LOJ 2483<\/a> \u3002<\/p>\n <\/p>\n
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\u5df2\u642c\u81f3 YZOJ P4254 \u3002<\/p>\n
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\n <\/p>\n
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\u6309\u7167\u60ef\u4f8b\uff0c\u8bbe \\(f_i\\) \u4e3a\u505a\u5230\u7b2c \\(i\\) \u6839\u67f1\u5b50\u65f6\u7684\u6700\u5c0f\u4ee3\u4ef7\u3002\u8ba1 \\(sumw\\) \u4e3a \\(w\\) \u7684\u524d\u7f00\u548c\u3002<\/p>\n
\u8f7b\u677e\u5f97\u5230\u8f6c\u79fb\u65b9\u7a0b \\(f_i=\\min\\limits_{j=1}^{i-1}\\{f_j+sumw_{i-1}-sumw_j+(h_i-h_j)^2\\}\\)<\/span> \uff0c\u8fd9\u6837\u5c31\u6709\u4e86 \\(30pts\\) \u3002<\/p>\n <\/p>\n
\u4e0d\u96be\u53d1\u73b0\u5176\u5b9e\u8fd9\u662f\u4e00\u4e2a\u659c\u7387\u4f18\u5316<\/strong>\u7684\u7ecf\u5178\u6a21\u578b\u3002\u5bf9\u8fd9\u4e2a\u8f6c\u79fb\u65b9\u7a0b\u8fdb\u884c\u4e00\u4e9b\u5904\u7406<\/p>\n\\(\\begin{align} f_i&=f_j+sumw_{i-1}-sumw_j+(h_i-h_j)^2 \\\\ \\underbrace{f_i}_{b_i}&=\\underbrace{f_j-sumw_j+h^2_j}_{y_j}-\\underbrace{h_i}_{k_i} \\cdot \\underbrace{2h_j}_{x_j}+\\underbrace{sumw_{i-1}+h^2_i}_{const_i} \\end{align}\\)<\/span><\/p>\n\u4f46\u662f\u95ee\u9898\u6765\u4e86\uff0c\u8fd9\u91cc\u7684 \\(k_i\\) \u548c \\(x_j\\) \u90fd\u4e0d\u662f\u5355\u8c03<\/strong>\u7684\u3002\u54e6\u8c41\uff0c\u5b8c\u86cb\uff0c\u8981\u5199 Splay \u7ef4\u62a4\u4e0b\u51f8\u58f3\u4e86\u3002<\/del>\u8fd9\u4e2a\u65f6\u5019\u5c31\u8981\u7528\u5230\u4e00\u79cd\u795e\u5947\u7684\u5206\u6cbb\u7b97\u6cd5 \u2014\u2014 CDQ\u5206\u6cbb<\/strong>\u3002<\/p>\n\u56de\u987e\u4e00\u4e0b\u659c\u7387\u4f18\u5316\u7684\u5199\u6cd5\uff1a\u628a\u53ef\u4ee5\u5b9e\u73b0\u628a\u4e00\u4e2a\u70b9\u52a0\u5165\u5355\u8c03\u961f\u5217\u7684\u64cd\u4f5c\u662f\u56e0\u4e3a\u52a0\u5165\u7684<\/strong> \\(x\\) \u662f\u5355\u8c03\u7684\uff0c\u53ef\u4ee5\u53d6\u961f\u5934\u5143\u7d20\u4e3a\u6700\u5c0f\u503c\u662f\u56e0\u4e3a\u8be2\u95ee\u7684<\/strong> \\(k\\) \u662f\u5355\u8c03\u7684\u3002CDQ \u5206\u6cbb\u5c31\u662f\u7262\u7262\u6293\u4f4f\u8fd9\u4e24\u70b9\u8fdb\u884c\u64cd\u4f5c\u7684\u3002<\/p>\n\u56e0\u4e3a\u662f\u5206\u6cbb\uff0c\u6240\u4ee5\u8981\u6709\u4e00\u4e2a\u5206\u6cbb\u533a\u95f4 \\([l, r]\\) \u8868\u793a\u7528 \\([l, mid]\\) \u4e2d\u6240\u6709\u70b9<\/strong>\u7684\u4fe1\u606f\u53bb\u66f4\u65b0 \\((mid, r]\\) <\/strong>\u7684\u6700\u5c0f\u503c\u3002\u90a3\u4e48\u5982\u4f55\u5feb\u901f\u505a\u5230\u8fd9\u4e2a\u66f4\u65b0\u64cd\u4f5c\u5462\uff1f\uff1f\uff1f\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u8981\u628a \\([l, mid]\\) \u4e2d\u7684\u70b9\u6309\u7167 \\(x\\) \u6392\u5e8f\uff0c\\((mid, r]\\) \u4e2d\u7684\u5143\u7d20\u6309\u7167\u659c\u7387 \\(k\\) \u6392\u5e8f\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u6309\u7167\u6700\u7b80\u5355\u659c\u7387\u4f18\u5316\u7684\u5199\u6cd5\u53bb\u66f4\u65b0\u4e86\u3002<\/p>\n\u76f4\u63a5\u5728\u91cc\u9762 std::sort()<\/span>\u00a0 \u590d\u6742\u5ea6\u4f1a\u591a\u4e2a \\(log\\)\uff08\u867d\u7136\u5bf9\u4e8e\u6570\u636e\u6c34\u7684\u672c\u9898\u4e5f\u80fd\u8fc7\uff09\uff0c\u4f46\u662f\u8fd0\u7528\u5f52\u5e76\u6392\u5e8f<\/strong>\u7684\u601d\u60f3\u5c31\u53ef\u4ee5\u8fbe\u5230 \\(O(Nlog^2N)\\) \u7684\u590d\u6742\u5ea6\u3002<\/p>\n\u9996\u5148\u5728\u5916\u9762\u628a\u5143\u7d20\u6309\u7167\u659c\u7387 \\(k\\) \u6392\u5e8f\uff0c\u7136\u540e\u5f00\u59cb CDQ\u5206\u6cbb\u3002<\/p>\n
\u5bf9\u4e8e\u533a\u95f4 \\([l, r]\\) \uff0c\u628a\u5176\u4e2d\u5143\u7d20\u7684\u7f16\u53f7\u6309\u7167\u4e0e \\(mid\\) \u7684\u5927\u5c0f\u8fdb\u884c\u5206\u7c7b\uff0c\\(\\leq mid\\) \u7684\u5206\u5230\u5de6\u8fb9\uff0c\\(>mid\\) \u7684\u5206\u5230\u53f3\u8fb9\uff0c\u56e0\u4e3a\u8981\u7528 \\([l, mid]\\) \u4e2d\u7684\u70b9\u66f4\u65b0 \\((mid, r]\\)\uff0c\u6240\u4ee5\u8981\u5148\u628a\u8fd9\u4e9b\u70b9\u627e\u51fa\u6765\u3002<\/p>\n
\u63a5\u7740\u9012\u5f52\u8fdb \\([l, mid]\\) \uff0c\u4fdd\u8bc1\u5de6\u533a\u95f4\u5df2\u7ecf\u88ab\u5904\u7406\u8fc7\u4e86\uff0c\u624d\u80fd\u66f4\u65b0\u53f3\u533a\u95f4\u7684\u5143\u7d20\u3002<\/p>\n
\u73b0\u5728\u5de6\u533a\u95f4\u7684\u70b9\u548c\u4fe1\u606f<\/strong>\u90fd\u88ab\u5904\u7406\u597d\u4e86\uff0c\u5e76\u4e14 \\(x\\) \u662f\u5355\u8c03\u7684\uff08\u5df2\u7ecf\u88ab\u5904\u7406\u597d\u4e86\uff0c\u8be6\u89c1\u540e\u8ff0\uff09\uff0c\u52a0\u4e0a\u53f3\u533a\u95f4\u7684\u5143\u7d20\u7684\u659c\u7387 \\(k\\) \u662f\u5355\u8c03\u7684\uff08\u8fd8\u6ca1\u6709\u88ab\u5904\u7406\u8fc7\uff09\uff0c\u6240\u4ee5\u5c31\u53ef\u4ee5\u628a\u5de6\u533a\u95f4\u7684\u70b9\u52a0\u5165\u5355\u8c03\u961f\u5217\uff0c\u7136\u540e\u53bb\u66f4\u65b0\u53f3\u533a\u95f4\u7684\u5143\u7d20\u3002<\/p>\n\u4f46\u662f\u73b0\u5728 \\((mid, r]\\) \u4e2d\u7684\u5143\u7d20\u53ea\u6709\u6765\u81ea \\([l, mid]\\) \u7684\u8d21\u732e\uff0c\u53ef\u80fd\u53d6\u4e0d\u5230\u6700\u4f18\u89e3\uff0c\u6240\u4ee5\u8fd8\u8981\u9012\u5f52\u8fdb \\((mid, r]\\) \u7ee7\u7eed\u5904\u7406\u53f3\u533a\u95f4\u3002<\/p>\n
\u6700\u540e\u4e0d\u8981\u5fd8\u4e86\uff0c\u8981\u5229\u7528\u5f52\u5e76\u6392\u5e8f<\/strong>\u5c06 \\([l ,r]\\) \u4e2d \u70b9\u7684 \\(x\\) \u53d8\u4e3a\u5355\u8c03\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u4fdd\u8bc1\u524d\u9762\u64cd\u4f5c\u7684\u6b63\u786e\u6027\u3002\uff08\u53ef\u4ee5\u7406\u89e3\u4e3a\u9012\u5f52\u8fdb\u4e00\u4e2a\u533a\u95f4\u540e\uff0c\u8fd9\u4e2a\u533a\u95f4\u5185\u70b9\u7684 \\(x\\) \u5c31\u4f1a\u88ab\u5904\u7406\u6210\u5355\u8c03\u7684\uff0c\u53ef\u4ee5\u8fdb\u884c\u540e\u7eed\u7684\u64cd\u4f5c\uff09<\/p>\n\u8fd9\u5c31\u662f CDQ\u5206\u6cbb \u7684\u5927\u81f4\u8fc7\u7a0b\u3002<\/p>\n
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#include <cstdio>\r\n#include <cstdlib>\r\n#include <cstring>\r\n#include <climits>\r\n#include <algorithm>\r\n\r\n#define _min(_a,_b) ((_a)<(_b)?(_a):(_b))\r\ntypedef long long LLONG;\r\n\r\ninline int getnum()\r\n{\r\n\tregister int sgn=1;\r\n\tregister char c=getchar();\r\n\t(c=='-' ? sgn=-1 : 0);\r\n\twhile(!(c>='0' && c<='9'))\r\n\t\tc=getchar(),(c=='-' ? sgn=-1 : 0);\r\n\tregister int a=0;\r\n\twhile(c>='0' && c<='9')\r\n\t{\r\n\t\ta*=10;a+=c-'0';\r\n\t\tc=getchar();\r\n\t}\r\n\treturn sgn*a;\r\n}\r\n\r\nint idx[105050],tidx[105050];\r\nLLONG h[105050],sw[105050],f[105050];\r\nstruct _comp\r\n{\r\n\tbool operator ()(const int&a,const int&b)const\r\n\t{\r\n\t\treturn (h[a]<h[b]);\r\n\t}\r\n};\r\n\r\nstruct _point\r\n{\r\n\tLLONG x,y;\r\n\tLLONG operator * (const _point&o)const\r\n\t{\r\n\t\treturn (x*o.y-y*o.x);\r\n\t}\r\n\t_point operator - (const _point&o)const\r\n\t{\r\n\t\treturn (_point){x-o.x,y-o.y};\r\n\t}\r\n}glist[105050],list[105050],tlist[105050];\r\n\r\nvoid CDQ_Divide(register int l,register int r)\r\n{\r\n\tif(l==r)\r\n\t{\r\n\t\tglist[l]=(_point){2*h[l],f[l]-sw[l]+(LLONG)h[l]*h[l]};\r\n\t\treturn;\r\n\t}\r\n\t\r\n\tregister int mid=(l+r)>>1,tl=l,tr=mid+1;\r\n\tfor(register int i=l;i<=r;i++)\r\n\t\tif(idx[i]<=mid)\r\n\t\t\ttidx[tl++]=idx[i];\r\n\t\telse\r\n\t\t\ttidx[tr++]=idx[i];\r\n\tfor(register int i=l;i<=r;i++)\r\n\t\tidx[i]=tidx[i];\r\n\tCDQ_Divide(l,mid);\r\n\t\r\n\tregister int front=0,rear=0;\r\n\tfor(register int i=l;i<=mid;i++)\r\n\t{\r\n\t\twhile(front+1<rear && (list[rear-1]-list[rear-2])*(glist[i]-list[rear-1])<=0)\r\n\t\t\trear--;\r\n\t\tlist[rear++]=glist[i];\r\n\t}\r\n\tfor(register int i=mid+1;i<=r;i++)\r\n\t{\r\n\t\tregister int x=idx[i];\r\n\t\tstruct _point ts=(_point){1,h[x]};\r\n\t\twhile(front+1<rear && (list[front+1]-list[front])*ts>=0)\r\n\t\t\tfront++;\r\n\t\tLLONG tf=list[front].y-(LLONG)h[x]*list[front].x+sw[x-1]+(LLONG)h[x]*h[x];\r\n\t\tf[x]=_min(f[x],tf);\r\n\t}\r\n\tCDQ_Divide(mid+1,r);\r\n\t\r\n\ttl=l,tr=mid+1;\r\n\tfor(register int i=l;i<=r;i++)\r\n\t\tif((tl<=mid && glist[tl].x<=glist[tr].x) || tr>r)\r\n\t\t\ttlist[i]=glist[tl++];\r\n\t\telse\r\n\t\t\ttlist[i]=glist[tr++];\r\n\tfor(register int i=l;i<=r;i++)\r\n\t\tglist[i]=tlist[i];\r\n}\r\n\r\nint main()\r\n{\r\n\tregister int N=getnum();\r\n\tfor(register int i=1;i<=N;i++)\r\n\t\th[i]=getnum(),idx[i]=i;\r\n\tfor(register int i=1;i<=N;i++)\r\n\t\tsw[i]=sw[i-1]+getnum();\r\n\t\r\n\t\/*for(register int i=2;i<=N;i++)\r\n\t{\r\n\t\tf[i]=LLONG_MAX;\r\n\t\tfor(register int j=1;j<i;j++)\r\n\t\t{\r\n\t\t\tLLONG ts=f[j]+sw[i-1]-sw[j]+(LLONG)(h[i]-h[j])*(h[i]-h[j]);\r\n\t\t\tif(ts<f[i])\r\n\t\t\t\tf[i]=ts;\r\n\t\t}\r\n\t}*\/\r\n\t\r\n\tstd::sort(&idx[1],&idx[N+1],_comp());\r\n\tmemset(f,0x7F,sizeof(f)),f[1]=0;\r\n\tCDQ_Divide(1,N);\r\n\t\r\n\tprintf(\"%lld\\n\",f[N]);\r\n\t\r\n\treturn 0;\r\n}<\/pre>\n <\/p>\n
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