n

set,map\u9019\u4e9b\u6771\u897f\u7684\u5e38\u6578\u771f\u7684\u6709\u5920\u9ad8\u963f<\/p>\n

\u60f3\u6cd5<\/h3>\n

$a_i$\u548c$a_j$\u4e32\u63a5\u8d77\u4f86\u662f$a_i\\times 10^{digit(a_j)}+a_j$\uff0c\u6211\u5011\u56fa\u5b9a\u4e00\u500b$a_i$\u548c$digit(???)$\uff0c\u8981\u770b\u770b\u54ea\u4e9b$a_j$\u7b26\u5408\u9577\u5ea6\u662f$digit(???)$\u548c\u4e32\u63a5\u8d77\u4f86\u53ef\u88ab\u6574\u9664\u3002<\/p>\n

\u5be6\u4f5c\u7d30\u7bc0:\u6211\u5011\u53ef\u4ee5\u5148\u628a\u76f8\u540c\u9577\u5ea6\u7684$a_j$\u653e\u5230\u4e00\u500b$vector$\u88e1\uff0c\u4e26\u4e14\u63a5\u8457$a_j%=k$\u3002
\n\u5982\u6b64\u4e00\u4f86\u6c7a\u5b9a\u4e86$a_i\\times 10^{digit(???)}\\mod k$\u662f\u591a\u5c11\u4ee5\u5f8c\uff0c\u53ea\u9700\u8981\u627e\u5230\u5728\u540c\u6a23\u9577\u5ea6\u7684$a_j$\u4e2d\uff0c\u54ea\u4e9b\u525b\u597d$=(k-(a_i\\times 10^{digit(???)}\\mod k))\\mod k$\uff0c\u800c\u53c8\u56e0\u70ba\u6211\u5011\u5df2\u7d93\u628a$a_j%=k$\u904e\u4e86\uff0c\u6240\u4ee5\u53ef\u4ee5\u5728$sort$\u904e\u540c\u6a23\u9577\u5ea6\u7684$a_j$\u5f8c\uff0c\u76f4\u63a5$upperbound-lowerbound$\u627e\u540c\u6a23\u503c\u7684\u5143\u7d20\u6709\u591a\u5c11\u500b\u3002
\n\u9084\u6709\uff0c$digit(a_j)$\u7684\u5176\u4e2d\u4e00\u7a2e\u6c42\u6cd5\u662f$digit(a_j)=1+log10(a_j)$<\/p>\n

\u5176\u5be6\u6211\u4e00\u958b\u59cb\u662f\u7528$map$\u5be6\u4f5c\u7684\uff0c\u60f3\u6cd5\u662f\u56fa\u5b9a$a_j$\u7136\u5f8c\u770b\u770b$map$\u88e1\u53c8\u591a\u5c11\u500b$a_i\\times 10^{digit(???)}$\u525b\u597d\u53ef\u4ee5\u7b26\u5408\u8981\u6c42\uff0c\u7136\u5f8c\u5f9e\u524d\u5f80\u5f8c+\u5f9e\u5f8c\u5f80\u524d\u5404\u770b\u4e00\u6b21\u3002\u7136\u800c\u9019\u984c\u7684$k\\le1e9$\uff0c\u56e0\u6b64$map$\u975e\u5e38\u53ef\u80fd\u9700\u8981$insert$\u975e\u5e38\u591a\u7a7a\u767d\u5143\u7d20\uff0c\u9020\u6210TLE\uff0c\u5176\u4e2d\u4e00\u90e8\u5206\u9020\u6210TLE\u7684\u539f\u56e0\u662f\u56e0\u70ba$insert$\u6642\u8981$allocate$\u65b0\u8a18\u61b6\u9ad4\u4f4d\u7f6e\uff0c\u9019\u6a23\u9020\u6210\u975e\u5e38\u5927\u7684\u5e38\u6578\u3002<\/p>\n

\u7a0b\u5f0f\u78bc(\u6b63\u5e38\u4f5c\u6cd5):<\/h3>\n
const int _n=2e5+10;  \nint t,n,k,a[_n],dig[_n],ten[11];  \nVI d2a[_n];  \nll ans;  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>k;rep(i,0,n)cin>>a[i];ten[0]=1;rep(i,1,11)ten[i]=1ll*ten[i-1]*10%k;  \n  rep(i,0,n)dig[i]=1+log10(a[i]),a[i]%=k,d2a[dig[i]].pb(a[i]);  \n  rep(i,1,11)sort(all(d2a[i]));  \n  rep(i,0,n)rep(j,1,11){  \n    t=k-1ll*a[i]*ten[j]%k;if(t==k)t=0;  \n    ans+=upper_bound(all(d2a[j]),t)-lower_bound(all(d2a[j]),t);  \n    if(dig[i]==j and a[i]==t)ans--;  \n  }cout<<ans<<'\\n';  \n  return 0;  \n}  \n<\/code><\/pre>\n
\u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b
\nSubmission<\/a><\/p>\n
\u7a0b\u5f0f\u78bc(\u6539\u6210unordered_map\u4ee5\u5f8c1800ms\u7684AC):<\/h3>\n
const int _n=2e5+10;  \nint t,n,k,a[_n];  \nll ans;  \nll ten[11]={1,10,100,1000,10000,100000,1000000,10000000,100000000,1000000000,10000000000ll};  \nstruct pair_hash  \n{  \n\ttemplate <class T1, class T2>  \n\tstd::size_t operator() (const std::pair<T1, T2> &pair) const  \n\t{  \n\t\treturn std::hash<T1>()(pair.first) ^ std::hash<T2>()(pair.second);  \n\t}  \n};  \nunordered_map<PII,int,pair_hash> mp;  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>k;rep(i,0,n)cin>>a[i];  \n  rep(i,0,n){  \n    ans+=mp[{(k-a[i]%k)%k,1+log10(a[i])}];  \n    rep(j,1,11)mp[{(1ull*a[i]*ten[j])%(1ull*k),j}]++;  \n  }mp.clear();  \n  per(i,0,n){  \n    ans+=mp[{(k-a[i]%k)%k,1+log10(a[i])}];  \n    rep(j,1,11)mp[{(1ull*a[i]*ten[j])%(1ull*k),j}]++;  \n  }cout<<ans<<'\\n';  \n  return 0;  \n}  \n<\/code><\/pre>\n
\u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b
\nSubmission<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[3],"tags":[10,9,8],"class_list":["post-86","post","type-post","status-publish","format-standard","hentry","category-ojcode","tag-codeforce","tag-ojcode","tag-8"],"jetpack_publicize_connections":[],"yoast_head":"\nD. 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