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2.聽
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pascal
1锛�鏁扮粍鍨�
Const
m=鏍堣〃鐩暟鐨勪笂闄愶紱
Type
stack=array[1..m] of stype; {鏍堢被鍨媫
Var
s:stack;{鏍坿
top:integer;
2锛庤褰曞瀷
const
m=鏍堣〃鐩暟鐨勪笂闄愶紱
type
stack=record
elem: array[1..m] of elemtp;
top:0..m; {鏍堥《鎸囬拡}
end;
Var
s:stack;{鏍坿
C++浠ｇ爜
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#include
classSqStack
{
private:
enum{MaxSize=100};
intdata[MaxSize];
inttop;
public:
SqStack();
~SqStack();
boolisEmpty();
voidpushInt(intx);
intpopInt();
intgetTop();
voiddisplay();
};
SqStack::SqStack()
{
top=-1;
}
SqStack::~SqStack(){}
boolSqStack::isEmpty()//鍒ゆ柇鏍堜负绌�
{
return(top==-1);
}
voidSqStack::pushInt(intx)//鍏冪礌杩涙爤
{
if(top==MaxSize-1)
{
std::cout<<"鏍堜笂婧㈠嚭锛�"<
}
else
{
++top;
data[top]=x;
}
}
intSqStack::popInt()//閫�鏍�
{
inttmp=0;
if(top==-1)
{
std::cout<<"鏍堝凡绌猴紒"<
}
else
{
tmp=data[top--];
}
returntmp;
}
intSqStack::getTop()//鑾峰緱鏍堥《鍏冪礌
{
inttmp=0;
if(top==-1)
{
std::cout<<"鏍堢┖锛�"<
}
else
{
tmp=data[top];
}
returntmp;
}
voidSqStack::display()//鎵撳嵃鏍堥噷鍏冪礌
{
std::cout<<"鏍堜腑鍏冪礌锛�"<
for(intindex=top;index>=0;--index)
{
std::cout<
}
}
intmain()
{
SqStackst;
std::cout<<"鏍堢┖锛�"<
for(inti=1;i<10;i++)
{
st.pushInt(i);
}
st.display();
std::cout<<"閫�涓�娆℃爤"<
st.popInt();
std::cout<<"鏍堥《鍏冪礌:"<
st.popInt();
st.display();
return0;
}
C浠ｇ爜
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#include
#include
#defineDataTypeint
#defineMAXSIZE1024
typedefstruct
{
DataTypedata[MAXSIZE];
inttop;
}SeqStack;
SeqStack*Init_SeqStack()//鏍堝垵濮嬪寲
{
SeqStack*s;
s=(SeqStack*)malloc(sizeof(SeqStack));
if(!s)
{
printf("绌洪棿涓嶈冻\n");
returnNULL;
}
else
{
s->top=-1;
returns;
}
}
intEmpty_SeqStack(SeqStack*s)//鍒ゆ爤绌�
{
if(s->top==-1)
return1;
else
return0;
}
intPush_SeqStack(SeqStack*s,DataTypex)//鍏ユ爤
{
if(s->top==MAXSIZE-1)
return0;//鏍堟弧涓嶈兘鍏ユ爤
else
{
s->top++;
s->data[s->top]=x;
return1;
}
}
intPop_SeqStack(SeqStack*s,DataType*x)//鍑烘爤
{
if(Empty_SeqStack(s))
return0;//鏍堢┖涓嶈兘鍑烘爤
else
{
*x=s->data[s->top];
s->top--;
return1;
}//鏍堥《鍏冪礌瀛樺叆*x锛岃繑鍥�
}
DataTypeTop_SeqStack(SeqStack*s)//鍙栨爤椤跺厓绱�
{
if(Empty_SeqStack(s))
return0;//鏍堢┖
else
returns->data[s->top];
}
intPrint_SeqStack(SeqStack*s)
{
inti;
printf("褰撳墠鏍堜腑鐨勫厓绱�:\n");
for(i=s->top;i>=0;i--)
printf("%3d",s->data[i]);
printf("\n");
return0;
}
intmain()
{
SeqStack*L;
intn,num,m;
inti;
L=Init_SeqStack();
printf("鍒濆鍖栧畬鎴怽n");
printf("鏍堢┖锛�%d\n",Empty_SeqStack(L));
printf("璇疯緭鍏ュ叆鏍堝厓绱犱釜鏁帮細\n");
scanf("%d",&n);
printf("璇疯緭鍏ヨ鍏ユ爤鐨�%d涓厓绱狅細\n",n);
for(i=0;i
{
scanf("%d",&num);
Push_SeqStack(L,num);
}
Print_SeqStack(L);
printf("鏍堥《鍏冪礌锛�%d\n",Top_SeqStack(L));
printf("璇疯緭鍏ヨ鍑烘爤鐨勫厓绱犱釜鏁帮紙涓嶈兘瓒呰繃%d涓級锛歕n",n);
scanf("%d",&n);
printf("渚濇鍑烘爤鐨�%d涓厓绱狅細\n",n);
for(i=0;i
{
Pop_SeqStack(L,&m);
printf("%3d",m);
}
printf("\n");
Print_SeqStack(L);
printf("鏍堥《鍏冪礌锛�%d\n",Top_SeqStack(L));
return0;
}
瀹氫箟stack鐨勭畝鍗曚唬鐮侊細
stack sta;
鍏ユ爤锛歴ta.push(x);
鍑烘爤锛歴ta.pop();
鍒ゆ柇鏍堢殑澶у皬锛毬爏ta.size();
鍒ゆ柇鏍堟槸鍚︿负绌猴細sta.empty();
闃熷垪
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椤哄簭闃熷垪
寤虹珛椤哄簭闃熷垪缁撴瀯蹇呴』涓哄叾闈欐�佸垎閰嶆垨鍔ㄦ�佺敵璇蜂竴鐗囪繛缁殑瀛樺偍绌洪棿锛屽苟璁剧疆涓や釜鎸囬拡杩涜绠＄悊銆備竴涓槸闃熷ご鎸囬拡front锛屽畠鎸囧悜闃熷ご鍏冪礌锛涘彟涓�涓槸闃熷熬鎸囬拡rear锛屽畠鎸囧悜涓嬩竴涓叆闃熷厓绱犵殑瀛樺偍浣嶇疆锛屽鍥炬墍绀�
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锛�1锛壜�"涓嬫孩"鐜拌薄锛氬綋闃熷垪涓虹┖鏃讹紝鍋氬嚭闃熻繍绠椾骇鐢熺殑婧㈠嚭鐜拌薄銆傗�滀笅婧⑩�濇槸姝ｅ父鐜拌薄锛屽父鐢ㄤ綔绋嬪簭鎺у埗杞Щ鐨勬潯浠躲��
锛�2锛�"鐪熶笂婧�"鐜拌薄锛氬綋闃熷垪婊℃椂锛屽仛杩涙爤杩愮畻浜х敓绌洪棿婧㈠嚭鐨勭幇璞°�傗�滅湡涓婃孩鈥濇槸涓�绉嶅嚭閿欑姸鎬侊紝搴旇娉曢伩鍏嶃��
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寰幆闃熷垪
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鍦ㄥ惊鐜槦鍒椾腑锛屽綋闃熷垪涓虹┖鏃讹紝鏈塮ront=rear锛岃�屽綋鎵�鏈夐槦鍒楃┖闂村叏鍗犳弧鏃讹紝涔熸湁front=rear銆備负浜嗗尯鍒繖涓ょ鎯呭喌锛岃瀹氬惊鐜槦鍒楁渶澶氬彧鑳芥湁MaxSize-1涓槦鍒楀厓绱狅紝褰撳惊鐜槦鍒椾腑鍙墿涓嬩竴涓┖瀛樺偍鍗曞厓鏃讹紝闃熷垪灏卞凡缁忔弧浜嗐�傚洜姝わ紝闃熷垪鍒ょ┖鐨勬潯浠舵椂front=rear锛岃�岄槦鍒楀垽婊＄殑鏉′欢鏃秄ront=锛坮ear+1锛�%MaxSize銆傞槦绌哄拰闃熸弧鐨勬儏鍐靛鍥撅細
闃熷垪鐨勬暟缁勫疄鐜�
闃熷垪鍙互鐢ㄦ暟缁凲[1鈥]鏉ュ瓨鍌紝鏁扮粍鐨勪笂鐣宮鍗虫槸闃熷垪鎵�瀹硅鐨勬渶澶у閲忋�傚湪闃熷垪鐨勮繍绠椾腑闇�璁句袱涓�鎸囬拡锛歨ead锛岄槦澶存寚閽堬紝鎸囧悜瀹為檯闃熷ご鍏冪礌锛泃ail锛岄槦灏炬寚閽堬紝鎸囧悜瀹為檯闃熷熬鍏冪礌鐨勪笅涓�涓綅缃�備竴鑸儏鍐典笅锛屼袱涓�鎸囬拡鐨勫垵鍊艰涓�0锛岃繖鏃堕槦鍒椾负绌猴紝娌℃湁鍏冪礌銆傛暟缁勫畾涔塓[1鈥�10]銆俀(i) i=3,4,5,6,7,8銆傚ご鎸囬拡head=2锛屽熬鎸囬拡tail=8銆傞槦鍒椾腑鎷ユ湁鐨勫厓绱犱釜鏁颁负:L=tail-head銆傜幇瑕佽鎺掑ご鐨勫厓绱犲嚭闃燂紝鍒欓渶灏嗗ご鎸囬拡鍔�1銆傚嵆head=head+1杩欐椂澶存寚閽堝悜涓婄Щ鍔ㄤ竴涓綅缃紝鎸囧悜Q(3)锛岃〃绀篞(3)宸插嚭闃熴�傚鏋滄兂璁╀竴涓柊鍏冪礌鍏ラ槦锛屽垯闇�灏�鎸囬拡鍚戜笂绉诲姩涓�涓綅缃�傚嵆tail=tail+1杩欐椂Q(9)鍏ラ槦銆傚綋闃熷熬宸茬粡澶勭悊鍦ㄦ渶涓婇潰鏃讹紝鍗硉ail=10锛屽鏋滆繕瑕佹墽琛屽叆闃熸搷浣滐紝鍒欒鍙戠敓"涓婃孩"锛屼絾瀹為檯涓婇槦鍒椾腑杩樻湁涓変釜绌轰綅缃紝鎵�浠ヨ繖绉嶆孩鍑虹О涓�"鍋囨孩鍑�"銆�
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闃熷垪鍜屾爤涓�鏍峰彧鍏佽鍦ㄦ柇鐐瑰鎻掑叆鍜屽垹闄ゅ厓绱犮��
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1銆乼ail=tail+1锛�
2銆佽嫢tail=n+1锛屽垯tail=1锛�
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4銆佸惁鍒欙紝Q(tail)=X锛岀粨鏉燂紙X涓烘柊鍏ュ嚭鍏冪礌锛夈��
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褰撴湁琛ㄥご缁撶偣鏃讹紝鍏紡鍙樹负cq.rear=(cq.front+1)/锛坢ax+1锛夈��
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闃熷垪閲囩敤鐨凢IFO(first in first out)锛屾柊鍏冪礌锛堢瓑寰呰繘鍏ラ槦鍒楃殑鍏冪礌锛夋�绘槸琚彃鍏ュ埌閾捐〃鐨勫熬閮紝鑰岃鍙栫殑鏃跺�欐�绘槸浠庨摼琛ㄧ殑澶撮儴寮�濮嬭鍙栥�傛瘡娆¤鍙栦竴涓厓绱狅紝閲婃斁涓�涓厓绱犮�傛墍璋撶殑鍔ㄦ�佸垱寤猴紝鍔ㄦ�侀噴鏀俱�傚洜鑰屼篃涓嶅瓨鍦ㄦ孩鍑虹瓑闂銆傜敱浜�閾捐〃鐢�缁撴瀯浣�闂存帴鑰屾垚锛岄亶鍘嗕篃鏂逛究銆�
闃熷垪鐨勫熀鏈繍绠�
锛�1锛夊垵濮嬪寲闃熷垪锛欼nit_Queue(q)聽锛屽垵濮嬫潯浠讹細闃焣聽涓嶅瓨鍦ㄣ�傛搷浣滅粨鏋滐細鏋勯�犱簡涓�涓┖闃燂紱
锛�2锛夊叆闃熸搷浣滐細聽In_Queue(q,x),鍒濆鏉′欢锛� 闃焣聽瀛樺湪銆傛搷浣滅粨鏋滐細 瀵瑰凡瀛樺湪鐨勯槦鍒梣锛屾彃鍏ヤ竴涓厓绱爔聽鍒伴槦灏撅紝闃熷彂鐢熷彉鍖栵紱
锛�3锛夊嚭闃熸搷浣滐細聽Out_Queue(q,x)锛屽垵濮嬫潯浠�:聽闃焣聽瀛樺湪涓旈潪绌猴紝鎿嶄綔缁撴灉锛� 鍒犻櫎闃熼鍏冪礌锛屽苟杩斿洖鍏跺�硷紝闃熷彂鐢熷彉鍖栵紱
锛�4锛夎闃熷ご鍏冪礌锛欶ront_Queue(q,x)锛屽垵濮嬫潯浠�:聽闃焣聽瀛樺湪涓旈潪绌猴紝鎿嶄綔缁撴灉锛� 璇婚槦澶村厓绱狅紝骞惰繑鍥炲叾鍊硷紝闃熶笉鍙橈紱
锛�5锛夊垽闃熺┖鎿嶄綔锛欵mpty_Queue(q)锛屽垵濮嬫潯浠讹細 闃焣聽瀛樺湪锛屾搷浣滅粨鏋滐細 鑻聽涓虹┖闃熷垯杩斿洖涓�1锛屽惁鍒欒繑鍥炰负0銆俒3]
鎿嶄綔绫诲瀷浣滅敤杩斿洖鍊间緥瀛�
length(s)鍑芥暟姹傚瓧绗︿覆s鐨勯暱搴︽暣鍨媠:='123456789';
l:=length(s);{l鐨勫�间负9}
copy锛坰,w,k)鍑芥暟澶嶅埗s涓粠w寮�濮嬬殑k浣嶅瓧绗︿覆s:='123456789';
s1:=copy(s,3,5);{s1鐨勫�兼槸'34567'}
val(s,k,code)杩囩▼灏嗗瓧绗︿覆s杞负鏁板�硷紝瀛樺湪k涓紱code鏄敊璇唬鐮乿ar s:string;k,code:integer;
begin
s:='1234';
val(s,k,code);
write(k);{k=1234}
str(i,s)杩囩▼灏嗘暟鍊糹杞负瀛楃涓瞫i:=1234;
str(i,s);
write(s);{s='1234'}
Delete(s,w,k)杩囩▼鍦╯涓垹闄や粠绗瑆浣嶅紑濮嬬殑k涓瓧绗 := 'Honest Abe Lincoln';
Delete(s,8,4);
Writeln(s); { 'Honest Lincoln' }
Insert(s1, S, w)杩囩▼灏唖1鎻掑埌s涓w浣峉 := 'Honest Lincoln';
Insert('Abe ', S, 8); { 'Honest Abe Lincoln' }
Pos(c, S)鍑芥暟姹傚瓧绗鍦╯涓殑浣嶇疆鏁村瀷S := ' 123.5';
i :=Pos(' ', S);{i鐨勫�间负1}
+杩愮畻绗﹀皢涓や釜瀛楃涓茶繛鎺ヨ捣鏉1:='1234';
s2:='5678';
s:=s1+s2;{'12345678'}
鍦⊿TL涓紝瀵归槦鍒楃殑浣跨敤寰堟槸杈冨畬缇�
涓嬮潰缁欏嚭寰幆闃熷垪鐨勮繍绠楃畻娉�:
(1)灏嗗惊鐜槦鍒楃疆涓虹┖
//灏嗛槦鍒楀垵濮嬪寲
SeQueue::SeQueue()
{ front=0;
rear=0;
cout<<"init!"<
}
(2)鍒ゆ柇寰幆闃熷垪鏄惁涓虹┖
int SeQueue::Empty()
{ if(rear==front) return(1);
else return(0);
}
(3)鍦ㄥ惊鐜槦鍒椾腑鎻掑叆鏂扮殑鍏冪礌x
void SeQueue::AddQ(ElemType x)
{ if((rear+1) % MAXSIZE==front) cout<<" QUEUE IS FULL! "<
else{ rear=(rear+1) % MAXSIZE;
elem[rear]=x;
cout<<" OK!";
}
}
(4)鍒犻櫎闃熷垪涓槦棣栧厓绱�
ElemType SeQueue::DelQ()
{ if(front==rear)
{ cout<<" QUEUE IS EMPTY! "<
else{ front=(front+1) % MAXSIZE;
return(elem[front]);
}
}
(5)鍙栭槦鍒椾腑鐨勯槦棣栧厓绱�
ElemType SeQueue::Front()
{ ElemType x;
if(front== rear)
cout<<"QUEUE IS EMPTY "<
else x= elem[(front+1)%MAXSIZE];
return (x);
}
閾捐〃
绾挎�ц〃鐨勯摼寮忓瓨鍌ㄨ〃绀虹殑鐗圭偣鏄敤涓�缁勪换鎰忕殑瀛樺偍鍗曞厓瀛樺偍绾挎�ц〃鐨�鏁版嵁鍏冪礌锛堣繖缁勫瓨鍌ㄥ崟鍏冨彲浠ユ槸杩炵画鐨勶紝涔熷彲浠ユ槸涓嶈繛缁殑锛夈�傚洜姝わ紝涓轰簡琛ㄧず姣忎釜鏁版嵁鍏冪礌涓庡叾鐩存帴鍚庣户鏁版嵁鍏冪礌 涔嬮棿鐨勯�昏緫鍏崇郴锛屽鏁版嵁鍏冪礌 鏉ヨ锛岄櫎浜�瀛樺偍鍏舵湰韬殑淇℃伅涔嬪锛岃繕闇�瀛樺偍涓�涓寚绀哄叾鐩存帴鍚庣户鐨勪俊鎭紙鍗崇洿鎺ュ悗缁х殑瀛樺偍浣嶇疆锛夈�傜敱杩欎袱閮ㄥ垎淇℃伅缁勬垚涓�涓�"缁撶偣"锛堝姒傝堪鏃佺殑鍥炬墍绀猴級锛岃〃绀虹嚎鎬ц〃涓竴涓�鏁版嵁鍏冪礌銆傜嚎鎬ц〃鐨勯摼寮忓瓨鍌ㄨ〃绀猴紝鏈変竴涓己鐐瑰氨鏄鎵句竴涓暟锛屽繀椤昏浠庡ご寮�濮嬫壘璧凤紝鍗佸垎楹荤儲銆�
鏍规嵁鎯呭喌锛屼篃鍙互鑷繁璁捐閾捐〃鐨勫叾瀹冩墿灞曘�備絾鏄竴鑸笉浼氬湪杈逛笂闄勫姞鏁版嵁锛屽洜涓洪摼琛ㄧ殑鐐瑰拰杈瑰熀鏈笂鏄竴涓�瀵瑰簲鐨勶紙闄や簡绗竴涓垨鑰呮渶鍚庝竴涓�鑺傜偣锛屼絾鏄篃涓嶄細浜х敓鐗规畩鎯呭喌锛夈�備笉杩囨湁涓�涓壒渚嬫槸濡傛灉閾捐〃鏀寔鍦ㄩ摼琛ㄧ殑涓�娈典腑鎶婂墠鍜屽悗鎸囬拡鍙嶅悜锛屽弽鍚戞爣璁板姞鍦ㄨ竟涓婂彲鑳戒細鏇存柟渚裤��
瀵逛簬闈炵嚎鎬х殑閾捐〃锛屽彲浠ュ弬瑙佺浉鍏崇殑鍏朵粬鏁版嵁缁撴瀯锛屼緥濡傛爲銆佸浘銆傚彟澶栨湁涓�绉嶅熀浜庡涓嚎鎬ч摼琛ㄧ殑鏁版嵁缁撴瀯锛氳烦琛紝鎻掑叆銆佸垹闄ゅ拰鏌ユ壘绛夊熀鏈搷浣滅殑閫熷害鍙互杈惧埌O(nlogn锛夛紝鍜�骞宠 浜屽弶鏍�涓�鏍枫��
鍏朵腑瀛樺偍鏁版嵁鍏冪礌淇℃伅鐨勫煙绉颁綔鏁版嵁鍩燂紙璁惧煙鍚嶄负data锛夛紝瀛樺偍鐩存帴鍚庣户瀛樺偍浣嶇疆鐨勫煙绉颁负鎸囬拡鍩燂紙璁惧煙鍚嶄负next锛夈��鎸囬拡鍩熶腑瀛樺偍鐨勪俊鎭張绉板仛鎸囬拡鎴栭摼銆�
鐢卞垎鍒〃绀�,锛屸�︼紝鐨凬聽涓粨鐐逛緷娆＄浉閾炬瀯鎴愮殑閾捐〃锛岀О涓�绾挎�ц〃鐨勯摼寮忓瓨鍌ㄨ〃绀猴紝鐢变簬姝ょ被閾捐〃鐨勬瘡涓粨鐐逛腑鍙寘鍚竴涓寚閽堝煙锛屾晠鍙堢О鍗曢摼琛�鎴栫嚎鎬ч摼琛ㄣ��
鍩烘湰鎿嶄綔
锛�pascal璇█锛�
寤虹珛
绗竴琛岃鍏锛岃〃绀簄涓暟
绗簩琛屽寘鎷琻涓暟
浠ラ摼琛ㄧ殑褰㈠紡瀛樺偍杈撳嚭杩欎簺鏁�
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programproject1;
type
point=^node;
node=record
data:longint;
next:point;
end;
var
i,n,e:longint;
p,q,head,last:point;
begin
write('Inputthenumbercount:');
readln(n);
i:=1;
new(head);
read(e);
head^.data:=e;
head^.next:=nil;
last:=head;
q:=head;
whilei
begin
inc(i);
read(e);
new(p);
q^.next:=p;
p^.data:=e;
p^.next:=nil;
last:=p;
q:=last
end;
//寤虹珛閾捐〃
q:=head;
whileq^.next<>nildo
begin
write(q^.data,'');
q:=q^.next;
end;
write(q^.data);
//杈撳嚭
readln;
readln
end.
鍒犻櫎
鍦ㄤ互z涓哄ご鐨勯摼琛ㄤ腑鎼滅储绗竴涓猲锛屽鏋滄壘鍒板垯鍒犲幓锛岃繑鍥炲�间负1锛屽惁鍒欒繑鍥�0
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functiondelete(n:longint;varz:point):longint;
var
t,s:point;
begin
t:=z;
while(t^.next<>nil)and(t^.data<>n)do
begin
s:=t;
t:=t^.next;
end;
ift^.data<>nthenexit(0);
s^.next:=t^.next;
dispose(t);
exit鈶�
end;
鏌ユ壘
绫讳技浜庡垹闄わ紝鍙渶瑕佹壘鍒颁笉鍒犲嵆鍙�
鎻掑叆
鎻掑叆锛屽湪浠z涓哄ご鐨勯摼琛ㄧw涓殑鍓嶉潰鎻掑叆nn鍏冪礌锛屽嚱鏁拌繑鍥炲�兼甯告槸0锛屽鏋渨瓒呰繃浜嗛摼琛ㄧ殑闀垮害锛屽嚱鏁拌繑鍥為摼琛ㄧ殑闀垮害
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functioninsert(w,nn:longint;varzz:point):longint;
vard:longint;v,vp,vs:point;
begin
v:=zz;
ford:=1towdo
ifv^.next=nil
thenexit(d)
else
begin
vp:=v;
v:=v^.next;
end;
new(vs);
vs^.data:=nn;
vp^.next:=vs;
vs^.next:=v;
exit(0)
end;
閾捐〃鍑芥暟
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#include
#include
#include
usingnamespacestd;
structNode
{
intdata;//鏁版嵁鍩�
structNode*next;//鎸囬拡鍩�
};
/*
Create
*鍑芥暟鍔熻兘锛氬垱寤洪摼琛�.
*杈撳叆锛氬悇鑺傜偣鐨刣ata
*杩斿洖鍊硷細鎸囬拡head
*/
Node*Create()
{
intn=0;
Node*head,*p1,*p2;
p1=p2=newNode;
cin>>p1->data;
head=NULL;
while(p1->data!=0)
{
if(n==0)
{
head=p1;
}
else
p2->next=p1;
p2=p1;
p1=newNode;
cin>>p1->data;
n++;
}
p2->next=NULL;
returnhead;
}
/*
insert
*鍑芥暟鍔熻兘锛氬湪閾捐〃涓彃鍏ュ厓绱�.
*杈撳叆锛歨ead閾捐〃澶存寚閽堬紝p鏂板厓绱犳彃鍏ヤ綅缃紝x鏂板厓绱犱腑鐨勬暟鎹煙鍐呭
*杩斿洖鍊硷細鏃�
*/
voidinsert(Node*head,intp,intx)
{
Node*tmp=head;//for寰幆鏄负浜嗛槻姝㈡彃鍏ヤ綅缃秴鍑轰簡閾捐〃闀垮害
for(inti=0;i
{
if(tmp==NULL)
return;
if(i
tmp=tmp->next;
}
Node*tmp2=newNode;
tmp2->data=x;
tmp2->next=tmp->next;
tmp->next=tmp2;
}
/*
del
*鍑芥暟鍔熻兘锛氬垹闄ら摼琛ㄤ腑鐨勫厓绱�
*杈撳叆锛歨ead閾捐〃澶存寚閽堬紝p琚垹闄ゅ厓绱犱綅缃�
*杩斿洖鍊硷細琚垹闄ゅ厓绱犱腑鐨勬暟鎹煙.濡傛灉鍒犻櫎澶辫触杩斿洖-1
*/
intdel(Node*head,intp)
{
Node*tmp=head;
for(inti=0;i
{
if(tmp==NULL)
return-1;
if(i
tmp=tmp->next;
}
intret=tmp->next->data;
tmp->next=tmp->next->next;
returnret;
}
voidprint(Node*head)
{
for(Node*tmp=head;tmp!=NULL;tmp=tmp->next)
printf("%d",tmp->data);
printf("\n");
}
intmain()
{
Node*head;
head=newNode;
head->data=-1;
head->next=NULL;
return0;
}
渚嬪瓙
#include
#defineNULL0
structstudent
{
longnum;
structstudent*next;
};
intmain()
{
inti,n;
student*p=(structstudent*)malloc(sizeof(structstudent));
student*q=p;
printf("杈撳叆鍑犱釜鍊�");
scanf("%d",&n);
for(i=1;i<=n;i++)
{
scanf("%d",&(q->num));
q->next=(structstudent*)malloc(sizeof(structstudent));
q=q->next;
}
printf("鍊肩鍑犱釜");
intrank;
scanf("%d%d",&(q->num),&rank);
student*w=p;
for(i=1;i
{
w=w->next;
}
q->next=w->next;
w->next=q;
for(i=1;i<=n+1;i++)
{
printf("%d",p->num);
p=p->next;
}
return0;
}//鎸囬拡鍚庣Щ楹荤儲閾捐〃褰㈠紡寰幆閾捐〃
寰幆閾捐〃鏄笌鍗曢摼琛ㄤ竴鏍凤紝鏄竴绉嶉摼寮忕殑瀛樺偍缁撴瀯锛屾墍涓嶅悓鐨勬槸锛屽惊鐜摼琛ㄧ殑鏈�鍚庝竴涓粨鐐圭殑鎸囬拡鏄寚鍚戣寰幆閾捐〃鐨勭涓�涓粨鐐规垨鑰呰〃澶寸粨鐐癸紝浠庤�屾瀯鎴愪竴涓幆褰㈢殑閾俱��
寰幆閾捐〃鐨勮繍绠椾笌鍗曢摼琛ㄧ殑杩愮畻鍩烘湰涓�鑷淬�傛墍涓嶅悓鐨勬湁浠ヤ笅鍑犵偣锛�
1銆佸湪寤虹珛涓�涓惊鐜摼琛ㄦ椂锛屽繀椤讳娇鍏舵渶鍚庝竴涓粨鐐圭殑鎸囬拡鎸囧悜琛ㄥご缁撶偣锛岃�屼笉鏄薄鍗曢摼琛ㄩ偅鏍风疆涓篘ULL銆傛绉嶆儏鍐佃繕浣跨敤浜庡湪鏈�鍚庝竴涓粨鐐瑰悗鎻掑叆涓�涓柊鐨勭粨鐐广��
2銆佸湪鍒ゆ柇鏄惁鍒拌〃灏炬椂锛屾槸鍒ゆ柇璇ョ粨鐐归摼鍩熺殑鍊兼槸鍚︽槸琛ㄥご缁撶偣锛屽綋閾惧煙鍊肩瓑浜庤〃澶�鎸囬拡鏃讹紝璇存槑宸插埌琛ㄥ熬銆傝�岄潪璞″崟閾捐〃閭ｆ牱鍒ゆ柇閾惧煙鍊兼槸鍚︿负NULL銆�
鍙屽悜閾捐〃
鍙屽悜閾捐〃鍏跺疄鏄崟閾捐〃鐨勬敼杩涖��
褰撴垜浠鍗曢摼琛ㄨ繘琛屾搷浣滄椂锛屾湁鏃朵綘瑕佸鏌愪釜缁撶偣鐨勭洿鎺ュ墠椹辫繘琛屾搷浣滄椂锛屽張蹇呴』浠庤〃澶村紑濮嬫煡鎵俱�傝繖鏄敱鍗曢摼琛ㄧ粨鐐圭殑缁撴瀯鎵�闄愬埗鐨勩�傚洜涓哄崟閾捐〃姣忎釜缁撶偣鍙湁涓�涓瓨鍌ㄧ洿鎺ュ悗缁х粨鐐瑰湴鍧�鐨勯摼鍩燂紝閭ｄ箞鑳戒笉鑳藉畾涔変竴涓棦鏈夊瓨鍌ㄧ洿鎺ュ悗缁х粨鐐瑰湴鍧�鐨勯摼鍩燂紝鍙堟湁瀛樺偍鐩存帴鍓嶉┍缁撶偣鍦板潃鐨勯摼鍩熺殑杩欐牱涓�涓弻閾惧煙缁撶偣缁撴瀯鍛紵杩欏氨鏄弻鍚戦摼琛ㄣ��
鍦ㄥ弻鍚戦摼琛ㄤ腑锛岀粨鐐归櫎鍚湁鏁版嵁鍩熷锛岃繕鏈変袱涓摼鍩燂紝涓�涓瓨鍌ㄧ洿鎺ュ悗缁х粨鐐瑰湴鍧�锛屼竴鑸О涔嬩负鍙抽摼鍩燂紱涓�涓瓨鍌ㄧ洿鎺ュ墠椹辩粨鐐瑰湴鍧�锛屼竴鑸О涔嬩负宸﹂摼鍩熴��
搴旂敤涓句緥姒傝堪
绾︾憻澶幆闂锛氬凡鐭涓汉锛堜互缂栧彿1锛�2锛�3...n鍒嗗埆琛ㄧず锛夊洿鍧愬湪涓�寮犲渾妗屽懆鍥淬�備粠缂栧彿涓簁鐨勪汉寮�濮嬫姤鏁帮紝鏁板埌m鐨勯偅涓汉鍑哄垪锛涗粬鐨勪笅涓�涓汉鍙堜粠1寮�濮嬫姤鏁帮紝鏁板埌m鐨勯偅涓汉鍙堝嚭鍒楋紱渚濇瑙勫緥閲嶅涓嬪幓锛岀洿鍒板渾妗屽懆鍥寸殑浜哄叏閮ㄥ嚭鍒椼�備緥濡傦細n = 9,k = 1,m = 5
鍙傝�冧唬鐮�
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#include
#include
#defineN41
#defineM5
typedefstructnode*link;
structnode
{
intitem;
linknext;
};
linkNODE(intitem,linknext)
{
linkt=malloc(sizeof*t);
t->item=item;
t->next=next;
returnt;
}
intmain(void)
{
inti;
linkt=NODE锛�1,NULL);
t->next=t;
for(i=2;i<=N;i++)
t=t->next=NODE(i,t->next);
while(t!=t->next)
{
for(i=1;i
t=t->next;
t->next=t->next->next;
}
printf("%d\n",t->item);
return0;
}
鍏朵粬鐩稿叧缁撹涓庝釜浜烘�荤粨
C璇█鏄涔�鏁版嵁缁撴瀯鐨勫緢濂界殑瀛︿範宸ュ叿銆傜悊瑙ｄ簡C涓敤缁撴瀯浣�鎻忚堪鏁版嵁缁撴瀯锛岄偅涔堝浜庣悊瑙ｅ叾C++鎻忚堪锛孞ava鎻忚堪閮藉氨杞昏�屾槗涓句簡锛�
閾捐〃鐨勬彁鍑轰富瑕佸湪浜庨『搴忓瓨鍌ㄤ腑鐨勬彃鍏ュ拰鍒犻櫎鐨�鏃堕棿澶嶆潅搴�鏄嚎鎬ф椂闂寸殑锛岃�岄摼琛ㄧ殑鎿嶄綔鍒欏彲浠ユ槸甯告暟鏃堕棿鐨勫鏉傚害銆傚浜庨摼琛ㄧ殑鎻掑叆涓庡垹闄ゆ搷浣滐紝涓汉鍋氫簡涓�鐐规�荤粨锛岄�傜敤浜庡悇绉嶉摼琛ㄥ涓嬶細
鎻掑叆鎿嶄綔澶勭悊椤哄簭锛氫腑闂磋妭鐐圭殑閫昏緫锛屽悗鑺傜偣閫昏緫锛屽墠鑺傜偣閫昏緫銆傛寜鐓ц繖涓『搴忓鐞嗗彲浠ュ畬鎴愪换浣曢摼琛ㄧ殑鎻掑叆鎿嶄綔銆�
鍒犻櫎鎿嶄綔鐨勫鐞嗛『搴忥細鍓嶈妭鐐归�昏緫锛屽悗鑺傜偣閫昏緫锛屼腑闂磋妭鐐归�昏緫銆�
鎸夌収姝ら『搴忓彲浠ュ鐞嗕换浣曢摼琛ㄧ殑鍒犻櫎鎿嶄綔銆�
濡傛灉涓嶅瓨鍦ㄥ叾涓殑鏌愪釜鑺傜偣鐣ヨ繃鍗冲彲銆�
涓婇潰鐨勬�荤粨锛屽ぇ瀹跺彲浠ョ湅鍒颁竴涓幇璞★紝灏辨槸鎻掑叆鐨勯『搴忓拰鍒犻櫎鐨勯『搴忔伆濂芥槸鐩稿弽鐨勶紝寰堟湁鎰忔�濓紒
鎿嶄綔
-----鎮夊凹澶у宸ョ▼瀛﹂櫌寮犲織鍒�锛圫tone Cold锛変綔鍝�
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#include
#include
#include
typedefstructSlist
{
intdata;
structSlist*next;
}
SLIST;
SLIST*InitList_Sq()/*鍒濆鍖栧嚱鏁�*/
{
inta;
SLIST*h,*s,*r;
h=(SLIST*)malloc(sizeof(SLIST));/*寤虹珛澶存寚閽堬紝澶存寚閽堜笉鍙互鏇存敼锛侊紒锛�*/
r=h;
if(!h)
{
printf("鍒嗛厤澶辫触");
exit(0);
}
scanf("%d",&a);
for(;a!=-1;)
{
s=(SLIST*)malloc(sizeof(SLIST));/*姣忔閮藉紑杈熶竴涓粨鐐圭┖闂村苟璧嬪��*/
s->data=a;
r->next=s;
r=s;
scanf("%d",&a);
}
r->next='\0';
returnh;
}
voidprint_list(SLIST*finder)/*鎵撳嵃鍑芥暟*/
{
while(finder!='\0')
{
printf("->%d",finder->data);
finder=finder->next;
}
printf("->end\n");
}
intDeleteNode(SLIST*killer)//鍒犻櫎鑺傜偣鍑芥暟
{
inti,j=0;
SLIST*p,*q;
intx;
p=killer;
q=killer->next;
printf("璇疯緭鍏ユ偍瑕佸垹闄ょ殑鑺傜偣搴忓彿锛�");
scanf("%d",&i);
while((p->next!='\0')&&(j
{
p=p->next;
j++;
q=p->next;
}
if(p->next=='\0'||j>i-1锛�
{
printf("\nerror");
return-1;
}
else
{
p->next=q->next;
x=q->data;
free(q);
returnx;
}
}
voidInsert_Node(SLIST*jumper)//鎻掑叆鍑芥暟锛屾湰绠楁硶涓哄墠鎻掔粨鐐规硶
{
intt,e,j=0;
SLIST*p,*q;
p=jumper;
printf("璇疯緭鍏ヨ鎻掑叆浣嶇疆鐨勫簭鍙凤細");
scanf("%d",&t);
printf("璇疯緭鍏ヨ鎻掑叆鐨勫厓绱狅細");
scanf("%d",&e);
while(p->next!='\0'&&j
{
j++;
p=p->next;
}
if(p=='\0'||j>t-1锛�
printf("鎻掑叆鐨勭洰鐨勪綅缃笉瀛樺湪");
else
{
q=(SLIST*)malloc(sizeof(SLIST));
q->data=e;
q->next=p->next;
p->next=q;
}
}
voidLocate_List(SLIST*reader)//鏌ユ壘鍊间负e鐨勫厓绱�
{
inte,i=0;
SLIST*p;
p=reader;
printf("璇疯緭鍏ヨ鏌ユ壘鐨勫厓绱狅細");
scanf("%d",&e);
while(p->next!='\0'&&p->data!=e)
{
i++;
p=p->next;
}
if(p->data==e)
printf("姝ゅ厓绱犲湪%d鍙蜂綅缃甛n",i);
else
printf("鏃犳鍏冪礌锛�");
}
voidmain()
{
inti,k,y;
SLIST*head;
printf("\n1.寤虹珛绾挎�ц〃");
printf("\n2.鍦╥浣嶇疆鎻掑叆鍏冪礌e");
printf("\n3.鍒犻櫎绗琲涓厓绱狅紝杩斿洖鍏跺��");
printf("\n4.鏌ユ壘鍊间负e鐨勫厓绱�");
printf("\n5.缁撴潫绋嬪簭杩愯");
printf("\n===================================================");
printf("璇疯緭鍏ユ偍鐨勯�夋嫨锛�");
scanf("%d",&k);
switch(k)
{
case1:
{
head=InitList_Sq();
print_list(head->next);
}break;
case2:
{
head=InitList_Sq();
print_list(head->next);
Insert_Node(head);
print_list(head->next);
}
break;
case3:
{
head=InitList_Sq();
print_list(head->next);
y=DeleteNode(head);
print_list(head->next);
if(y!=-1锛�
printf("琚垹闄ゅ厓绱犱负锛�%d",y);
}break;//澶寸粨鐐逛笉绠楋紝浠庢湁鏁版嵁鐨勫紑濮嬬畻绗竴涓�
case4:
{
head=InitList_Sq();
print_list(head->next);
Locate_List(head);
}break;
}
}
鏈▼搴忓彲鍦�寰蒋VC++涓嬬紪璇戦�氳繃骞朵笖杩愯
浣跨敤鏂规硶绠�浠嬶細杩愯绋嬪簭鍚庯紝鍏堟墦鏁板瓧1锛岀劧鍚庡洖杞︼紝杩欐牱灏卞彲浠ュ厛鍒涘缓涓�涓柊鐨勯摼琛紝姣斿浣犺鍒涘缓涓�涓�
4->5->6->7杩欐牱涓�涓摼琛紝浣犲氨杈撳叆鏁板瓧4鍥炶溅锛岃緭鍏�5鍥炶溅锛岃緭鍏�6鍥炶溅锛岃緭鍏�7鍥炶溅锛屾渶鍚庤緭鍏�-1鍥炶溅锛岃繖涓�-1灏辨槸鍛婅瘔绋嬪簭鍒版涓烘鐨勬爣蹇�
鍋囧浣犺浣跨敤鎻掑叆鐨勫姛鑳戒綅缃彃鍏ワ紝灏辫緭鍏�3锛屽洖杞︼紝绋嬪簭浼氶棶浣犳彃鍏ョ殑鏁板�兼槸浠�涔堬紝姣斿浣犺鎻掑叆999锛岀劧鍚庡洖杞︼紝999灏辫鎻掕繘鍘讳簡
鍏朵粬鐨勫姛鑳介兘澶у悓灏忓紓
Data_structures
鈻�闆嗗悎鈻�瀹瑰櫒聽聽
鈻�鏁扮粍鈻�鍏宠仈鏁扮粍鈻�Multimap鈻�闆�
鈻�澶氶噸闆�鈻�鏁ｅ垪琛�鈻�鏍戠姸鏁扮粍聽
鈻�鍒楄〃鈻摼琛ㄢ柂闃熷垪鈻�鍫嗘爤
鈻�寰幆闃熷垪鈻�璺宠穬鍒楄〃聽聽
鈻�鏍�鈻�浜屽弶鏌ユ壘鏍�鈻�鍫�鈻�绾挎鏍�
鈻�绾㈤粦鏍�鈻�AVL鏍�聽聽
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绉戝鐧剧淇℃伅绉戝鍒嗙被聽锛屄�涓浗鐢靛瓙瀛︿細聽锛屄�
浜屽弶鏍�
鍦ㄨ绠楁満绉戝涓紝浜屽弶鏍戞槸姣忎釜缁撶偣鏈�澶氭湁涓や釜瀛愭爲鐨勬爲缁撴瀯銆傞�氬父瀛愭爲琚О浣溾�滃乏瀛愭爲鈥濓紙left subtree锛夊拰鈥滃彸瀛愭爲鈥濓紙right subtree锛夈�備簩鍙夋爲甯歌鐢ㄤ簬瀹炵幇浜屽弶鏌ユ壘鏍戝拰浜屽弶鍫嗐��
浜屽弶鏍戠殑姣忎釜缁撶偣鑷冲鍙湁浜屾５瀛愭爲(涓嶅瓨鍦ㄥ害澶т簬2鐨勭粨鐐�)锛屼簩鍙夋爲鐨勫瓙鏍戞湁宸﹀彸涔嬪垎锛屾搴忎笉鑳介鍊掋�備簩鍙夋爲鐨勭i灞傝嚦澶氭湁2^{i-1}涓粨鐐癸紱娣卞害涓簁鐨勪簩鍙夋爲鑷冲鏈�2^k-1涓粨鐐癸紱瀵逛换浣曚竴妫典簩鍙夋爲T锛屽鏋滃叾缁堢缁撶偣鏁颁负n_0锛屽害涓�2鐨勭粨鐐规暟涓簄_2锛屽垯n_0=n_2+1銆�
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涓枃鍚�
浜屽弶鏍�
澶栨枃鍚�
Binary Tree
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璁＄畻鏈轰腑鏁版嵁缁撴瀯鐨勪竴绉�
绠�浠�
姣忎釜缁撶偣鏈�澶氭湁涓や釜瀛愭爲鐨勬爲缁撴瀯
鐩綍
00001.聽1聽瀹氫箟
00002.聽2聽鍩烘湰姒傚康
00003.聽鈻�绫诲瀷
00004.聽鈻�鐩稿叧鏈
00005.聽鈻�浜屽弶鏍戞�ц川
00001.聽鈻�瀛樺偍缁撴瀯
00002.聽鈻�杈ㄦ瀽
00003.聽3聽閬嶅巻椤哄簭
00004.聽鈻�鍏堝簭閬嶅巻
00005.聽鈻�涓簭閬嶅巻
00001.聽鈻�鍚庡簭閬嶅巻
00002.聽鈻�灞傛閬嶅巻
00003.聽鈻�绾跨储浜屽弶鏍�
00004.聽4聽瀹炵幇婕旂ず
瀹氫箟
缂栬緫
浜屽弶鏍戝湪鍥捐涓槸杩欐牱瀹氫箟鐨勶細浜屽弶鏍戞槸涓�涓繛閫氱殑鏃犵幆鍥撅紝骞朵笖姣忎竴涓《鐐圭殑搴︿笉澶т簬3銆傛湁鏍逛簩鍙夋爲杩樿婊¤冻鏍圭粨鐐圭殑搴︿笉澶т簬2銆傛湁浜嗘牴缁撶偣涔嬪悗锛屾瘡涓《鐐瑰畾涔変簡鍞竴鐨勭埗缁撶偣锛屽拰鏈�澶�2涓瓙缁撶偣銆傜劧鑰岋紝娌℃湁瓒冲鐨勪俊鎭潵鍖哄垎宸︾粨鐐瑰拰鍙崇粨鐐广�傚鏋滀笉鑰冭檻杩為�氭�э紝鍏佽鍥句腑鏈夊涓繛閫氬垎閲忥紝杩欐牱鐨勭粨鏋勫彨鍋氭．鏋椼��
鍩烘湰姒傚康
缂栬緫
浜屽弶鏍戞槸閫掑綊瀹氫箟鐨勶紝鍏剁粨鐐规湁宸﹀彸瀛愭爲涔嬪垎锛岄�昏緫涓婁簩鍙夋爲鏈変簲绉嶅熀鏈舰鎬侊細
(1)绌轰簩鍙夋爲鈥斺�斿鍥�(a)锛�
(2)鍙湁涓�涓牴缁撶偣鐨勪簩鍙夋爲鈥斺�斿鍥�(b)锛�
(3)鍙湁宸﹀瓙鏍戔�斺�斿鍥�(c)锛�
(4)鍙湁鍙冲瓙鏍戔�斺�斿鍥�(d)锛�
(5)瀹屽叏浜屽弶鏍�鈥斺�斿鍥�(e)銆�
娉ㄦ剰锛氬敖绠′簩鍙夋爲涓庢爲鏈夎澶氱浉浼间箣澶勶紝浣嗕簩鍙夋爲涓嶆槸鏍戠殑鐗规畩鎯呭舰銆俒1]
绫诲瀷
(1)瀹屽叏浜屽弶鏍�鈥斺�旇嫢璁句簩鍙夋爲鐨勯珮搴︿负h锛岄櫎绗爃聽灞傚锛屽叾瀹冨悇灞偮�(1锝瀐-1)聽鐨勭粨鐐规暟閮借揪鍒版渶澶т釜鏁帮紝绗琱灞傛湁鍙跺瓙缁撶偣锛屽苟涓斿彾瀛愮粨鐐归兘鏄粠宸﹀埌鍙充緷娆℃帓甯冿紝杩欏氨鏄�瀹屽叏浜屽弶鏍�銆�
(2)婊′簩鍙夋爲鈥斺�旈櫎浜嗗彾缁撶偣澶栨瘡涓�涓粨鐐归兘鏈夊乏鍙冲瓙鍙朵笖鍙跺瓙缁撶偣閮藉鍦ㄦ渶搴曞眰鐨勪簩鍙夋爲銆�
(3)骞宠 浜屽弶鏍�鈥斺��骞宠 浜屽弶鏍�鍙堣绉颁负AVL鏍戯紙鍖哄埆浜�AVL绠楁硶锛夛紝瀹冩槸涓�妫典簩鍙夋帓搴忔爲锛屼笖鍏锋湁浠ヤ笅鎬ц川锛氬畠鏄竴妫电┖鏍戞垨瀹冪殑宸﹀彸涓や釜瀛愭爲鐨勯珮搴﹀樊鐨勭粷瀵瑰�间笉瓒呰繃1锛屽苟涓斿乏鍙充袱涓瓙鏍戦兘鏄竴妫�骞宠 浜屽弶鏍�銆俒2]
鐩稿叧鏈
鏍戠殑缁撶偣锛氬寘鍚竴涓暟鎹厓绱犲強鑻ュ共鎸囧悜瀛愭爲鐨勫垎鏀紱
瀛╁瓙缁撶偣锛氱粨鐐圭殑瀛愭爲鐨勬牴绉颁负璇ョ粨鐐圭殑瀛╁瓙锛�
鍙屼翰缁撶偣锛欱聽缁撶偣鏄疉聽缁撶偣鐨勫瀛愶紝鍒橝缁撶偣鏄疊聽缁撶偣鐨勫弻浜诧紱
鍏勫紵缁撶偣锛氬悓涓�鍙屼翰鐨勫瀛愮粨鐐癸紱聽鍫傚厔缁撶偣锛氬悓涓�灞備笂缁撶偣锛�
绁栧厛缁撶偣:聽浠庢牴鍒拌缁撶偣鐨勬墍缁忓垎鏀笂鐨勬墍鏈夌粨鐐瑰瓙瀛欑粨鐐癸細浠ユ煇缁撶偣涓烘牴鐨勫瓙鏍戜腑浠讳竴缁撶偣閮界О涓鸿缁撶偣鐨勫瓙瀛�
缁撶偣灞傦細鏍圭粨鐐圭殑灞傚畾涔変负1锛涙牴鐨勫瀛愪负绗簩灞傜粨鐐癸紝渚濇绫绘帹锛�
鏍戠殑娣卞害锛氭爲涓渶澶х殑缁撶偣灞�
缁撶偣鐨勫害锛氱粨鐐瑰瓙鏍戠殑涓暟
鏍戠殑搴︼細聽鏍戜腑鏈�澶х殑缁撶偣搴︺��
鍙跺瓙缁撶偣锛氫篃鍙粓绔粨鐐癸紝鏄害涓郝�0聽鐨勭粨鐐癸紱
鍒嗘灊缁撶偣锛氬害涓嶄负0鐨勭粨鐐癸紱
鏈夊簭鏍戯細瀛愭爲鏈夊簭鐨勬爲锛屽锛氬鏃忔爲锛�
鏃犲簭鏍戯細涓嶈�冭檻瀛愭爲鐨勯『搴忥紱[3]
浜屽弶鏍戞�ц川
(1)聽鍦�闈炵┖浜屽弶鏍�涓紝绗琲灞傜殑缁撶偣鎬绘暟涓嶈秴杩�
, i>=1锛�
(2)聽娣卞害涓篽鐨勪簩鍙夋爲鏈�澶氭湁
涓粨鐐�(h>=1)锛屾渶灏戞湁h涓粨鐐癸紱
(3)聽瀵逛簬浠绘剰涓�妫典簩鍙夋爲锛屽鏋滃叾鍙剁粨鐐规暟涓篘0锛岃�屽害鏁颁负2鐨勭粨鐐规�绘暟涓篘2锛屽垯N0=N2+1锛�
(4)聽鍏锋湁n涓粨鐐圭殑瀹屽叏浜屽弶鏍�鐨勬繁搴︿负
锛堟敞锛歔 ]琛ㄧず鍚戜笅鍙栨暣锛�
(5)鏈塏涓粨鐐圭殑瀹屽叏浜屽弶鏍�鍚勭粨鐐瑰鏋滅敤椤哄簭鏂瑰紡瀛樺偍锛屽垯缁撶偣涔嬮棿鏈夊涓嬪叧绯伙細
鑻涓虹粨鐐圭紪鍙峰垯 濡傛灉I>1锛屽垯鍏剁埗缁撶偣鐨勭紪鍙蜂负I/2锛�
濡傛灉2*I<=N锛屽垯鍏跺乏鍎垮瓙锛堝嵆宸﹀瓙鏍戠殑鏍圭粨鐐癸級鐨勭紪鍙蜂负2*I锛涜嫢2*I>N锛屽垯鏃犲乏鍎垮瓙锛�
濡傛灉2*I+1<=N锛屽垯鍏跺彸鍎垮瓙鐨勭粨鐐圭紪鍙蜂负2*I+1锛涜嫢2*I+1>N锛屽垯鏃犲彸鍎垮瓙銆�
(6)缁欏畾N涓妭鐐癸紝鑳芥瀯鎴恏(N)绉嶄笉鍚岀殑浜屽弶鏍戙��
h(N)涓�鍗＄壒鍏版暟鐨勭N椤广�俬(n)=C(2*n锛宯)/(n+1)銆�
(7)璁炬湁i涓灊鐐癸紝I涓烘墍鏈夋灊鐐圭殑閬撹矾闀垮害鎬诲拰锛孞涓哄彾鐨勯亾璺暱搴︽�诲拰J=I+2i[4]
瀛樺偍缁撴瀯
(1锛夐『搴忓瓨鍌ㄦ柟寮�
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typenode=record
data:datatype
l,r:integer;
end;
vartr:array[1..n]ofnode;
(2)閾捐〃瀛樺偍鏂瑰紡锛屽锛�
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typebtree=^node锛�
node=record
data:datatye;
lchild,rchild:btree;
end;
1聽
杈ㄦ瀽
浜屽弶鏍戜笉鏄爲鐨勪竴绉嶇壒娈婃儏褰紝灏界鍏朵笌鏍戞湁璁稿鐩镐技涔嬪锛屼絾鏍戝拰浜�
浜屽弶鏍�(3寮�)
鍙夋爲鏈変袱涓富瑕佸樊鍒細
1.聽鏍戜腑缁撶偣鐨勬渶澶у害鏁版病鏈夐檺鍒讹紝鑰屼簩鍙夋爲缁撶偣鐨勬渶澶у害鏁颁负2锛�
2.聽鏍戠殑缁撶偣鏃犲乏銆佸彸涔嬪垎锛岃�屼簩鍙夋爲鐨勭粨鐐规湁宸︺�佸彸涔嬪垎銆�
閬嶅巻椤哄簭
缂栬緫
閬嶅巻鏄鏍戠殑涓�绉嶆渶鍩烘湰鐨勮繍绠楋紝鎵�璋撻亶鍘嗕簩鍙夋爲锛屽氨鏄寜涓�瀹氱殑瑙勫垯鍜岄『搴忚蛋閬嶄簩鍙夋爲鐨勬墍鏈夌粨鐐癸紝浣挎瘡涓�涓粨鐐归兘琚闂竴娆★紝鑰屼笖鍙璁块棶涓�娆°�傜敱浜庝簩鍙夋爲鏄潪绾挎�х粨鏋勶紝鍥犳锛�鏍戠殑閬嶅巻瀹炶川涓婃槸灏嗕簩鍙夋爲鐨勫悇涓粨鐐硅浆鎹㈡垚涓轰竴涓嚎鎬у簭鍒楁潵琛ㄧず銆�
璁綥銆丏銆丷鍒嗗埆琛ㄧず閬嶅巻宸﹀瓙鏍戙�佽闂牴缁撶偣鍜岄亶鍘嗗彸瀛愭爲锛� 鍒欏涓�妫典簩鍙夋爲鐨勯亶鍘嗘湁涓夌鎯呭喌锛欴LR锛堢О涓哄厛鏍规搴忛亶鍘嗭級锛孡DR锛堢О涓轰腑鏍规搴忛亶鍘嗭級锛孡RD聽锛堢О涓哄悗鏍规搴忛亶鍘嗭級銆�
鍏堝簭閬嶅巻
棣栧厛璁块棶鏍癸紝鍐嶅厛搴忛亶鍘嗗乏锛堝彸锛夊瓙鏍戯紝鏈�鍚庡厛搴忛亶鍘嗗彸锛堝乏锛夊瓙鏍戯紝C璇█浠ｇ爜濡備笅锛�
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voidXXBL(tree*root){
//DoSomethingwithroot
if(root->lchild!=NULL)
XXBL(root->lchild);
if(root->rchild!=NULL)
XXBL(root->rchild);
}
涓簭閬嶅巻
棣栧厛涓簭閬嶅巻宸︼紙鍙筹級瀛愭爲锛屽啀璁块棶鏍癸紝鏈�鍚庝腑搴忛亶鍘嗗彸锛堝乏锛夊瓙鏍戯紝C璇█浠ｇ爜濡備笅
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voidZXBL(tree*root)
{
if(root->lchild!=NULL)
ZXBL(root->lchild);//DoSomethingwithroot
if(root->rchild!=NULL)
ZXBL(root->rchild);
}
鍚庡簭閬嶅巻
棣栧厛鍚庡簭閬嶅巻宸︼紙鍙筹級瀛愭爲锛屽啀鍚庡簭閬嶅巻鍙筹紙宸︼級瀛愭爲锛屾渶鍚庤闂牴锛孋璇█浠ｇ爜濡備笅
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voidHXBL(tree*root){
if(root->lchild!=NULL)
HXBL(root->lchild);
if(root->rchild!=NULL)
HXBL(root->rchild);//DoSomethingwithroot
}
灞傛閬嶅巻
鍗虫寜鐓у眰娆¤闂紝閫氬父鐢�闃熷垪鏉ュ仛銆傝闂牴锛岃闂瓙濂筹紝鍐嶈闂瓙濂崇殑瀛愬コ锛堣秺寰�鍚庣殑灞傛瓒婁綆锛夛紙涓や釜瀛愬コ鐨勭骇鍒浉鍚岋級
绾跨储浜屽弶鏍�
绾跨储浜屽弶鏍�(淇濈暀閬嶅巻鏃剁粨鐐瑰湪浠讳竴搴忓垪鐨勫墠椹卞拰鍚庣户鐨勪俊鎭�)锛氳嫢缁撶偣鏈夊乏瀛愭爲锛屽垯鍏秎child鍩熸寚绀哄叾宸﹀瀛愶紝鍚﹀垯浠child鍩熸寚绀哄叾鍓嶉┍锛涜嫢缁撶偣鏈夊彸瀛愭爲锛屽垯鍏秗child鍩熸寚绀哄叾鍙冲瀛愶紝鍚﹀垯浠child鎸囩ず鍏跺悗缁с�傝繕闇�鍦ㄧ粨鐐圭粨鏋勪腑澧炲姞涓や釜鏍囧織鍩烲Tag鍜孯Tag銆侺Tag=0鏃讹紝lchild鍩熸寚绀虹粨鐐圭殑宸﹀瀛愶紝LTag=1鏃讹紝lchild鍩熸寚绀虹粨鐐圭殑鍓嶉┍锛汻Tag=0鏃讹紝rchild鍩熸寚绀虹粨鐐圭殑鍙冲瀛愶紝RTag=1鏃讹紝rchild鍩熸寚绀虹粨鐐圭殑鍚庣户銆備互杩欑缁撶偣缁撴瀯鏋勬垚鐨勪簩鍙夌嚎绱�閾捐〃锛岄摼琛ㄤ綔涓轰簩鍙夋爲鐨�瀛樺偍缁撴瀯锛屽彨鍋氬叾涓寚鍚戠粨鐐瑰墠椹卞拰鍚庣户鐨�鎸囬拡鍙仛绾跨储锛屽姞涓婄嚎绱㈢殑浜屽弶鏍戠О涓虹嚎绱簩鍙夋爲銆傚浜屽弶鏍戜互鏌愮娆″簭閬嶅巻浣垮叾鍙樹负绾跨储浜屽弶鏍戠殑杩囩▼鍙仛绾跨储鍖栥�傝嫢瀵逛簩鍙夋爲杩涜涓簭閬嶅巻锛屽垯鎵�寰楃殑绾跨储浜屽弶鏍戠О涓轰腑搴忕嚎绱簩鍙夋爲锛岀嚎绱�閾捐〃绉颁负涓轰腑搴忕嚎绱㈤摼琛ㄣ�傜嚎绱簩鍙夋爲鏄竴绉�鐗╃悊缁撴瀯銆�
绾跨储浜屽弶鏍戠殑瀛樺偍缁撴瀯
鍦ㄤ腑搴忕嚎绱㈡爲鎵剧粨鐐瑰悗缁х殑瑙勫緥鏄細鑻ュ叾鍙虫爣蹇椾负1锛屽垯鍙抽摼涓虹嚎绱紝鎸囩ず鍏跺悗缁э紝鍚﹀垯閬嶅巻鍏跺彸瀛愭爲鏃惰闂殑绗竴涓粨鐐癸紙鍙冲瓙鏍戞渶宸︿笅鐨勭粨鐐癸級涓哄叾鍚庣户锛涙壘缁撶偣鍓嶉┍鐨勮寰嬫槸锛氳嫢鍏跺乏鏍囧織涓�1锛屽垯宸﹂摼涓虹嚎绱紝鎸囩ず鍏跺墠椹憋紝鍚﹀垯閬嶅巻宸﹀瓙鏍戞椂鏈�鍚庤闂殑涓�涓粨鐐癸紙宸﹀瓙鏍戜腑鏈�鍙充笅鐨勭粨鐐癸級涓哄叾鍓嶉┍銆�
鍦ㄥ悗搴忕嚎绱㈡爲涓壘鍒扮粨鐐圭殑鍚庣户鍒嗕笁绉嶆儏鍐碉細
鑻ョ粨鐐规槸浜屽弶鏍戠殑鏍癸紝鍒欏叾鍚庣户涓虹┖锛涜嫢缁撶偣鏄叾鍙屼翰鐨勫彸瀛╁瓙锛屾垨鏄叾鍙屼翰鐨勫乏瀛╁瓙涓斿叾鍙屼翰娌℃湁鍙冲瓙鏍戯紝鍒欏叾鍚庣户鍗充负鍙屼翰缁撶偣锛涜嫢缁撶偣鏄叾鍙屼翰鐨勫乏瀛╁瓙锛屼笖鍏跺弻浜叉湁鍙冲瓙鏍戯紝鍒欏叾鍚庣户涓哄弻浜插彸瀛愭爲涓婃寜鍚庡簭閬嶅巻鍒楀嚭鐨勭涓�涓粨鐐广��
鏁版嵁缁撴瀯瀹氫箟涓猴細
/*浜屽弶绾跨储瀛樺偍琛ㄧず*/typedefenum{Link,Thread}PointerTag;/* Link(0)锛氭寚閽堬紝Thread(1)锛氱嚎绱�*/typedefstruct BiThrNode{ TElemType data;struct BiThrNode *lchild,*rchild;/*宸﹀彸瀛╁瓙鎸囬拡*/PointerTag LTag,RTag;/*聽宸﹀彸鏍囧織聽*/}BiThrNode,*BiThrTree;
瀹炵幇婕旂ず
缂栬緫
鑼冧緥浜屽弶鏍戯細
A
B C
D E
姝ゆ爲鐨勯『搴忕粨鏋勪负锛欰BCD##E
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intmain()
{
node*p=newnode;
node*p=head;
head=p;
stringstr;
cin聽>>str;
creat(p,str,0)//榛樿鏍硅妭鐐瑰湪str涓嬫爣0鐨勪綅缃�
return0;
}
//p涓烘爲鐨勬牴鑺傜偣锛堝凡寮�杈熷姩鎬佸唴瀛橈級,str涓轰簩鍙夋爲鐨勯『搴忓瓨鍌ㄦ暟缁凙BCD##E鎴栧叾浠栭『搴忓瓨鍌ㄦ暟缁勶紝r褰撳墠缁撶偣鎵�鍦ㄩ『搴忓瓨鍌ㄦ暟缁勪綅缃�
intmain()
{
node*聽p聽=聽newnode;
node*聽p聽=聽head;
head聽=聽p;
string聽str;
cin聽>>聽str;
creat(p,聽str,聽0)//榛樿鏍硅妭鐐瑰湪str涓嬫爣0鐨勪綅缃�
return0;
}
//p涓烘爲鐨勬牴鑺傜偣锛堝凡寮�杈熷姩鎬佸唴瀛橈級,str涓轰簩鍙夋爲鐨勯『搴忓瓨鍌ㄦ暟缁凙BCD##E鎴栧叾浠栭『搴忓瓨鍌ㄦ暟缁勶紝r褰撳墠缁撶偣鎵�鍦ㄩ『搴忓瓨鍌ㄦ暟缁勪綅缃�
voidcreat(node*聽p,聽string聽str,聽intr)
{
p->data聽=聽str[r];
if(str[r聽*聽2聽+聽1]聽==聽'#'||聽r聽*聽2聽+聽1聽>聽str.size()聽-聽1)p->lch聽=聽NULL;
else
{
p->lch聽=聽newnode;
creat(p->lch,聽str,聽r聽*聽2聽+聽1);
}
if(str[r聽*聽2聽+聽2]聽==聽'#'||聽r聽*聽2聽+聽2聽>聽str.size()聽-聽1)p->rch聽=聽NULL;
else
{
p->rch聽=聽newnode;
creat(p->rch,聽str,聽r聽*聽2聽+聽2);
}
}
璇嶆潯鍥惧唽鏇村鍥惧唽
璇嶆潯鍥剧墖(3)
浜屽弶鏍�(3)
璁＄畻鏈虹瀛︿腑鐨勬爲
浜屽弶鏍戔柂浜屽弶鏍戔柂浜屽弶鏌ユ壘鏍�鈻�绗涘崱灏旀爲鈻�Top tree
鈻�T鏍�聽聽聽
鑷钩琛′簩鍙夋煡鎵炬爲鈻�AA鏍�鈻�AVL鏍�鈻�绾㈤粦鏍�鈻�浼稿睍鏍�
鈻�鏍戝爢鈻�鑺傜偣澶у皬骞宠 鏍�聽聽
B鏍戔柂B鏍�鈻�B+鏍�鈻�B*鏍�鈻�Bx鏍�
鈻�UB鏍�鈻�2-3鏍�鈻�2-3-4鏍�鈻�(a,b)-鏍�
鈻�Dancing tree鈻�H鏍�聽聽
Trie鈻�鍓嶇紑鏍�鈻�鍚庣紑鏍�鈻�鍩烘暟鏍�聽
绌洪棿鍒掑垎鏍戔柂鍥涘弶鏍�鈻�鍏弶鏍�鈻�k-d鏍�鈻�vp-鏍�
鈻�R鏍�鈻�R*鏍�鈻�R+鏍�鈻�X鏍�
鈻�M鏍�鈻�绾挎鏍�鈻�甯屽皵浼壒R鏍�鈻�浼樺厛R鏍�
闈炰簩鍙夋爲鈻�Exponential tree鈻�Fusion tree鈻�鍖洪棿鏍�鈻�PQ tree
鈻�Range tree鈻�SPQR tree鈻�Van Emde Boas tree聽
鍏朵粬绫诲瀷鈻�鍫�鈻�鏁ｅ垪鏍�鈻�Finger tree鈻�Metric tree
鈻�Cover tree鈻�BK-tree鈻�Doubly-chained tree鈻�iDistance
鈻�Link-cut tree鈻�鏍戠姸鏁扮粍聽聽
鍙傝�冭祫鏂�
绾㈤粦鏍�
绾㈤粦鏍戯紙Red Black Tree锛� 鏄竴绉嶈嚜骞宠 浜屽弶鏌ユ壘鏍戯紝鏄湪璁＄畻鏈�绉戝涓敤鍒扮殑涓�绉�鏁版嵁缁撴瀯锛屽吀鍨嬬殑鐢ㄩ�旀槸瀹炵幇鍏宠仈鏁扮粍銆�
瀹冩槸鍦�1972骞寸敱Rudolf Bayer鍙戞槑鐨勶紝褰撴椂琚О涓哄钩琛′簩鍙塀鏍戯紙symmetric binary B-trees锛夈�傚悗鏉ワ紝鍦�1978骞磋聽Leo J. Guibas聽鍜屄燫obert Sedgewick聽淇敼涓哄浠婄殑鈥滅孩榛戞爲鈥濄��
绾㈤粦鏍戝拰AVL鏍戠被浼硷紝閮芥槸鍦ㄨ繘琛屾彃鍏ュ拰鍒犻櫎鎿嶄綔鏃堕�氳繃鐗瑰畾鎿嶄綔淇濇寔浜屽弶鏌ユ壘鏍戠殑骞宠 锛屼粠鑰岃幏寰楄緝楂樼殑鏌ユ壘鎬ц兘銆�
瀹冭櫧鐒舵槸澶嶆潅鐨勶紝浣嗗畠鐨勬渶鍧忔儏鍐佃繍琛屾椂闂翠篃鏄潪甯歌壇濂界殑锛屽苟涓斿湪瀹炶返涓槸楂樻晥鐨勶細聽瀹冨彲浠ュ湪O(log n)鏃堕棿鍐呭仛鏌ユ壘锛屾彃鍏ュ拰鍒犻櫎锛岃繖閲岀殑n聽鏄爲涓厓绱犵殑鏁扮洰銆�
涓枃鍚�
绾㈤粦鏍�
澶栨枃鍚�
RED BLACK TREE
鎬ц川
鑷钩琛′簩鍙夋煡鎵炬爲
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瀹炵幇鍏宠仈鏁扮粍
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椴侀亾澶疯礉灏�
鍙戞槑鏃堕棿
1972骞�
鍒悕
瀵圭О浜屽弶B鏍�
鐩綍
00001.聽1聽鏁版嵁缁撴瀯
00002.聽2聽鏍戠殑鏃嬭浆
00001.聽3聽鎬ц川
00002.聽4聽鏈
00001.聽5聽鐢ㄩ��
00002.聽6聽鎿嶄綔
鏁版嵁缁撴瀯
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鏍戠殑鏃嬭浆
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鏍戠殑宸︽棆(2寮�)
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濡傚彸鍥俱��
鎬ц川
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鎬ц川1.聽鑺傜偣鏄孩鑹叉垨榛戣壊銆�
鎬ц川2.聽鏍硅妭鐐规槸榛戣壊銆�
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鎬ц川4聽姣忎釜绾㈣壊鑺傜偣鐨勪袱涓瓙鑺傜偣閮芥槸榛戣壊銆�(浠庢瘡涓彾瀛愬埌鏍圭殑鎵�鏈夎矾寰勪笂涓嶈兘鏈変袱涓繛缁殑绾㈣壊鑺傜偣)
鎬ц川5.聽浠庝换涓�鑺傜偣鍒板叾姣忎釜鍙跺瓙鐨勬墍鏈夎矾寰勯兘鍖呭惈鐩稿悓鏁扮洰鐨勯粦鑹茶妭鐐广��
杩欎簺绾︽潫寮哄埗浜嗙孩榛戞爲鐨勫叧閿�ц川:聽浠庢牴鍒板彾瀛愮殑鏈�闀跨殑鍙兘璺緞涓嶅浜庢渶鐭殑鍙兘璺緞鐨勪袱鍊嶉暱銆傜粨鏋滄槸杩欎釜鏍戝ぇ鑷翠笂鏄钩琛＄殑銆傚洜涓烘搷浣滄瘮濡傛彃鍏ャ�佸垹闄ゅ拰鏌ユ壘鏌愪釜鍊肩殑鏈�鍧忔儏鍐垫椂闂撮兘瑕佹眰涓庢爲鐨勯珮搴︽垚姣斾緥锛岃繖涓湪楂樺害涓婄殑鐞嗚涓婇檺鍏佽绾㈤粦鏍戝湪鏈�鍧忔儏鍐典笅閮芥槸楂樻晥鐨勶紝鑰屼笉鍚屼簬鏅�氱殑浜屽弶鏌ユ壘鏍戙��
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绾㈤粦鏍戞槸2-3-4鏍�鐨勪竴绉嶇瓑鍚屻�傛崲鍙ヨ瘽璇达紝瀵逛簬姣忎釜聽2-3-4聽鏍戯紝閮藉瓨鍦ㄨ嚦灏戜竴涓�鏁版嵁鍏冪礌鏄悓鏍锋搴忕殑绾㈤粦鏍戙�傚湪聽2-3-4聽鏍戜笂鐨勬彃鍏ュ拰鍒犻櫎鎿嶄綔涔熺瓑鍚屼簬鍦ㄧ孩榛戞爲涓鑹茬炕杞拰鏃嬭浆銆傝繖浣垮緱聽2-3-4聽鏍戞垚涓虹悊瑙ｇ孩榛戞爲鑳屽悗鐨勯�昏緫鐨勯噸瑕佸伐鍏凤紝杩欎篃鏄緢澶氫粙缁嶇畻娉曠殑鏁欑涔﹀湪绾㈤粦鏍戜箣鍓嶄粙缁嵚�2-3-4聽鏍戠殑鍘熷洜锛屽敖绠÷�2-3-4聽鏍戝湪瀹炶返涓笉缁忓父浣跨敤銆�
鎿嶄綔
鍦ㄧ孩榛戞爲涓婂彧璇绘搷浣滀笉闇�瑕佸鐢ㄤ簬浜屽弶鏌ユ壘鏍戠殑鎿嶄綔鍋氬嚭淇敼锛屽洜涓哄畠涔熸槸浜屽弶鏌ユ壘鏍戙�備絾鏄紝鍦ㄦ彃鍏ュ拰鍒犻櫎涔嬪悗锛岀孩榛戝睘鎬у彲鑳藉彉寰楄繚瑙勩�傛仮澶嶇孩榛戝睘鎬ч渶瑕佸皯閲�(O(log n))鐨勯鑹插彉鏇�(杩欏湪瀹炶返涓槸闈炲父蹇�熺殑)骞朵笖涓嶈秴杩囦笁娆�鏍戞棆杞�(瀵逛簬鎻掑叆鏄袱娆�)銆傝繖鍏佽鎻掑叆鍜屽垹闄や繚鎸佷负聽O(log n)聽娆★紝浣嗘槸瀹冨鑷翠簡闈炲父澶嶆潅鐨勬搷浣溿�俒1]
璇嶆潯鍥惧唽鏇村鍥惧唽
璇嶆潯鍥剧墖(3)
鏍戠殑宸︽棆(2)
Data_structures
鈻�闆嗗悎鈻�瀹瑰櫒聽聽
鈻�鏁扮粍鈻�鍏宠仈鏁扮粍鈻�Multimap鈻�闆�
鈻�澶氶噸闆�鈻�鏁ｅ垪琛�鈻�鏍戠姸鏁扮粍聽
鈻�鍒楄〃鈻�閾捐〃鈻�闃熷垪鈻�鍫嗘爤
鈻�寰幆闃熷垪鈻�璺宠穬鍒楄〃聽聽
鈻�鏍�鈻�浜屽弶鏌ユ壘鏍�鈻�鍫�鈻�绾挎鏍�
鈻孩榛戞爲鈻�AVL鏍�聽聽
鈻�鍥�鈻�鏈夊悜鏃犵幆鍥�鈻�浜屽厓鍐崇瓥鍥�鈻�鏃犲悜鍥�
璁＄畻鏈虹瀛︿腑鐨勬爲
浜屽弶鏍戔柂浜屽弶鏍�鈻�浜屽弶鏌ユ壘鏍�鈻�绗涘崱灏旀爲鈻�Top tree
鈻�T鏍�聽聽聽
鑷钩琛′簩鍙夋煡鎵炬爲鈻�AA鏍�鈻�AVL鏍�鈻孩榛戞爲鈻�浼稿睍鏍�
鈻�鏍戝爢鈻�鑺傜偣澶у皬骞宠 鏍�聽聽
B鏍戔柂B鏍�鈻�B+鏍�鈻�B*鏍�鈻�Bx鏍�
鈻�UB鏍�鈻�2-3鏍�鈻�2-3-4鏍�鈻�(a,b)-鏍�
鈻�Dancing tree鈻�H鏍�聽聽
Trie鈻�鍓嶇紑鏍�鈻�鍚庣紑鏍�鈻�鍩烘暟鏍�聽
绌洪棿鍒掑垎鏍戔柂鍥涘弶鏍�鈻�鍏弶鏍�鈻�k-d鏍�鈻�vp-鏍�
鈻�R鏍�鈻�R*鏍�鈻�R+鏍�鈻�X鏍�
鈻�M鏍�鈻�绾挎鏍�鈻�甯屽皵浼壒R鏍�鈻�浼樺厛R鏍�
闈炰簩鍙夋爲鈻�Exponential tree鈻�Fusion tree鈻�鍖洪棿鏍�鈻�PQ tree
鈻�Range tree鈻�SPQR tree鈻�Van Emde Boas tree聽
鍏朵粬绫诲瀷鈻�鍫�鈻�鏁ｅ垪鏍�鈻�Finger tree鈻�Metric tree
鈻�Cover tree鈻�BK-tree鈻�Doubly-chained tree鈻�iDistance
鈻�Link-cut tree鈻�鏍戠姸鏁扮粍聽聽
鍙傝�冭祫鏂�
B鏍�
鍦˙-鏍戜腑鏌ユ壘缁欏畾鍏抽敭瀛�鐨勬柟娉曟槸锛岄鍏堟妸鏍圭粨鐐瑰彇鏉ワ紝鍦ㄦ牴缁撶偣鎵�鍖呭惈鐨勫叧閿瓧K1,鈥�,Kn鏌ユ壘缁欏畾鐨勫叧閿瓧锛堝彲鐢ㄩ『搴忔煡鎵炬垨浜屽垎鏌ユ壘娉曪級锛岃嫢鎵惧埌绛変簬缁欏畾鍊肩殑鍏抽敭瀛楋紝鍒欐煡鎵炬垚鍔燂紱鍚﹀垯锛屼竴瀹氬彲浠ョ‘瀹氳鏌ユ壘鐨勫叧閿瓧鍦↘i涓嶬i+1涔嬮棿锛孭i涓烘寚鍚戝瓙鏍戞牴鑺傜偣鐨勬寚閽堬紝姝ゆ椂鍙�鎸囬拡Pi鎵�鎸囩殑缁撶偣缁х画鏌ユ壘锛岀洿鑷虫壘鍒帮紝鎴栨寚閽圥i涓虹┖鏃舵煡鎵惧け璐ャ��
涓枃鍚�
B鏍�
澶栨枃鍚�
B tree
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B-鏍戙�丅_鏍�
鎻愬嚭鑰�
R.Bayer鍜孍.mccreight
鎻愬嚭鏃堕棿
1970骞�
搴旂敤瀛︾
璁＄畻鏈�
閫傜敤棰嗗煙鑼冨洿
杞欢
閫傜敤棰嗗煙鑼冨洿
缂栫▼
鐩綍
00001.聽1聽瀹氫箟
00002.聽2聽鎬ц兘鍒嗘瀽
00003.聽3聽绀轰緥
瀹氫箟
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8聽A[i+1] =key
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绀轰緥浠ｇ爜涓篊璇█锛岃緭鍏ュ弬鏁颁腑锛岄渶瑕佹帓搴忕殑鏁扮粍涓篴rray[ ]锛岃捣濮嬬储寮曚负first锛堟暟缁勬湁搴忛儴鍒嗘渶鍚庝竴涓殑涓嬫爣锛屾垨鑰呯洿鎺ユ爣0锛夛紝缁堟绱㈠紩涓簂ast锛堟暟缁勫厓绱犵殑鏈�鍚庝竴涓殑涓嬫爣锛夈�傜ず渚嬩唬鐮佺殑鍑芥暟閲囩敤insert-place鎺掑簭锛岃皟鐢ㄥ畬鎴愬悗锛宎rray[]涓粠first鍒發ast澶勪簬鍗囧簭鎺掑垪銆�
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voidinsertion_sort(intarray[],intfirst,intlast)
{
inti,j;
inttemp;
for(i=first+1;i
{
temp=array[i];
j=i-1;
//涓庡凡鎺掑簭鐨勬暟閫愪竴姣旇緝锛屽ぇ浜巘emp鏃讹紝璇ユ暟绉诲悗
while((j>=0)&&(array[j]>temp))
{
array[j+1]=array[j];
j--;
}
//瀛樺湪澶т簬temp鐨勬暟
if(j!=i-1)
{array[j+1]=temp;}
}
}
杩欎釜鏇村ソ锛�
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voidinsert_sort(int*array,unsignedintn)
{
inti,j;
inttemp;
for(i=1;i
{
temp=*(array+i);
for(j=i;j>0&&*(array+j-1)>temp;j--)
{
*(array+j)=*(array+j-1);
}
*(array+j)=temp;
}
}
c++鐗堟湰
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#include
template
voidinsertion_sort(biIterbegin,biIterend)
{
typedeftypenamestd::iterator_traits::value_typevalue_type;
biIterbond=begin;
std::advance(bond,1);
for(;bond!=end;std::advance(bond,1)){
value_typekey=*bond;
biIterins=bond;
biIterpre=ins;
std::advance(pre,-1);
while(ins!=begin&&*pre>key){
*ins=*pre;
std::advance(ins,-1);
std::advance(pre,-1);
}
*ins=key;
}
}
PHP鐗堟湰
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functioninsertSort($arr){
for($i=1;$i
$tmp=$arr[$i];
$key=$i-1;
while($key>=0&&$tmp<$arr[$key]){
$arr[$key+1]=$arr[$key];
$key--;
}
if(($key+1)!=$i)
$arr[$key+1]=$tmp;
}
return$arr;
}
Java鐗堟湰
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/**
*鎻掑叆鎺掑簭
*@paramarr
*@return
*/
privatestaticint[]聽insertSort(int[]arr){
if(arr聽==聽null||聽arr.length聽<聽2){
returnarr;
}
for(inti=1;i
for(intj=i;j>0;j--){
if(arr[j]
//TODO:
inttemp=arr[j];
arr[j]=arr[j-1];
arr[j-1]=temp;
}else{
//鎺ヤ笅鏉ユ槸鏃犵敤鍔�
break;
}
}
}
returnarr;
}
杩愯杞欢锛歟clipse
C#鐗堟湰
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classProgram
{
staticvoidMain(string[]args)
{
InsertionSort();
}
///
///鎻掑叆鎺掑簭娉�
///
privatestaticvoidInsertionSort()
{
Console.WriteLine("鎻掑叆鎺掑簭娉�");
inttemp=0;
int[]arr={23,44,66,76,98,11,3,9,7};
Console.WriteLine("鎺掑簭鍓嶇殑鏁扮粍锛�");
foreach(intiteminarr)
{
Console.Write(item+",");
}
Console.WriteLine();
varlength=arr.Length;
for(inti=1;i
{
for(intj=i;j>0;j--)
{
if(arr[j]>arr[j-1])
{
temp=arr[j];
arr[j]=arr[j-1];
arr[j-1]=temp;
}
}
//姣忔鎺掑簭鍚庢暟缁�
PrintResult(arr);
}
Console.ReadKey();
}
///
///鎵撳嵃缁撴灉
///
///
privatestaticvoidPrintResult(IEnumerablearr)
{
foreach(intiteminarr)
{
Console.Write(item+",");
}
Console.WriteLine();
}
}
Pascal鐗堟湰
procedure insertsort(var R:filetype);
//瀵筊[1..N]鎸夐�掑搴忚繘琛屾彃鍏ユ帓搴忥紝R[0]鏄洃瑙嗗摠
begin
for I := 2 To n do //渚濇鎻掑叆R[2],...,R[n]
begin
R[0]:=R[I];j:=l-1;
while R[0] < R[J] Do //鏌ユ壘R[I]鐨勬彃鍏ヤ綅缃�
begin
R[j+1]:=R[j]; //灏嗗ぇ浜嶳[I]鐨勫厓绱犲悗绉�
j:=j-1;
end;
R[j+1]:=R[0] ; //鎻掑叆R[I]
end;
end; //InsertSort
(杩愯杞欢锛欶ree Pascal IDE 2.6.4)
Ruby鐗堟湰
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definsertion_sort(array)聽聽
array.each_with_index聽do|element,聽index|聽聽聽聽
nextifindex聽==聽0#绗竴涓厓绱犻粯璁ゅ凡鎺掑簭聽聽聽聽
j聽=聽index聽-聽1
whilej聽>=聽0&&聽array[j]聽>聽element聽聽聽聽聽聽
array[j聽+聽1]聽=聽array[j]聽
j聽-=聽1
end
array[j聽+聽1]聽=聽element聽聽聽
end
array
end
Scala鐗堟湰
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definsertSort(ilist:Array[Int]){
for(i<-1untililist.length){
for(j<-(0untili).reverse){
if(ilist(j+1)
valtmp=ilist(j+1)
ilist(j+1)=ilist(j)
ilist(j)=tmp
}
}
}
}
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鍏紡
Data=rand()/10000+10000
鏁版嵁缁撴瀯璁捐
閾捐〃鍙互閲囩敤寰堝绉嶆柟寮忓疄鐜�
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骞冲潎鎯呭喌涓嬫《鎺掑簭浠ョ嚎鎬ф椂闂磋繍琛屻�傚儚鍩烘暟鎺掑簭涓�鏍凤紝妗舵帓搴忎篃瀵硅緭鍏ヤ綔浜嗘煇绉嶅亣璁撅紝聽鍥犺�岃繍琛屽緱寰堝揩銆傚叿聽浣撴潵璇达紝鍩烘暟鎺掑簭鍋囪杈撳叆鏄敱涓�涓皬鑼冨洿鍐呯殑鏁存暟鏋勬垚锛岃�屾《鎺掑簭鍒櫬犲亣璁捐緭鍏ョ敱涓�涓殢鏈鸿繃绋嬩骇鐢燂紝璇ヨ繃绋嬪皢鍏冪礌涓�鑷村湴鍒嗗竷鍦ㄥ尯闂碵0锛�1)涓娿�� 妗舵帓搴忕殑鎬濇兂灏辨槸鎶婂尯闂碵0锛�1)鍒掑垎鎴恘涓浉鍚屽ぇ灏忕殑瀛愬尯闂达紝鎴栫О妗讹紝鐒跺悗灏唍涓緭鍏ユ暟鍒嗗竷鍒板悇涓《涓幓銆傚洜涓鸿緭鍏ユ暟鍧囧寑鍒嗗竷鍦╗0锛�1)涓婏紝鎵�浠ヤ竴鑸笉浼氭湁寰堝鏁拌惤鍦ㄤ竴涓《涓殑鎯呭喌銆備负寰楀埌缁撴灉锛屽厛瀵瑰悇涓《涓殑鏁拌繘琛屾帓搴忥紝鐒跺悗鎸夋搴忔妸鍚勬《涓殑鍏冪礌鍒楀嚭鏉ュ嵆鍙��
鍦ㄦ《鎺掑簭绠楁硶鐨勪唬鐮佷腑锛屽亣璁捐緭鍏ユ槸鍚玭涓厓绱犵殑鏁扮粍A锛屼笖姣忎釜鍏冪礌婊¤冻0鈮� A[i]<1銆傚彟澶栬繕闇�瑕佷竴涓緟鍔�鏁扮粍B[O..n-1]鏉ュ瓨鏀�閾捐〃瀹炵幇鐨勬《锛屽苟鍋囪鍙互鐢ㄦ煇绉嶆満鍒舵潵缁存姢杩欎簺琛ㄣ��
妗舵帓搴忕殑绠楁硶濡備笅(浼唬鐮�琛ㄧず)锛屽叾涓璮loor(x)鏄湴鏉垮嚱鏁帮紝琛ㄧず涓嶈秴杩噚鐨勬渶澶ф暣鏁般��
procedure Bin_Sort(var A:List);begin
n:=length(A);
for i:=1 to n do
灏咥[i]鎻掑埌琛˙[floor(n*A[i])]涓�;
for i:=0 to n-1 do
鐢�鎻掑叆鎺掑簭瀵硅〃B[i]杩涜鎺掑簭;
灏嗚〃B[0],B[1],...,B[n-1]鎸夐『搴忓悎骞�;
end;
鍙冲浘婕旂ず浜嗘《鎺掑簭浣滅敤浜庢湁10涓暟鐨勮緭鍏�鏁扮粍涓婄殑鎿嶄綔杩囩▼銆�(a)杈撳叆鏁扮粍A[1..10]銆�(b)鍦ㄨ绠楁硶鐨勭5琛屽悗鐨勬湁搴忚〃(妗�)鏁扮粍B[0..9]銆傛《i涓瓨鏀句簡鍖洪棿[i/10锛�(i+1)/10]涓婄殑鍊笺�傛帓搴忚緭鍑虹敱琛˙[O]銆丅[1]銆�...銆丅[9]鐨勬寜搴忓苟缃瀯鎴愩��
瑕佽鏄庤繖涓畻娉曡兘姝ｇ‘鍦板伐浣滐紝鐪嬩袱涓厓绱燗[i]鍜孉[j]銆傚鏋滃畠浠惤鍦ㄥ悓涓�涓《涓紝鍒欏畠浠湪杈撳嚭搴忓垪涓湁鐫�姝ｇ‘鐨勭浉瀵规搴忥紝鍥犱负瀹冧滑鎵�鍦ㄧ殑妗舵槸閲囩敤鎻掑叆鎺掑簭鐨勩�傜幇鍋囪瀹冧滑钀藉埌涓嶅悓鐨勬《涓紝璁惧垎鍒负B[i']鍜孊[j']銆備笉澶变竴鑸�э紝鍋囪i' i'=floor(n*A[i])鈮loor(n*A[j])=j'聽寰楃煕鐩韭�(鍥犱负i'聽鏉ュ垎鏋愮畻娉曠殑杩愯鏃堕棿銆傞櫎绗�5琛屽锛屾墍鏈夊悇琛屽湪鏈�鍧忔儏鍐电殑鏃堕棿閮芥槸O(n)銆傜5琛屼腑妫�鏌ユ墍鏈夋《鐨勬椂闂存槸O(n)銆傚垎鏋愪腑鍞竴鏈夎叮鐨勯儴鍒嗗氨鍦ㄤ簬绗�5琛屼腑鎻掍汉鎺掑簭鎵�鑺辩殑鏃堕棿銆�
涓哄垎鏋愭彃浜烘帓搴忕殑鏃堕棿浠ｄ环锛岃ni涓鸿〃绀烘《B[i]涓厓绱犱釜鏁扮殑闅忔満鍙橀噺銆傚洜涓�鎻掑叆鎺掑簭浠ヤ簩娆℃椂闂磋繍琛岋紝鏁呬负鎺掑簭妗禕[i]涓厓绱犵殑鏈熸湜鏃堕棿涓篍[O(ni2)]=O(E[ni2])锛屽鍚勪釜妗朵腑鐨勬墍鏈夊厓绱犳帓搴忕殑鎬绘湡鏈涙椂闂翠负锛歄(n)銆�
(1)聽涓轰簡姹傝繖涓拰寮忥紝瑕佺‘瀹氭瘡涓殢鏈哄彉閲弉i鐨勫垎甯冦�傛垜浠叡鏈塶涓厓绱狅紝n涓《銆傛煇涓厓绱犺惤鍒版《B[i]鐨勬鐜囦负l/n锛屽洜涓烘瘡涓《瀵瑰簲浜庡尯闂碵0锛�1)鐨刲/n銆傝繖绉嶆儏鍐典笌鎶曠悆鐨勪緥瀛愬緢绫讳技锛氭湁n涓悆聽(鍏冪礌)鍜宯涓洅瀛惵�(妗�)锛屾瘡娆℃姇鐞冮兘鏄嫭绔嬬殑锛屼笖浠ユ鐜噋=1/n钀藉埌浠讳竴妗朵腑銆傝繖鏍凤紝ni=k鐨勬鐜囧氨鏈嶄粠浜岄」鍒嗗竷B(k;n,p)锛屽叾鏈熸湜鍊间负E[ni]=np=1锛屾柟宸甐[ni]=np(1-p)=1-1/n銆傚浠绘剰闅忔満鍙橀噺X锛屾湁鍙冲浘鎵�绀�琛ㄨ揪寮�銆�
(2)灏嗚繖涓晫鐢ㄥ埌(1)寮忎笂锛屽緱鍑烘《鎺掑簭涓殑鎻掍汉鎺掑簭鐨勬湡鏈涜繍琛屾椂闂翠负O(n)銆傚洜鑰岋紝鏁翠釜妗舵帓搴忕殑鏈熸湜杩愯鏃堕棿灏辨槸绾挎�х殑銆�
浠ｄ环
妗舵帓搴忓埄鐢ㄥ嚱鏁扮殑鏄犲皠鍏崇郴锛屽噺灏戜簡鍑犱箮鎵�鏈夌殑姣旇緝宸ヤ綔銆傚疄闄呬笂锛屾《鎺掑簭鐨刦(k)鍊肩殑璁＄畻锛屽叾浣滅敤灏辩浉褰撲簬蹇帓涓垝鍒嗭紝宸茬粡鎶婂ぇ閲�鏁版嵁鍒嗗壊鎴愪簡鍩烘湰鏈夊簭鐨�鏁版嵁鍧�(妗�)銆傜劧鍚庡彧闇�瑕佸妗朵腑鐨勫皯閲忔暟鎹仛鍏堣繘鐨勬瘮杈冩帓搴忓嵆鍙��
瀵筃涓�鍏抽敭瀛�杩涜妗舵帓搴忕殑鏃堕棿澶嶆潅搴�鍒嗕负涓や釜閮ㄥ垎锛�
(1)聽寰幆璁＄畻姣忎釜鍏抽敭瀛楃殑妗舵槧灏勫嚱鏁帮紝杩欎釜鏃堕棿澶嶆潅搴�鏄疧(N)銆�
(2)聽鍒╃敤鍏堣繘鐨勬瘮杈�鎺掑簭绠楁硶瀵规瘡涓《鍐呯殑鎵�鏈夋暟鎹繘琛屾帓搴忥紝鍏�鏃堕棿澶嶆潅搴�涓郝犫垜 O(Ni*logNi)聽銆傚叾涓璑i聽涓虹i涓《鐨勬暟鎹噺銆�
寰堟樉鐒讹紝绗�(2)閮ㄥ垎鏄《鎺掑簭鎬ц兘濂藉潖鐨勫喅瀹氬洜绱犮�傚敖閲忓噺灏戞《鍐呮暟鎹殑鏁伴噺鏄彁楂樻晥鐜囩殑鍞竴鍔炴硶(鍥犱负鍩轰簬姣旇緝鎺掑簭鐨勬渶濂藉钩鍧�鏃堕棿澶嶆潅搴�鍙兘杈惧埌O(N*logN)浜�)銆傚洜姝わ紝鎴戜滑闇�瑕佸敖閲忓仛鍒颁笅闈袱鐐癸細
(1)聽鏄犲皠鍑芥暟f(k)鑳藉灏哊涓暟鎹钩鍧囩殑鍒嗛厤鍒癕涓《涓紝杩欐牱姣忎釜妗跺氨鏈塠N/M]涓暟鎹噺銆�
(2)聽灏介噺鐨勫澶ф《鐨勬暟閲忋�傛瀬闄愭儏鍐典笅姣忎釜妗跺彧鑳藉緱鍒颁竴涓暟鎹紝杩欐牱灏卞畬鍏ㄩ伩寮�浜嗘《鍐呮暟鎹殑鈥滄瘮杈冣�濇帓搴忔搷浣溿�� 褰撶劧锛屽仛鍒拌繖涓�鐐瑰緢涓嶅鏄擄紝鏁版嵁閲忓法澶х殑鎯呭喌涓嬶紝f(k)鍑芥暟浼氫娇寰楁《闆嗗悎鐨勬暟閲忓法澶э紝绌洪棿娴垂涓ラ噸銆傝繖灏辨槸涓�涓椂闂翠唬浠峰拰绌洪棿浠ｄ环鐨勬潈琛￠棶棰樹簡銆�
瀵逛簬N涓緟鎺掓暟鎹紝M涓《锛屽钩鍧囨瘡涓《[N/M]涓暟鎹殑妗舵帓搴忓钩鍧�鏃堕棿澶嶆潅搴�涓猴細
O(N)+O(M*(N/M)*log(N/M))=O(N+N*(logN-logM))=O(N+N*logN-N*logM)
褰揘=M鏃讹紝鍗虫瀬闄愭儏鍐典笅姣忎釜妗跺彧鏈変竴涓暟鎹椂銆傛《鎺掑簭鐨勬渶濂芥晥鐜囪兘澶熻揪鍒癘(N)銆�
鎬荤粨锛氭《鎺掑簭鐨勫钩鍧�鏃堕棿澶嶆潅搴�涓虹嚎鎬х殑O(N+C)锛屽叾涓瑿=N*(logN-logM)銆傚鏋滅浉瀵逛簬鍚屾牱鐨凬锛屾《鏁伴噺M瓒婂ぇ锛屽叾鏁堢巼瓒婇珮锛屾渶濂界殑鏃堕棿澶嶆潅搴�杈惧埌O(N)銆傚綋鐒舵《鎺掑簭鐨�绌洪棿澶嶆潅搴�涓篛(N+M)锛屽鏋滆緭鍏ユ暟鎹潪甯稿簽澶э紝鑰屾《鐨勬暟閲忎篃闈炲父澶氾紝鍒欑┖闂翠唬浠锋棤鐤戞槸鏄傝吹鐨勩�傛澶栵紝妗舵帓搴忔槸绋冲畾鐨勩�俒1]
婧愮爜
io.open();//鎵撳紑鎺у埗鍙�
/**-------------------------------------------------------*聽妗舵帓搴�**------------------------------------------------------*/
/*
妗舵帓搴忓亣璁捐緭鍏ュ厓绱犲潎鍖�鑰岀嫭绔嬪垎甯冨湪鍖洪棿[0,1)聽鍗陈�0 <= x and x < 1;灏嗗尯闂村垝鍒嗘垚n涓浉鍚屽ぇ灏忕殑瀛愬尯闂�(妗�)锛岀劧鍚庡皢n涓緭鍏ユ寜澶у皬鍒嗗竷鍒板悇涓《涓幓锛屽姣忎釜妗跺唴閮ㄨ繘琛屾帓搴忋�傛渶鍚庡皢鎵�鏈夋《鐨勬帓搴忕粨鏋滃悎骞惰捣鏉�.
*/
//鎻掑叆鎺掑簭绠楁硶
insert_sort = function( array ){
for( right=2;#array ) {
var top = array[right];
//Insert array[right] into the sorted seqquence array[1....right-1]
var left = right -1;
while( left and array[left]>top){
array[left+1] = array[left];
left--;
}
array[left+1] = top;
}
return array;
}
//妗�鎺掑簭绠楁硶
bucket_sort = function( array ){
var n = #array;
var bucket ={}
for(i=0;n;1){
bucket[i] = {} //鍒涘缓涓�涓《
}
var bucket_index
for(i=1;n;1){
bucket_index = math.floor(n * array[i]);
table.push( bucket [ bucket_index ],array[i] );//鏀惧埌妗堕噷鍘�
}
for(i=1;n;1){
insert_sort( bucket[i] ); //瀵规瘡涓《杩涜鎻掑叆鎺掑簭
}
return table.concat( table.unpack(bucket) )
}
io.print("----------------")
io.print("妗舵帓搴�")
io.print("----------------")
array={};
//妗舵帓搴忓亣璁捐緭鍏ユ槸鐢变竴涓殢鏈鸿繃绋嬩骇鐢熺殑灏忔暟.
math.randomize()
for(i=1;20;1){
table.push( array,math.random() )
}
//鎺掑簭
array = bucket_sort( array )
//杈撳嚭缁撴灉
for(i=1;#array;1){
io.print( array[i] )
}
execute("pause") //鎸�浠绘剰閿�缁х画
io.close();//鍏抽棴鎺у埗鍙�
C++
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#include
usingnamespace聽std;
inta[]={1,255,8,6,25,47,14,35,58,75,96,158,657};
constintlen=sizeof(a)/sizeof(int);
intb[10][len+1]={0};//灏哹鍏ㄩ儴缃�0
voidbucketSort(inta[]);//妗舵帓搴忓嚱鏁�
voiddistribute聽Elments(inta[],intb[10][len+1],intdigits);
voidcollectElments(inta[],intb[10][len+1]);
intnumOfDigits(inta[]);
voidzeroBucket(intb[10][len+1]);//灏哹鏁扮粍涓殑鍏ㄩ儴鍏冪礌缃�0
intmain()
{
cout<<"鍘熷鏁扮粍锛�";
for(inti=0;i
cout<
cout<
bucketSort(a);
cout<<"鎺掑簭鍚庢暟缁勶細";
for(inti=0;i
cout<
cout<
return0;
}
voidbucketSort(inta[])
{
intdigits=numOfDigits(a);
for(inti=1;i<=digits;i++)
{
distributeElments(a,b,i);
collectElments(a,b);
if(i!=digits)
zeroBucket(b);
}
}
intnumOfDigits(inta[])
{
intlargest=0;
for(inti=0;i
if(a[i]>largest)
largest=a[i];
intdigits=0;//digits涓烘渶澶у�肩殑浣嶆暟
while(largest)
{
digits++;
largest/=10;
}
returndigits;
}
voiddistributeElments(inta[],intb[10][len+1],intdigits)
{
intdivisor=10;//闄ゆ暟
for(inti=1;i
divisor*=10;
for(intj=0;j
{
intnumOfDigist=(a[j]%divisor-a[j]%(divisor/10))/(divisor/10);
//numOfDigits涓虹浉搴旂殑(divisor/10)浣嶇殑鍊�,濡傚綋divisor=10鏃�,姹傜殑鏄釜浣嶆暟
intnum=++b[numOfDigist][0];//鐢╞涓涓�鍒楃殑鍏冪礌鏉ュ偍瀛樻瘡琛屼腑鍏冪礌鐨勪釜鏁�
b[numOfDigist][num]=a[j];
}
}
voidcollectElments(inta[],intb[10][len+1])
{
intk=0;
for(inti=0;i<10;i++)
for(intj=1;j<=b[i][0];j++)
a[k++]=b[i][j];
}
voidzeroBucket(intb[][len+1])
{
for(inti=0;i<10;i++)
for(intj=0;j
b[i][j]=0;
}
JAVA
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publicstaticvoidbasket(intdata[])//data涓哄緟鎺掑簭鏁扮粍
{
intn=data.length;
intbask[][]=newint[10][n];
intindex[]=newint[10];
intmax=Integer.MIN_VALUE;
for(inti=0;i
{
max=max>(Integer.toString(data[i]).length())?max:(Integer.toString(data[i]).length());
}
String聽str;
for(inti=max-1;i>=0;i--)
{
for(intj=0;j
{
str="";
if(Integer.toString(data[j]).length()
{
for(intk=0;k
str+="0";
}
str+=Integer.toString(data[j]);
bask[str.charAt(i)-'0'][index[str.charAt(i)-'0']++]=data[j];
}
intpos=0;
for(intj=0;j<10;j++)
{
for(intk=0;k
{
data[pos++]=bask[j][k];
}
}
for(intx=0;x<10;x++)index[x]=0;
}
}
PAS
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programbarrel_sort;
var
a:array[0..100]聽ofinteger;
i,k,n:integer;
begin
fori:=0to100doa[i]:=0;{鍒濆鍖杴
readln(n);
fori:=1ton聽do
begin
read(k);
a[k]:=a[k]+1;{灏嗙瓑浜巏鐨勫�煎叏閮ㄨ鍏ョk妗朵腑}
end;
fori:=0to100do
whilea[i]>0do
begin
write(i:6);
a[i]:=a[i]-1;
end;{杈撳嚭鎺掑簭缁撴瀯}
end.
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浠ｄ环锛�(1) 10G鏁版嵁渚濇璇诲叆鍐呭瓨鐨処O浠ｄ环(杩欎釜鏄棤娉曢伩鍏嶇殑锛孋PU涓嶈兘鐩存帴鍦�纾佺洏涓婅繍绠�)銆�(2)鍦ㄥ唴瀛樹腑閬嶅巻536,870,912涓暟鎹紝杩欐槸涓�涓狾(n)鐨勭嚎鎬�鏃堕棿澶嶆潅搴�銆�(3)鎶�256涓《鍐欏洖鍒�256涓鐩樻枃浠剁┖闂翠腑锛岃繖涓唬浠锋槸棰濆鐨勶紝涔熷氨鏄浠樺嚭涓�鍊嶇殑10G鏁版嵁杞Щ鐨勬椂闂淬��
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#include聽
voidswap(int*a,聽int*b);
voidadjustHeap(intparam1,intj,聽intinNums[]);
voidHeapSort(intnums,聽intinNums[]);
//澶ф牴鍫嗚繘琛岃皟鏁�
voidadjustHeap(intparam1,聽intj,聽intinNums[])
{
inttemp=inNums[param1];
for(intk=param1*2+1;k
{
//濡傛灉鍙宠竟鍊煎ぇ浜庡乏杈瑰�硷紝鎸囧悜鍙宠竟
if(k+1
{
k++;
}
//濡傛灉瀛愯妭鐐瑰ぇ浜庣埗鑺傜偣锛屽皢瀛愯妭鐐瑰�艰祴缁欑埗鑺傜偣,骞朵互鏂扮殑瀛愯妭鐐逛綔涓虹埗鑺傜偣锛堜笉鐢ㄨ繘琛屼氦鎹級
if(inNums[k]>temp)
{
inNums[param1]=inNums[k];
param1=k;
}
else
break;
}
//put聽the聽value聽in聽the聽final聽position
inNums[param1]=temp;
}
//鍫嗘帓搴忎富瑕佺畻娉�
voidHeapSort(intnums,intinNums[])
{
//1.鏋勫缓澶ч《鍫�
for(inti=nums/2-1;i>=0;i--)
{
//put聽the聽value聽in聽the聽final聽position
adjustHeap(i,nums,inNums);
}
//2.璋冩暣鍫嗙粨鏋�+浜ゆ崲鍫嗛《鍏冪礌涓庢湯灏惧厓绱�
for(intj=nums-1;j>0;j--)
{
//鍫嗛《鍏冪礌鍜屾湯灏惧厓绱犺繘琛屼氦鎹�
inttemp=inNums[0];
inNums[0]=inNums[j];
inNums[j]=temp;
adjustHeap(0,j,inNums);//閲嶆柊瀵瑰爢杩涜璋冩暣
}
}
intmain()聽{
intdata[]聽=聽{6,5,8,4,7,9,1,3,2};
intlen聽=聽sizeof(data)聽/聽sizeof(int);
HeapSort(len,data);
return0;
}
C++
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3148
//鍫嗘帓搴�
//鏁寸悊鑺傜偣time:O(lgn)
template
voidMinHeapify(T*arry,intsize,intelement)
{
intlchild=element*2+1,rchild=lchild+1;//宸﹀彸瀛愭爲
while(rchild
{
if(arry[element]<=arry[lchild]&&arry[element]<=arry[rchild])//濡傛灉姣斿乏鍙冲瓙鏍戦兘灏忥紝瀹屾垚鏁寸悊
{
return;
}
if(arry[lchild]<=arry[rchild])//濡傛灉宸﹁竟鏈�灏�
{
swap(arry[element],arry[lchild]);//鎶婂乏闈㈢殑鎻愬埌涓婇潰
element=lchild;//寰幆鏃舵暣鐞嗗瓙鏍�
}
else//鍚﹀垯鍙抽潰鏈�灏�
{
swap(arry[element],arry[rchild]);//鍚岀悊
element=rchild;
}
lchild=element*2+1;
rchild=lchild+1;//閲嶆柊璁＄畻瀛愭爲浣嶇疆
}
if(lchild
{
swap(arry[lchild],arry[element]);
}
return;
}
//鍫嗘帓搴弔ime:O(nlgn)
template
voidHeapSort(T*arry,intsize)
{
inti;
for(i=size-1;i>=0;i--)//浠庡瓙鏍戝紑濮嬫暣鐞嗘爲
{
MinHeapify(arry,size,i);
}
while(size>0)//鎷嗛櫎鏍�
{
swap(arry[size-1],arry[0]);//灏嗘牴锛堟渶灏忥級涓庢暟缁勬渶鏈氦鎹�
size--;//鏍戝ぇ灏忓噺灏�
MinHeapify(arry,size,0);//鏁寸悊鏍�
}
return;
}
Java
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publicclassHeapSort{
privatestaticint[]聽sort=newint[]{1,0,10,20,3,5,6,4,9,8,12,
17,34,11};
publicstaticvoidmain(String[]聽args){
buildMaxHeapify(sort);
heapSort(sort);
print(sort);
}
privatestaticvoidbuildMaxHeapify(int[]聽data){
//娌℃湁瀛愯妭鐐圭殑鎵嶉渶瑕佸垱寤烘渶澶у爢锛屼粠鏈�鍚庝竴涓殑鐖惰妭鐐瑰紑濮�
intstartIndex=getParentIndex(data.length-1);
//浠庡熬绔紑濮嬪垱寤烘渶澶у爢锛屾瘡娆￠兘鏄纭殑鍫�
for(inti=startIndex;i>=0;i--){
maxHeapify(data,data.length,i);
}
}
/**
*鍒涘缓鏈�澶у爢
*
*@paramdata
*@paramheapSize闇�瑕佸垱寤烘渶澶у爢鐨勫ぇ灏忥紝涓�鑸湪sort鐨勬椂鍊欑敤鍒帮紝鍥犱负鏈�澶氬�兼斁鍦ㄦ湯灏撅紝鏈熬灏变笉鍐嶅綊鍏ユ渶澶у爢浜�
*@paramindex褰撳墠闇�瑕佸垱寤烘渶澶у爢鐨勪綅缃�
*/
privatestaticvoidmaxHeapify(int[]聽data,intheapSize,intindex){
//褰撳墠鐐逛笌宸﹀彸瀛愯妭鐐规瘮杈�
intleft=getChildLeftIndex(index);
intright=getChildRightIndex(index);
intlargest=index;
if(left
largest=left;
}
if(right
largest=right;
}
//寰楀埌鏈�澶у�煎悗鍙兘闇�瑕佷氦鎹紝濡傛灉浜ゆ崲浜嗭紝鍏跺瓙鑺傜偣鍙兘灏变笉鏄渶澶у爢浜嗭紝闇�瑕侀噸鏂拌皟鏁�
if(largest!=index){
inttemp=data[index];
data[index]=data[largest];
data[largest]=temp;
maxHeapify(data,heapSize,largest);
}
}
/**
*鎺掑簭锛屾渶澶у�兼斁鍦ㄦ湯灏撅紝data铏界劧鏄渶澶у爢锛屽湪鎺掑簭鍚庡氨鎴愪簡閫掑鐨�
*
*@paramdata
*/
privatestaticvoidheapSort(int[]聽data){
//鏈熬涓庡ご浜ゆ崲锛屼氦鎹㈠悗璋冩暣鏈�澶у爢
for(inti=data.length-1;i>0;i--){
inttemp=data[0];
data[0]=data[i];
data[i]=temp;
maxHeapify(data,i,0);
}
}
/**
*鐖惰妭鐐逛綅缃�
*
*@paramcurrent
*@return
*/
privatestaticintgetParentIndex(intcurrent){
return(current-1)>>1;
}
/**
*宸﹀瓙鑺傜偣position娉ㄦ剰鎷彿锛屽姞娉曚紭鍏堢骇鏇撮珮
*
*@paramcurrent
*@return
*/
privatestaticintgetChildLeftIndex(intcurrent){
return(current<<1)+1;
}
/**
*鍙冲瓙鑺傜偣position
*
*@paramcurrent
*@return
*/
privatestaticintgetChildRightIndex(intcurrent){
return(current<<1)+2;
}
privatestaticvoidprint(int[]聽data){
intpre=-2;
for(inti=0;i
if(pre<(int)getLog(i+1)){
pre=(int)getLog(i+1);
System.out.println();
}
System.out.print(data[i]+"|");
}
}
/**
*浠�2涓哄簳鐨勫鏁�
*
*@paramparam
*@return
*/
privatestaticdoublegetLog(doubleparam){
returnMath.log(param)/Math.log(2);
}
}
Python
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#!/usr/bin/envpython
#-*-coding:utf-8-*-
defheap_sort(lst):
forstartinrange((len(lst)-2)/2,-1,-1):
sift_down(lst,start,len(lst)-1)
forendinrange(len(lst)-1,0,-1):
lst[0],lst[end]=lst[end],lst[0]
sift_down(lst,0,end-1)
returnlst
defsift_down(lst,start,end):
root=start
whileTrue:
child=2*root+1
ifchild>end:
break
ifchild+1<=end聽andlst[child]
child+=1
iflst[root]
lst[root],lst[child]=lst[child],lst[root]
root=child
else:
break
defmain():
l=[9,2,1,7,6,8,5,3,4]
heap_sort(l)
if__name__=="__main__":
main()
C#
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///
///鏋勫缓鍫�
///
staticvoidHeapAdjust(List聽list,intparent,intlength)
{
inttemp=list[parent];
intchild=2*parent+1;
while(child
{
if(child+1
if(temp>=list[child])
break;
list[parent]=list[child];
parent=child;
child=2*parent+1;
}
list[parent]=temp;
}
///
///鍫嗘帓搴�
///
publicstaticList聽HeapSort(List聽list,inttop)
{
List聽topNode=newList();
for(inti=list.Count/2-1;i>=0;i--)
{
HeapAdjust(list,i,list.Count);
}
for(inti=list.Count-1;i>=list.Count-top;i--)
{
inttemp=list[0];
list[0]=list[i];
list[i]=temp;
topNode.Add(temp);
HeapAdjust(list,0,i);
}
returntopNode;
}
JavaScript
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/**
*AuthorJenaszhang
*/
Array.prototype.buildMaxHeap=function(){
for(vari=Math.floor(this.length/2)-1;i>=0;i--){
this.heapAdjust(i,this.length);
}
};
Array.prototype.swap=function(i,j){
vartmp=this[i];
this[i]=this[j];
this[j]=tmp;
};
Array.prototype.heapSort=function(){
this.buildMaxHeap();
for(vari=this.length-1;i>0;i--){
this.swap(0,i);
this.heapAdjust(0,i);
}
returnthis;
};
Array.prototype.heapAdjust=function(i,j){
varlargest=i;
varleft=2*i+1;
varright=2*i+2;
if(left
largest=left;
}
if(right
largest=right;
}
if(largest!=i){
this.swap(i,largest);
this.heapAdjust(largest,j);
}
};
vara=newArray();
[].push.apply(a,[2,3,89,57,23,72,43,105]);
console.log(a.heapSort());
Pascal浠ｇ爜
绀轰緥1锛�
const max=100000;
var
a:array[0..max] of longint;
n,i,tot,t:longint;
procedure down(i:longint);
var j,t:longint;
begin
while i<=tot shr 1 do
begin
j:=2*i;
if (j
if a[i]>a[j] then
begin
t:=a[i];
a[i]:=a[j];
a[j]:=t;
i:=j;
end
else break;
end;
end;
begin
readln(n);
for i:=1 to n do read(a[i]);
tot:=n;
for i:=n shr 1 downto 1 do down(i);
for i:=1 to n do
begin
t:=a[1]; a[1]:=a[tot]; a[tot]:=t;
dec(tot);
down(1);
end;
for i:=n downto 1 do write(a[i],' ');
end.
绀轰緥2:
const max=100000;
var
a:array[0..max] of longint;
n,i,tot,t:longint;
procedure down(i:longint);
var j,t:longint;
begin
while i<=tot shr 1 do
begin
j:=2*i;
if (j
if a[i]>a[j] then
begin
t:=a[i];
a[i]:=a[j];
a[j]:=t;
i:=j;
end
else break;
end;
end;
procedure up(i:longint);
var j,t:longint;
begin
j:=i;
while (j>1) and (a[j]
begin
t:=a[j];
a[j]:=a[j shr 1];
a[j shr 1]:=t;
j:=j shr 1;
end;
end;
begin
readln(n);
for i:=1 to n do
begin
read(a[i]);
up(i);
end;
tot:=n;
for i:=1 to n do
begin
t:=a[1]; a[1]:=a[tot]; a[tot]:=t;
dec(tot);
down(1);
end;
for i:=n downto 1 do write(a[i],' ');
end.
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1962
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00007.聽鈻�Ruby
00008.聽鈻�Erlang璇█
00001.聽鈻�Haskell璇█
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00005.聽鈻�Java
00006.聽鈻�C#
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鎺掑簭婕旂ず
绀轰緥
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鏁版嵁627389
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鏁版嵁327689
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i=2 j=3 k=6
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璋冪敤鍑芥暟
鍦╟++涓彲浠ョ敤鍑芥暟qsort锛堬級鍙互鐩存帴涓�鏁扮粍杩涜鎺掑簭銆俒1]
鐢犳硶:
void qsort(void *base, int nelem, int width, int (*fcmp)(const void *,const void *));
鍙傛暟锛�1聽寰�鎺掑簭鏁扮粍棣栧湴鍧�2聽鏁扮粍涓緟鎺掑簭鍏冪礌鏁伴噺3聽鍚勫厓绱犵殑鍗犵敤绌洪棿澶у皬4聽鎸囧悜鍑芥暟鐨�鎸囬拡锛岀敤浜庣‘瀹氭帓搴忕殑椤哄簭
绀轰緥浠ｇ爜
GO
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//聽绗竴绉嶅啓娉�
func聽quickSort(values聽[]int,聽left,聽right聽int)聽{
temp聽:=聽values[left]
p聽:=聽left
i,聽j聽:=聽left,聽right
fori聽<=聽j聽{
forj聽>=聽p聽&&聽values[j]聽>=聽temp聽{
j--
}
ifj聽>=聽p聽{
values[p]聽=聽values[j]
p聽=聽j
}
ifvalues[i]聽<=聽temp聽&&聽i聽<=聽p聽{
i++
}
ifi聽<=聽p聽{
values[p]聽=聽values[i]
p聽=聽i
}
}
values[p]聽=聽temp
ifp-left聽>聽1聽{
quickSort(values,聽left,聽p-1)
}
ifright-p聽>聽1聽{
quickSort(values,聽p+1,聽right)
}
}
func聽QuickSort(values聽[]int)聽{
iflen(values)聽<=聽1聽{
return
}
quickSort(values,聽0,聽len(values)-1)
}
//聽绗簩绉嶅啓娉�
func聽Quick2Sort(values聽[]int)聽{
iflen(values)聽<=聽1聽{
return
}
mid,聽i聽:=聽values[0],聽1
head,聽tail聽:=聽0,聽len(values)-1
forhead聽<聽tail聽{
fmt.Println(values)
ifvalues[i]聽>聽mid聽{
values[i],聽values[tail]聽=聽values[tail],聽values[i]
tail--
}聽else{
values[i],聽values[head]聽=聽values[head],聽values[i]
head++
i++
}
}
values[head]聽=聽mid
Quick2Sort(values[:head])
Quick2Sort(values[head+1:])
}
Ruby
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defquick_sort(a)聽聽
(x=a.pop)聽?聽quick_sort(a.select聽{聽|i|聽i聽<=聽x聽})聽+聽[x]聽+聽quick_sort(a.select聽{聽|i|聽i聽>聽x聽})聽:聽[]
end
Erlang璇█
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瓒呯畝鐭疄鐜帮細
q_sort([])->
[];
q_sort([H|R])->
q_sort([X||X<-R,X
q_sort([X||X<-R,X>=H]).
Haskell璇█
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q_sort聽n=casen聽of
[]->[]
(x:xs)->q_sort聽[a|a<-xs,a<=x]++[x]++q_sort聽[a|a<-xs,a>x]
C++璇█
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#include聽
usingnamespacestd;
voidQsort(inta[],聽intlow,聽inthigh)
{
if(low聽>=聽high)
{
return;
}
intfirst聽=聽low;
intlast聽=聽high;
intkey聽=聽a[first];/*鐢ㄥ瓧琛ㄧ殑绗竴涓褰曚綔涓烘灑杞�*/
while(first聽<聽last)
{
while(first聽<聽last聽&&聽a[last]聽>=聽key)
{
--last;
}
a[first]聽=聽a[last];/*灏嗘瘮绗竴涓皬鐨勭Щ鍒颁綆绔�*/
while(first聽<聽last聽&&聽a[first]聽<=聽key)
{
++first;
}
a[last]聽=聽a[first];聽聽聽聽
/*灏嗘瘮绗竴涓ぇ鐨勭Щ鍒伴珮绔�*/
}
a[first]聽=聽key;/*鏋㈣酱璁板綍鍒颁綅*/
Qsort(a,聽low,聽first-1);
Qsort(a,聽first+1,聽high);
}
intmain()
{
inta[]聽=聽{57,聽68,聽59,聽52,聽72,聽28,聽96,聽33,聽24};
Qsort(a,聽0,聽sizeof(a)聽/聽sizeof(a[0])聽-聽1);/*杩欓噷鍘熸枃绗笁涓弬鏁拌鍑�1鍚﹀垯鍐呭瓨瓒婄晫*/
for(inti聽=聽0;聽i聽<聽sizeof(a)聽/聽sizeof(a[0]);聽i++)
{
cout聽<<聽a[i]聽<<聽"";
}
return0;
}/*鍙傝�冩暟鎹粨鏋刾274(娓呭崕澶у鍑虹増绀撅紝涓ヨ敋鏁�)*/
C璇█鐗堟湰
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voidsort(int*a,聽intleft,聽intright)
{
if(left聽>=聽right)/*濡傛灉宸﹁竟绱㈠紩澶т簬鎴栬�呯瓑浜庡彸杈圭殑绱㈠紩灏变唬琛ㄥ凡缁忔暣鐞嗗畬鎴愪竴涓粍浜�*/
{
return;
}
inti聽=聽left;
intj聽=聽right;
intkey聽=聽a[left];
while(i聽<聽j)聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽/*鎺у埗鍦ㄥ綋缁勫唴瀵绘壘涓�閬�*/
{
while(i聽<聽j聽&&聽key聽<=聽a[j])
/*鑰屽鎵剧粨鏉熺殑鏉′欢灏辨槸锛�1锛屾壘鍒颁竴涓皬浜庢垨鑰呭ぇ浜巏ey鐨勬暟锛堝ぇ浜庢垨灏忎簬鍙栧喅浜庝綘鎯冲崌
搴忚繕鏄檷搴忥級2锛屾病鏈夌鍚堟潯浠�1鐨勶紝骞朵笖i涓巎鐨勫ぇ灏忔病鏈夊弽杞�*/
{
j--;/*鍚戝墠瀵绘壘*/
}
a[i]聽=聽a[j];
/*鎵惧埌涓�涓繖鏍风殑鏁板悗灏辨妸瀹冭祴缁欏墠闈㈢殑琚嬁璧扮殑i鐨勫�硷紙濡傛灉绗竴娆″惊鐜笖key鏄�
a[left]锛岄偅涔堝氨鏄粰key锛�*/
while(i聽<聽j聽&&聽key聽>=聽a[i])
/*杩欐槸i鍦ㄥ綋缁勫唴鍚戝墠瀵绘壘锛屽悓涓婏紝涓嶈繃娉ㄦ剰涓巏ey鐨勫ぇ灏忓叧绯诲仠姝㈠惊鐜拰涓婇潰鐩稿弽锛�
鍥犱负鎺掑簭鎬濇兂鏄妸鏁板線涓よ竟鎵旓紝鎵�浠ュ乏鍙充袱杈圭殑鏁板ぇ灏忎笌key鐨勫叧绯荤浉鍙�*/
{
i++;
}
a[j]聽=聽a[i];
}
a[i]聽=聽key;/*褰撳湪褰撶粍鍐呮壘瀹屼竴閬嶄互鍚庡氨鎶婁腑闂存暟key鍥炲綊*/
sort(a,聽left,聽i聽-聽1);/*鏈�鍚庣敤鍚屾牱鐨勬柟寮忓鍒嗗嚭鏉ョ殑宸﹁竟鐨勫皬缁勮繘琛屽悓涓婄殑鍋氭硶*/
sort(a,聽i聽+聽1,聽right);/*鐢ㄥ悓鏍风殑鏂瑰紡瀵瑰垎鍑烘潵鐨勫彸杈圭殑灏忕粍杩涜鍚屼笂鐨勫仛娉�*/
/*褰撶劧鏈�鍚庡彲鑳戒細鍑虹幇寰堝鍒嗗乏鍙筹紝鐩村埌姣忎竴缁勭殑i聽=聽j聽涓烘*/
}
JavaScript
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functionquickSort(array){
functionsort(prev,聽numsize){
varnonius聽=聽prev;
varj聽=聽numsize聽-1;
varflag聽=聽array[prev];
if((numsize聽-聽prev)聽>聽1)聽{
while(nonius聽<聽j){
for(;聽nonius聽<聽j;聽j--){
if(array[j]聽<聽flag)聽{
array[nonius++]聽=聽array[j];銆�//a[i]聽=聽a[j];聽i聽+=聽1;
break;
};
}
for(聽;聽nonius聽<聽j;聽nonius++){
if(array[nonius]聽>聽flag){
array[j--]聽=聽array[nonius];
break;
}
}
}
array[nonius]聽=聽flag;
sort(0,聽nonius);
sort(nonius聽+聽1,聽numsize);
}
}
sort(0,聽array.length);
returnarray;
}
Java
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class銆�Quick
{
publicvoid銆�sort(int銆�arr[],int銆�low,int銆�high)
銆�{
int銆�l=low;
int銆�h=high;
int銆�povit=arr[low];
while(l
銆�{
while(l=povit)
銆�h--;
if(l
int銆�temp=arr[h];
銆�arr[h]=arr[l];
銆�arr[l]=temp;
銆�l++;
銆�}
while(l
銆�l++;
if(l
int銆�temp=arr[h];
銆�arr[h]=arr[l];
銆�arr[l]=temp;
銆�h--;
銆�}
銆�}
銆�print(arr);
銆�System.out.print("l="+(l+1)+"h="+(h+1)+"povit="+povit+"\n");
if(l>low)sort(arr,low,l-1);
if(h
銆�}
}
/*//////////////////////////鏂瑰紡浜�////////////////////////////////*/
鏇撮珮鏁堢偣鐨勪唬鐮侊細
public>
T[]quickSort(T[]targetArr,intstart,intend)
{
inti=start+1,j=end;
Tkey=targetArr[start];
SortUtilsUtil=newSortUtil();
if(start>=end)return(targetArr);
/*浠巌++鍜宩--涓や釜鏂瑰悜鎼滅储涓嶆弧瓒虫潯浠剁殑鍊煎苟浜ゆ崲
*
*鏉′欢涓猴細i++鏂瑰悜灏忎簬key锛宩--鏂瑰悜澶т簬key
*/
while(true)
{
while(targetArr[j].compareTo(key)>0)j--;
while(targetArr[i].compareTo(key)<0&&i
if(i>=j)break;
sUtil.swap(targetArr,i,j);
if(targetArr[i]==key)
{
j--;
}else{
i++;
}
}
/*鍏抽敭鏁版嵁鏀惧埌鈥樹腑闂粹��*/
sUtil.swap(targetArr,start,j);
if(start
{
this.quickSort(targetArr,start,i-1);
}
if(j+1
{
this.quickSort(targetArr,j+1,end);
}
returntargetArr;
}
/*//////////////鏂瑰紡涓夛細鍑忓皯浜ゆ崲娆℃暟锛屾彁楂樻晥鐜�/////////////////////*/
private>
voidquickSort(T[]targetArr,intstart,intend)
{
inti=start,j=end;
Tkey=targetArr[start];
while(i
{
/*鎸塲--鏂瑰悜閬嶅巻鐩爣鏁扮粍锛岀洿鍒版瘮key灏忕殑鍊间负姝�*/
while(j>i&&targetArr[j].compareTo(key)>=0)
{
j--;
}
if(i
{
/*targetArr[i]宸茬粡淇濆瓨鍦╧ey涓紝鍙皢鍚庨潰鐨勬暟濉叆*/
targetArr[i]=targetArr[j];
i++;
}
/*鎸塱++鏂瑰悜閬嶅巻鐩爣鏁扮粍锛岀洿鍒版瘮key澶х殑鍊间负姝�*/
while(i
/*姝ゅ涓�瀹氳灏忎簬绛変簬闆讹紝鍋囪鏁扮粍涔嬪唴鏈変竴浜夸釜1锛�0浜ゆ浛鍑虹幇鐨勮瘽锛岃�宬ey鐨勫�煎張鎭板阀鏄�1鐨勮瘽锛岄偅涔堣繖涓皬浜庣瓑浜庣殑浣滅敤灏变細浣夸笅闈㈢殑if璇彞灏戞墽琛屼竴浜挎銆�*/
{
i++;
}
if(i
{
/*targetArr[j]宸蹭繚瀛樺湪targetArr[i]涓紝鍙皢鍓嶉潰鐨勫�煎～鍏�*/
targetArr[j]=targetArr[i];
j--;
}
}
/*姝ゆ椂i==j*/
targetArr[i]=key;
/*閫掑綊璋冪敤锛屾妸key鍓嶉潰鐨勫畬鎴愭帓搴�*/
this.quickSort(targetArr,start,i-1);
/*閫掑綊璋冪敤锛屾妸key鍚庨潰鐨勫畬鎴愭帓搴�*/
this.quickSort(targetArr,j+1,end);
}
C#
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usingSystem;聽
usingSystem.Collections.Generic;聽
usingSystem.Linq;聽
usingSystem.Text;
namespacetest
{
classQuickSort
{
staticvoidMain(string[]聽args)
{
int[]聽array聽=聽{聽49,聽38,聽65,聽97,聽76,聽13,聽27聽};
sort(array,聽0,聽array.Length聽-聽1);
Console.ReadLine();
}
/**涓�娆℃帓搴忓崟鍏冿紝瀹屾垚姝ゆ柟娉曪紝key宸﹁竟閮芥瘮key灏忥紝key鍙宠竟閮芥瘮key澶с��
**@param聽array鎺掑簭鏁扮粍聽
**@param聽low鎺掑簭璧峰浣嶇疆聽
**@param聽high鎺掑簭缁撴潫浣嶇疆
**@return鍗曞厓鎺掑簭鍚庣殑鏁扮粍聽*/
privatestaticintsortUnit(int[]聽array,聽intlow,聽inthigh)
{
intkey聽=聽array[low];
while(low聽<聽high)
{
/*浠庡悗鍚戝墠鎼滅储姣攌ey灏忕殑鍊�*/
while(array[high]聽>=聽key聽&&聽high聽>聽low)
--high;聽
/*姣攌ey灏忕殑鏀惧乏杈�*/
array[low]聽=聽array[high];聽聽聽
/*浠庡墠鍚戝悗鎼滅储姣攌ey澶х殑鍊硷紝姣攌ey澶х殑鏀惧彸杈�*/
while(array[low]聽<=聽key聽&&聽high聽>聽low)
++low;聽
/*姣攌ey澶х殑鏀惧彸杈�*/
array[high]聽=聽array[low];
}
/*宸﹁竟閮芥瘮key灏忥紝鍙宠竟閮芥瘮key澶с��//灏唊ey鏀惧湪娓告爣褰撳墠浣嶇疆銆�//姝ゆ椂low绛変簬high聽*/
array[low]聽=聽key;
foreach(inti聽inarray)
{
Console.Write("{0}\t",聽i);
}
Console.WriteLine();
returnhigh;
}聽聽聽聽
/**蹇�熸帓搴徛�
*@paramarry聽
*@return聽*/
publicstaticvoidsort(int[]聽array,聽intlow,聽inthigh)
{
if(low聽>=聽high)
return;聽
/*瀹屾垚涓�娆″崟鍏冩帓搴�*/
intindex聽=聽sortUnit(array,聽low,聽high);聽
/*瀵瑰乏杈瑰崟鍏冭繘琛屾帓搴�*/
sort(array,聽low,聽index聽-聽1);
/*瀵瑰彸杈瑰崟鍏冭繘琛屾帓搴�*/
sort(array,聽index聽+聽1,聽high);
}
}
}聽
杩愯缁撴灉锛�27 38 13 49 76 97 65
13 27 38 49 76 97 6513 27 38 49 65 76 97
蹇�熸帓搴忓氨鏄�閫掑綊璋冪敤姝よ繃绋嬧�斺�斿湪浠�49涓轰腑鐐瑰垎鍓茶繖涓暟鎹簭鍒楋紝鍒嗗埆瀵瑰墠闈竴閮ㄥ垎鍜屽悗闈竴閮ㄥ垎杩涜绫讳技鐨勫揩閫熸帓搴忥紝浠庤�屽畬鎴愬叏閮ㄦ暟鎹簭鍒楃殑蹇�熸帓搴忥紝鏈�鍚庢妸姝ゆ暟鎹簭鍒楀彉鎴愪竴涓湁搴忕殑搴忓垪锛屾牴鎹繖绉嶆�濇兂瀵逛簬涓婅堪鏁扮粍A鐨勫揩閫熸帓搴忕殑鍏ㄨ繃绋嬪鍥�6鎵�绀猴細
鍒濆鐘舵�伮爗49 38 65 97 76 13 27}聽杩涜涓�娆″揩閫熸帓搴忎箣鍚庡垝鍒嗕负聽{27 38 13} 49 {76 97 65}聽鍒嗗埆瀵瑰墠鍚庝袱閮ㄥ垎杩涜蹇�熸帓搴弡27 38 13}聽缁忕涓夋鍜岀鍥涙浜ゆ崲鍚庡彉鎴惵爗13 27 38}聽瀹屾垚鎺掑簭銆倇76 97 65}聽缁忕涓夋鍜岀鍥涙浜ゆ崲鍚庡彉鎴惵爗65 76 97}聽瀹屾垚鎺掑簭銆傚浘绀�
F#
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let聽rec聽qsort聽=
function
[]聽->聽[]
|x::xs聽->
qsort聽[fori聽in聽xs聽doifi聽<聽x聽then聽yieldi]@
x::qsort聽[fori聽in聽xs聽doifi聽>=x聽then聽yieldi]
PHP
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functionquickSort(&$arr){
if(count($arr)>1){
$k=$arr[0];
$x=array();
$y=array();
$_size=count($arr);
for($i=1;$i<$_size;$i++){
if($arr[$i]<=$k){
$x[]=$arr[$i];
}elseif($arr[$i]>$k){
$y[]=$arr[$i];
}
}
$x=quickSort($x);
$y=quickSort($y);
returnarray_merge($x,array($k),$y);
}else{
return$arr;
}
}
?>
Pascal
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杩欓噷鏄畬鍏ㄧ▼搴忥紝杩囩▼閮ㄥ垎涓哄揩鎺�
programqsort;
varn,p:integer;
a:array[0..100000]聽ofinteger;
procedureqs(l,r:integer);//鍋囪琚帓搴忕殑鏁扮粍鏄痑,涓斿揩鎺掑悗鎸夊崌搴忔帓鍒楋級
vari,j,m,t:integer;
begin
i:=l;
j:=r;//(l锛坙eft锛�,r锛坮ight锛夎〃绀哄揩鎺掔殑宸﹀彸鍖洪棿)
m:=a[(l+r)div2];//娉ㄦ剰锛氭湰鍙ヤ笉鑳藉啓鎴愶細m:=(l+r)div2;
repeat
whilea[i]
whilea[j]>m聽dodec(j);//鑻ユ槸闄嶅簭鎶�'<'涓庘��>'浜掓崲;
ifi<=j聽then
begin
t:=a[i];
a[i]:=a[j];
a[j]:=t;
inc(i);
dec(j);
end;
untili>j;
ifl
ifi
end;
begin
readln(n);//鏈塶涓暟鎹澶勭悊
forp:=1ton聽doread(a[p]);//杈撳叆鏁版嵁
qs(1,n);
forp:=1ton聽dowrite(a[p],'');//杈撳嚭蹇帓鍚庣殑鏁版嵁
end.
鎴栬��
procedurequickSort(vara:arrayofinteger;l,r:Integer);
vari,j,x:integer;
begin
ifl>=r聽thenexit;
i:=l;
j:=r;
x:=a[i];
whilei<=j聽do
begin
while(ix)聽dodec(j);
ifi
begin
a[i]:=a[j];
inc(i);
end;
while(i
ifi
begin
a[j]:=a[i];
dec(j);
end;
a[i]:=x;
quicksort(a,l,i-1);
quicksort(a,i+1,r);
end;
end;
Python閫掑綊
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#quick聽sort
defquickSort(L,聽low,聽high):
i聽=聽low聽
j聽=聽high
ifi聽>=聽j:
returnL
key聽=聽L[i]
whilei聽<聽j:
whilei聽<聽j聽andL[j]聽>=聽key:
j聽=聽j-1
L[i]聽=聽L[j]
whilei聽<聽j聽andL[i]聽<=聽key:聽聽聽聽
i聽=聽i+1
L[j]聽=聽L[i]
L[i]聽=聽key聽
quickSort(L,聽low,聽i-1)
quickSort(L,聽j+1,聽high)
returnL
浼樺寲
涓夊钩鍧囧垎鍖烘硶
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骞宠 蹇帓
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涓夎矾鍩烘暟蹇帓
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QUICKSORT(A锛�p锛�r)
1if聽p<r
2then聽q 鈫�PARTITION(A锛�p锛�r)
3QUICKSORT(A锛�p锛�q-1)
4QUICKSORT(A锛�q+1锛�r)
涓烘帓搴忎竴涓畬鏁寸殑鏁扮粍A锛屾渶鍒濈殑璋冪敤鏄疩UICKSORT(A锛�1锛�length[A]锛夈��
蹇��鎺掑簭绠楁硶鐨勫叧閿槸PARTITION杩囩▼锛屽畠瀵瑰瓙鏁扮粍A[p..r]杩涜灏卞湴閲嶆帓锛�
PARTITION(A锛�p锛�r)
1x鈫�A[r]
2i鈫�p-1
3for聽j鈫�p聽to聽r-1
4do if聽A[j]鈮�x
5then聽i鈫�i+1
6exchange聽A[i]鈫愨啋A[j]
7exchange聽A[i+1]鈫愨啋A[r]
8return聽i+1[2]
闅忔満
瀵筆ARTITION鍜孮UICKSORT鎵�浣滅殑鏀瑰姩姣旇緝灏忋�傚湪鏂扮殑鍒掑垎杩囩▼涓紝鎴戜滑鍦ㄧ湡姝ｈ繘琛屽垝鍒嗕箣鍓嶅疄鐜颁氦鎹細
锛堝叾涓璓ARTITION杩囩▼鍚屽揩閫�鎺掑簭浼唬鐮�锛堥潪闅忔満锛夛級
RANDOMIZED-PARTITION(A锛�p锛�r)
1i鈫� RANDOM(p锛�r)
2exchange聽A[r]鈫愨啋A[i]
3return聽PARTITION(A锛�p锛�r)
鏂扮殑蹇�熸帓搴忚繃绋嬩笉鍐嶈皟鐢≒ARTITION锛岃�屾槸璋冪敤RANDOMIZED-PARTITION銆�
RANDOMIZED-QUICKSORT(A锛�p锛�r)
1if聽p<r
2then聽q鈫� RANDOMIZED-PARTITION(A锛�p锛�r)
3RANDOMIZED-QUICKSORT(A锛�p锛�q-1)
4RANDOMIZED-QUICKSORT(A锛�q+1锛�r)聽[2]聽
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杩欓噷涓烘柟渚胯捣瑙侊紝鎴戜滑鍋囪绠楁硶Quick_Sort鐨勮寖鍥撮槇鍊间负1锛堝嵆涓�鐩村皢绾挎�ц〃鍒嗚В鍒板彧鍓╀竴涓厓绱狅級锛岃繖瀵硅绠楁硶澶嶆潅鎬�鐨勫垎鏋愭病鏈夋湰璐ㄧ殑褰卞搷銆�
鎴戜滑鍏堝垎鏋愬嚱鏁�partition鐨勬�ц兘锛岃鍑芥暟瀵逛簬纭畾鐨勮緭鍏ュ鏉傛�ф槸纭畾鐨勩�傝瀵熻鍑芥暟锛屾垜浠彂鐜帮紝瀵逛簬鏈塶涓厓绱犵殑纭畾杈撳叆L[p..r]锛岃鍑芥暟杩愯鏃堕棿鏄剧劧涓何革紙n锛夈��
鏈�鍧忔儏鍐�
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鎴戜滑浠庣洿瑙変笂鍙互鍒ゆ柇鍑烘渶鍧忔儏鍐靛彂鐢熷湪姣忔鍒掑垎杩囩▼浜х敓鐨勪袱涓尯闂村垎鍒寘鍚玭-1涓厓绱犲拰1涓厓绱犵殑鏃跺�欙紙璁捐緭鍏ョ殑琛ㄦ湁n涓厓绱狅級銆備笅闈㈡垜浠殏鏃惰涓鸿鐚滄祴姝ｇ‘锛屽湪鍚庢枃鎴戜滑鍐嶈缁嗚瘉鏄庤鐚滄祴銆�
瀵逛簬鏈塶涓厓绱犵殑琛↙[p..r]锛岀敱浜庡嚱鏁癙artition鐨勮绠楁椂闂翠负胃锛坣锛夛紝鎵�浠ュ揩閫熸帓搴忓湪搴忓潖鎯呭喌涓嬬殑澶嶆潅鎬ф湁閫掑綊寮忓涓嬶細
T(1)=胃(1),T(n)=T(n-1)+T(1)+胃(n) (1)
鐢ㄨ凯浠ｆ硶鍙互瑙ｅ嚭涓婂紡鐨勮В涓篢(n)=胃锛坣2锛夈��
杩欎釜鏈�鍧忔儏鍐佃繍琛屾椂闂翠笌鎻掑叆鎺掑簭鏄竴鏍风殑銆�
涓嬮潰鎴戜滑鏉ヨ瘉鏄庤繖绉嶆瘡娆″垝鍒嗚繃绋嬩骇鐢熺殑涓や釜鍖洪棿鍒嗗埆鍖呭惈n-1涓厓绱犲拰1涓厓绱犵殑鎯呭喌灏辨槸鏈�鍧忔儏鍐点��
璁綯(n锛夋槸杩囩▼Quick_Sort浣滅敤浜庤妯′负n鐨勮緭鍏ヤ笂鐨勬渶鍧忔儏鍐电殑鏃堕棿锛屽垯
T(n)=max(T(q)+T(n-q))+胃锛坣)锛屽叾涓�1鈮鈮-1 (2)
鎴戜滑鍋囪瀵逛簬浠讳綍k
灏嗗綊绾冲亣璁句唬鍏ワ紙2锛夛紝寰楀埌锛�
T(n锛夆墹max(cq2+c(n-q)2)+胃锛坣)=c*max(q2+(n-q)2)+胃锛坣)
鍥犱负鍦╗1,n-1]涓妐2+(n-q)2鍏充簬q閫掑噺锛屾墍浠ュ綋q=1鏃秖2+(n-q)2鏈夋渶澶у�糿2-2(n-1锛夈�備簬鏄湁锛�
T(n锛夆墹cn2-2c(n-1)+胃锛坣锛夆墹cn2
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杩欐牱锛�鎺掑簭绠楁硶鐨勬渶鍧忔儏鍐佃繍琛屾椂闂翠负胃锛坣2锛夛紝涓旀渶鍧忔儏鍐靛彂鐢熷湪姣忔鍒掑垎杩囩▼浜х敓鐨勪袱涓尯闂村垎鍒寘鍚玭-1涓厓绱犲拰1涓厓绱犵殑鏃跺�欍��
鏃堕棿澶嶆潅搴�涓簅锛坣2锛夈��
鏈�濂芥儏鍐�
濡傛灉姣忔鍒掑垎杩囩▼浜х敓鐨勫尯闂村ぇ灏忛兘涓簄/2锛屽垯蹇�熸帓搴忔硶杩愯灏卞揩寰楀浜嗐�傝繖鏃舵湁锛�
T(n)=2T(n/2)+胃锛坣),T(1)=胃锛�1) (3)
瑙ｅ緱锛歍(n)=胃锛坣logn)
蹇�熸帓搴忔硶鏈�浣虫儏鍐典笅鎵ц杩囩▼鐨勯�掑綊鏍戝涓嬪浘鎵�绀猴紝鍥句腑lgn琛ㄧず浠�10涓哄簳鐨勫鏁帮紝鑰屾湰鏂囦腑鐢╨ogn琛ㄧず浠�2涓哄簳鐨勫鏁�.
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浣嗘槸锛屾槸鍚︿竴瀹氳姣忔骞冲潎鍒掑垎鎵嶈兘杈惧埌鏈�濂芥儏鍐靛憿锛熻鐞嗚В杩欎竴鐐瑰氨蹇呴』鐞嗚В瀵圭О鎬ф槸濡備綍鍦ㄦ弿杩拌繍琛屾椂闂寸殑閫掑綊寮忎腑鍙嶆槧鐨勩�傛垜浠亣璁炬瘡娆″垝鍒嗚繃绋嬮兘浜х敓9:1鐨勫垝鍒嗭紝涔嶄竴鐪嬭鍒掑垎寰堜笉瀵圭О銆傛垜浠彲浠ュ緱鍒伴�掑綊寮忥細
T(n)=T(n/10)+T(9n/10)+胃锛坣),T(1)=胃锛�1) (4)
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路聽2.绠楁硶瀵艰(Introduction to Algorithms Second Edition)涓枃鐗� 鏈烘宸ヤ笟鍑虹増绀韭�2010
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Dijkstra绠楁硶
SPFA绠楁硶\Bellman-Ford绠楁硶
Floyd绠楁硶\Floyd-Warshall绠楁硶
Johnson绠楁硶
A*绠楁硶
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Dijkstra绠楁硶
Dijkstra绠楁硶锛�杩澃鏂壒鎷�锛夋槸鍏稿瀷鐨勬渶鐭矾寰�璺敱绠楁硶锛岀敤浜庤绠椾竴涓妭鐐瑰埌鍏朵粬鎵�鏈夎妭鐐圭殑鏈�鐭矾寰勩�備富瑕佺壒鐐规槸浠ヨ捣濮嬬偣涓轰腑蹇冨悜澶栧眰灞傛墿灞曪紝鐩村埌鎵╁睍鍒扮粓鐐逛负姝€��Dijkstra绠楁硶鑳藉緱鍑烘渶鐭矾寰勭殑鏈�浼樿В锛屼絾鐢变簬瀹冮亶鍘嗚绠楃殑鑺傜偣寰堝锛屾墍浠ユ晥鐜囦綆銆傚彲浠ョ敤鍫嗕紭鍖栥��
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Dijkstra涓�鑸殑琛ㄨ堪閫氬父鏈変袱绉嶆柟寮忥紝涓�绉嶇敤姘镐箙鍜屼复鏃舵爣鍙锋柟寮忥紝涓�绉嶆槸鐢∣PEN, CLOSE琛ㄦ柟寮忥紝Drew涓轰簡鍜屼笅闈㈣浠嬬粛鐨劼燗*聽绠楁硶鍜屄燚*聽绠楁硶琛ㄨ堪涓�鑷达紝杩欓噷鍧囬噰鐢∣PEN,CLOSE琛ㄧ殑鏂瑰紡銆�
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2锛� 浠嶰PEN琛ㄤ腑鎵惧嚭璺濊捣濮嬬偣鏈�杩戠殑鐐癸紝鎵惧嚭杩欎釜鐐圭殑鎵�鏈夊瓙鑺傜偣锛屾妸杩欎釜鐐规斁鍒癈LOSE琛ㄤ腑銆�
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4锛� 閲嶅绗�2鍜岀3姝�,鐩村埌OPEN琛ㄤ负绌猴紝鎴栨壘鍒扮洰鏍囩偣銆�
C++浠ｇ爜
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#include
#include
usingnamespacestd;
voiddijkstra(constint&beg,//鍑哄彂鐐�
constvector聽>聽&adjmap,//閭绘帴鐭╅樀锛岄�氳繃浼犲紩鐢ㄩ伩鍏嶆嫹璐�
vector聽&dist,//鍑哄彂鐐瑰埌鍚勭偣鐨勬渶鐭矾寰勯暱搴�
vector聽&path)//璺緞涓婂埌杈捐鐐圭殑鍓嶄竴涓偣
//璐熻竟琚浣滀笉鑱旈��
//绂忓埄锛氳繖涓嚱鏁版病鏈夌敤浠讳綍鍏ㄥ眬閲忥紝鍙互鐩存帴澶嶅埗锛�
{
constint&NODE=adjmap.size();//鐢ㄩ偦鎺ョ煩闃电殑澶у皬浼犻�掗《鐐逛釜鏁帮紝鍑忓皯鍙傛暟浼犻��
dist.assign(NODE,-1);//鍒濆鍖栬窛绂讳负鏈煡
path.assign(NODE,-1);//鍒濆鍖栬矾寰勪负鏈煡
vector聽flag(NODE,0);//鏍囧織鏁扮粍锛屽垽鏂槸鍚﹀鐞嗚繃
dist[beg]=0;//鍑哄彂鐐瑰埌鑷韩璺緞闀垮害涓�0
while(1)
{
intv=-1;//鍒濆鍖栦负鏈煡
for(inti=0;聽i!=NODE;聽++i)
if(!flag[i]&&dist[i]>=0)//瀵绘壘鏈澶勭悊杩囦笖
if(v<0||dist[i]
v=i;
if(v<0)return;//鎵�鏈夎仈閫氱殑鐐归兘琚鐞嗚繃
flag[v]=1;//鏍囪
for(inti=0;聽i!=NODE;聽++i)
if(adjmap[v][i]>=0)//鏈夎仈閫氳矾寰勪笖
if(dist[i]<0||dist[v]+adjmap[v][i]
{
dist[i]=dist[v]+adjmap[v][i];//鏇存柊
path[i]=v;//璁板綍璺緞
}
}
}
intmain()
{
intn_num,e_num,beg;//鍚箟瑙佷笅
cout<<"杈撳叆鐐规暟銆佽竟鏁般�佸嚭鍙戠偣锛�";
cin>>n_num>>e_num>>beg;
vector聽>聽adjmap(n_num,vector(n_num,-1));//榛樿鍒濆鍖栭偦鎺ョ煩闃�
for(inti=0,p,q;聽i!=e_num;聽++i)
{
cout<<"杈撳叆绗�"<
cin>>p>>q;
cin>>adjmap[p][q];
}
vector聽dist,path;//鐢ㄤ簬鎺ユ敹鏈�鐭矾寰勯暱搴﹀強璺緞鍚勭偣
dijkstra(beg,adjmap,dist,path);
for(inti=0;聽i!=n_num;聽++i)
{
cout<
for(intw=i;聽path[w]>=0;聽w=path[w])
cout<
cout<
}
}
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璐濆彾鏂垎绫绘妧鏈湪浼楀鍒嗙被鎶�鏈腑鍗犳湁閲嶈鍦颁綅锛屼篃灞炰簬缁熻瀛�鍒嗙被鐨勮寖鐣达紝鏄竴绉嶉潪瑙勫垯鐨勫垎绫绘柟娉曪紝璐濆彾鏂垎绫绘妧鏈�氳繃瀵瑰凡鍒嗙被鐨勬牱鏈瓙闆嗚繘琛岃缁冿紝瀛︿範褰掔撼鍑哄垎绫诲嚱鏁�(瀵圭鏁ｅ彉閲忕殑棰勬祴绉颁綔鍒嗙被锛屽杩炵画鍙橀噺鐨勫垎绫荤О涓�鍥炲綊)锛屽埄鐢ㄨ缁冨緱鍒扮殑鍒嗙被鍣ㄥ疄鐜板鏈垎绫绘暟鎹殑鍒嗙被銆傞�氳繃瀵规瘮鍒嗘瀽涓嶅悓鐨勫垎绫荤畻娉曪紝鍙戠幇鏈寸礌璐濆彾鏂垎绫荤畻娉�(Naive Bayes)锛屼竴绉嶇畝鍗曠殑璐濆彾鏂垎绫荤畻娉曪紝鍏跺簲鐢ㄦ晥鏋滄瘮绁炵粡缃戠粶鍒嗙被绠楁硶鍜屽垽瀹氭爲鍒嗙被绠楁硶杩樿濂斤紝鐗瑰埆鏄緟鍒嗙被鏁版嵁閲忛潪甯稿ぇ鏃讹紝璐濆彾鏂垎绫绘柟娉曠浉杈冨叾浠栧垎绫荤畻娉曞叿鏈夐珮鍑嗙‘鐜嘯1]聽銆�
涓枃鍚�
璐濆彾鏂垎绫�
澶栨枃鍚�
Bayesian聽classification
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灞炴��
鍒嗙被鏂规硶
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00001.聽1聽鑳屾櫙浠嬬粛
00002.聽2聽璐濆彾鏂垎绫荤殑鍘熺悊
00003.聽3聽璐濆彾鏂垎绫荤壒鐐�
鑳屾櫙浠嬬粛
缂栬緫
鏁版嵁鍒嗙被(Classification)鍦ㄥ晢涓氬簲鐢ㄤ笂鍏锋湁閲嶈鎰忎箟锛屾槸鏁版嵁鎸栨帢涓潪甯搁噸瑕佺殑涓�椤圭爺绌跺唴瀹广�傞�氬父鏁版嵁鍒嗙被鐨勫仛娉曟槸锛屽熀浜庢牱鏈暟鎹厛璁粌鏋勫缓鍒嗙被鍑芥暟鎴栬�呭垎绫绘ā鍨�(涔熺О涓哄垎绫诲櫒)锛岃鍒嗙被鍣ㄧ殑鍏锋湁灏嗗緟鍒嗙被鏁版嵁椤规槧灏勫埌鏌愪竴鐗圭偣绫诲埆鐨勫姛鑳斤紝鏁版嵁鍒嗙被鍜屽洖褰掑垎鏋愰兘鍙敤浜庨娴嬶紝棰勬祴鏄寚浠庡熀浜庢牱鏈暟鎹褰曪紝鏍规嵁鍒嗙被鍑嗗垯鑷姩缁欏嚭瀵规湭鐭ユ暟鎹殑鎺ㄥ箍鎻忚堪锛屼粠鑰屽疄鐜板鏈煡鏁版嵁杩涜棰勬祴聽[1]聽銆�
璐濆彾鏂垎绫绘槸缁熻瀛︾殑鍒嗙被鏂规硶锛屽叾鍒嗘瀽鏂规硶鐨勭壒鐐规槸浣跨敤姒傜巼鏉ヨ〃绀烘墍鏈夊舰寮忕殑涓嶇‘瀹氭�э紝瀛︿範鎴栨帹鐞嗛兘瑕佺敤姒傜巼瑙勫垯鏉ュ疄鐜般��
璐濆彾鏂垎绫荤殑鍘熺悊
缂栬緫
鍩轰簬缁熻瀛︾殑璐濆彾鏂垎绫绘柟娉曚互璐濆彾鏂悊璁轰负鍩虹锛岄�氳繃姹傝В鍚庨獙姒傜巼鍒嗗竷锛岄娴嬫牱鏈睘浜庢煇涓�绫诲埆鐨勬鐜囥�傝礉鍙舵柉鍏紡鍙啓鎴愬涓嬪舰寮�:
P(y|x)=P(x|y)*P(A)*P(y)/(P(x) (4-1)鍏朵腑锛孭(y I x)涓哄悗楠屾鐜囧垎甯冿紝P(y)涓哄厛楠屽垎甯冿紝P(x)閫氬父涓哄父鏁般��
涓轰簡绠�鍖栬繍绠楋紝鏈寸礌璐濆彾鏂垎绫荤畻娉曞亣瀹氫换鎰忓睘鎬у绫诲埆鐨勫奖鍝嶄笌鍏朵粬灞炴�у绫诲埆鐨勫奖鍝嶆棤鍏筹紝杩欑鍋囧畾绉颁负绫绘潯浠剁嫭绔嬫湸绱犲亣瀹氥�傚浘4-3灞曠ず浜嗘湸鏍戣礉鍙舵柉鍒嗙被涓睘鎬у拰绫讳箣闂寸殑鍏崇郴锛屽鍥�4-3鎵�绀猴紝C琛ㄧず寰呭垎绫诲埆锛孉1, ..., A4琛ㄧず鏍锋湰灞炴�э紝绠ご琛ㄧず灞炴�у彉閲忓拰绫诲埆鍙橀噺涔嬮棿鐨勪緷瀛樺叧绯伙紝浠庡浘涓彲浠ョ湅鍑猴紝鍦ㄦ湸绱犺礉鍙舵柉鍒嗙被妯″瀷涓紝鏍锋湰灞炴�i鍜孉j( i涓嶇瓑浜巎)涔嬮棿涓嶅瓨鍦ㄧ浉浜掍緷璧栧叧绯伙紝浠栦滑浠呬笌鑺傜偣绫籆鏈夊叧聽[1]聽銆�
宸茬煡鏍锋湰鏁版嵁x =< x1聽, . .. , xn>(鏍锋湰鏁版嵁x鍏辨湁n绉嶅睘鎬э紝鍏朵腑xi琛ㄧず绗琲涓睘鎬i鐨勫��)灞炰簬浠绘剰绫伙紝(y鈭� { c1,,...,ck})(鎬诲叡k涓被鍒紝cj琛ㄧず绗琷涓被)鐨勬鐜囥�傜粰瀹氫竴涓湭鍒嗙被鐨勬暟鎹牱鏈琗锛屽簲鐢ㄦ湸绱犺礉鍙舵柉鍒嗙被绠楁硶锛岄娴嬫牱鏈暟鎹甔灞炰簬鍏锋湁鏈�楂樺悗楠屾鐜囩殑绫伙紝鏈煡鏍锋湰X灞炰簬绫诲埆c;鐨勬潯浠舵槸锛屽綋涓斾粎褰�
P(ciIX)>P (cjIX)锛�1鈮鈮, (4-2)
璐濆彾鏂垎绫�(10寮�)
鍥犳锛屽皢鏈�澶у寲鍚庨獙姒傜巼P(ciIX)鎴栬�呭叾瀵规暟褰㈠紡绉颁负鏈�澶у悗楠屽亣瀹氾紝璁颁负arg maxyP( y IX)銆�
鏍规嵁鍏ㄦ鐜囧叕寮忥紝瀵逛簬浠绘剰绫诲埆ci銆�
鍦ㄤ换鎰忎竴娆″垎绫讳腑鍙栧�煎潎鐩哥瓑锛屼篃灏辨槸璇达紝鏁版嵁鏍锋湰X浜х敓鐨勬鐜囩浉鍚�(P(X)瀹氫箟涓哄父鏁�)锛屽洜姝わ紝鍙互灏嗗悗楠屾鐜嘝(yl X)琛ㄧず鎴愭鐜囦箻绉姣斿叧绯诲紡:
P(yIX)鈭漃(XIy)*P(y)
鍥犳锛屾眰鍙朼rg maxyP( y IX)鐩稿綋浜庢眰鍙朼rg maxyP(XIy)锛涜�宎rg maxyP(XIy)鐨勮绠楄鐩稿瀹规槗寰堝锛屾墍浠ワ紝鍦ㄥ疄闄呭簲鐢ㄤ腑閫氬父鏍规嵁寮�(4-4 )鏉ユ眰瑙ｅ悗楠屾鐜囥��
鏍规嵁鏈寸礌璐濆彾鏂垎绫荤畻娉曠殑绫绘潯浠剁嫭绔嬪亣璁撅紝缁欏畾鏍锋湰鏁版嵁鐨勭被鏍囧彿锛屽悇灞炴�у�紉i涔嬮棿鐩镐簰鏉′欢鐙珛锛屽郊姝や笉瀛樺湪鐩镐簰渚濊禆鍏崇郴銆�
涔熷氨鏄锛屼负瀵规湭鐭ユ牱鏈琗鍒嗙被锛屽姣忎釜绫籧i璁＄畻P(xl ci)P(ci);褰撲笖浠呭綋P(Xlci)P(ci)>P(Xlcj)P(cj)锛�1鈮鈮锛宩鈮爄 (4-7)
瀹氫箟鏍锋湰X灞炰簬绫诲埆ci锛屽嵆X琚寚娲惧埌P(X I ci)P(ci)鏈�澶х殑绫籧i聽[1]聽銆�
璐濆彾鏂垎绫荤壒鐐�
缂栬緫
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(1)璐濆彾鏂垎绫讳笉鎶婁竴涓疄渚嬬粷瀵圭殑鎸囨淳缁欐煇涓�绉嶅垎绫伙紝鑰屾槸閫氳繃璁＄畻寰楀埌瀹炰緥灞炰簬鏌愪竴鍒嗙被鐨勬鐜囷紝鍏锋湁鏈�澶ф鐜囩殑绫诲氨鏄瀹炰緥鎵�灞炵殑鍒嗙被;
(2)涓�鑸儏鍐典笅鍦ㄨ礉鍙舵柉鍒嗙被涓墍鏈夊睘鎬ч兘娼滃湪鐨勫鍒嗙被缁撴灉鍙戞尌浣滅敤锛岃兘澶熶娇鎵�鏈夌殑灞炴�ч兘鍙備笌鍒板垎绫讳腑;
(3)璐濆彾鏂垎绫诲疄渚嬬殑灞炴�у彲浠ユ槸绂绘暎鐨勩�佽繛缁殑锛屼篃鍙互鏄贩鍚堢殑銆�
璐濆彾鏂柟娉曞洜鍏跺湪鐞嗚涓婄粰鍑轰簡鏈�灏忓寲璇樊鐨勬渶浼樿В鍐虫柟娉曡�岃骞挎硾搴旂敤浜庡垎绫婚棶棰樸�傚湪璐濆彾鏂柟娉曠殑鍩虹涓婏紝鎻愬嚭浜嗚礉鍙舵柉缃戠粶((Bayesian Network, BN)鏂规硶銆傛湸绱犺礉鍙舵柉鍒嗙被灏辨槸鍋囧畾涓�涓睘鎬у浜庣粰瀹氬垎绫荤殑褰卞搷鐙珛浜庡叾浠栧睘鎬с�傝繖涓�鍋囧畾琚О浣滄潯浠剁嫭绔嬶紝瀵瑰疄鍔涘睘鎬х殑杩欑鍋囪澶уぇ绠�鍖栦簡鍒嗙被鎵�闇�鐨勮绠楅噺銆傚ぇ閲忕殑鐮旂┒缁撴灉琛ㄦ槑锛岃櫧鐒禕N绠楁硶瀵瑰睘鎬х粨鐐逛箣闂寸殑杩炴帴缁撴瀯杩涜浜嗛檺鍒讹紝浣嗘槸鏈寸礌璐濆彾鏂殑鍒嗙被鍣ㄧ殑鍒嗙被鎬ц兘浼樹簬鏍囧噯鐨勮礉鍙舵柉缃戠粶鍒嗙被鍣╗2]聽銆�
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璇嶆潯鍥剧墖(2)
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鎽勫奖娴嬮噺涓庨仴鎰熷
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绁炵粡缃戠粶
绁炵粡缃戠粶鍙互鎸囧悜涓ょ锛屼竴涓槸鐢熺墿绁炵粡缃戠粶锛屼竴涓槸浜哄伐绁炵粡缃戠粶銆�
鐢熺墿绁炵粡缃戠粶锛氫竴鑸寚鐢熺墿鐨�澶ц剳绁炵粡鍏�锛�缁嗚優锛�瑙︾偣绛夌粍鎴愮殑缃戠粶锛岀敤浜庝骇鐢熺敓鐗╃殑鎰忚瘑锛屽府鍔╃敓鐗╄繘琛�鎬濊��鍜�琛屽姩銆�
浜哄伐绁炵粡缃戠粶锛圓rtificial Neural Networks锛岀畝鍐欎负ANNs锛変篃绠�绉颁负绁炵粡缃戠粶锛圢Ns锛夋垨绉颁綔杩炴帴妯″瀷锛圕onnection Model锛夛紝瀹冩槸涓�绉嶆ā浠垮姩鐗╃缁忕綉缁滆涓虹壒寰侊紝杩涜鍒嗗竷寮忓苟琛屼俊鎭鐞嗙殑绠楁硶鏁板妯″瀷銆傝繖绉嶇綉缁滀緷闈犵郴缁熺殑澶嶆潅绋嬪害锛岄�氳繃璋冩暣鍐呴儴澶ч噺鑺傜偣涔嬮棿鐩镐簰杩炴帴鐨勫叧绯伙紝浠庤�岃揪鍒板鐞嗕俊鎭殑鐩殑銆�
浜哄伐绁炵粡缃戠粶锛氭槸涓�绉嶅簲鐢ㄧ被浼间簬澶ц剳绁炵粡绐佽Е鑱旀帴鐨勭粨鏋勮繘琛屼俊鎭鐞嗙殑鏁板妯″瀷銆傚湪宸ョ▼涓庡鏈晫涔熷父鐩存帴绠�绉颁负鈥滅缁忕綉缁溾�濇垨绫荤缁忕綉缁溿��
涔﹀悕
绁炵粡缃戠粶
浣滆��
渚獩褰紝鏉滀含涔夛紝姹
ISBN
聽9787560619026
椤垫暟
聽223
瀹氫环
锟�26.00
鍑虹増绀�
瑗垮畨鐢靛瓙绉戞妧澶у鍑虹増绀�
鍑虹増鏃堕棿
2007-8-1
绉嶇被
浜哄伐绁炵粡缃戠粶锛岀敓鐗╃缁忕綉缁�
鐩綍
00001.聽1聽鐮旂┒鍐呭
00002.聽鈻�鐢熺墿鍘熷瀷
00003.聽鈻�寤虹珛妯″瀷
00004.聽鈻�绠楁硶
00005.聽鈻�搴旂敤
00006.聽2聽鏈哄櫒瀛︿範鍜岀浉鍏抽鍩燂細
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1銆佸垝鍒嗘柟娉�(partitioning methods)
缁欏畾涓�涓湁N涓厓缁勬垨鑰呯邯褰曠殑鏁版嵁闆嗭紝鍒嗚娉曞皢鏋勯�燢涓垎缁勶紝姣忎竴涓垎缁勫氨浠ｈ〃涓�涓仛绫伙紝K鑱氱被绠楁硶涓彲浠ユ斁瀹斤級锛涘浜庣粰瀹氱殑K锛岀畻娉曢鍏堢粰鍑轰竴涓垵濮嬬殑鍒嗙粍鏂规硶锛屼互鍚庨�氳繃鍙嶅杩唬鐨勬柟娉曟敼鍙樺垎缁勶紝浣垮緱姣忎竴娆℃敼杩涗箣鍚庣殑鍒嗙粍鏂规閮借緝鍓嶄竴娆″ソ锛岃�屾墍璋撳ソ鐨勬爣鍑嗗氨鏄細鍚屼竴鍒嗙粍涓殑璁板綍瓒婅繎瓒婂ソ锛岃�屼笉鍚屽垎缁勪腑鐨勭邯褰曡秺杩滆秺濂姐�備娇鐢ㄨ繖涓熀鏈�濇兂鐨勭畻娉曟湁锛�K-MEANS绠楁硶銆並-MEDOIDS绠楁硶銆丆LARANS绠楁硶锛�
澶ч儴鍒嗗垝鍒嗘柟娉曟槸鍩轰簬璺濈鐨勩�傜粰瀹氳鏋勫缓鐨勫垎鍖烘暟k锛屽垝鍒嗘柟娉曢鍏堝垱寤轰竴涓垵濮嬪寲鍒掑垎銆傜劧鍚庯紝瀹冮噰鐢ㄤ竴绉嶈凯浠ｇ殑閲嶅畾浣嶆妧鏈紝閫氳繃鎶婂璞′粠涓�涓粍绉诲姩鍒板彟涓�涓粍鏉ヨ繘琛屽垝鍒嗐�備竴涓ソ鐨勫垝鍒嗙殑涓�鑸噯澶囨槸锛氬悓涓�涓皣涓殑瀵硅薄灏藉彲鑳界浉浜掓帴杩戞垨鐩稿叧锛岃�屼笉鍚岀殑绨囦腑鐨勫璞″敖鍙兘杩滅鎴栦笉鍚屻�傝繕鏈夎澶氳瘎鍒ゅ垝鍒嗚川閲忕殑鍏朵粬鍑嗗垯銆備紶缁熺殑鍒掑垎鏂规硶鍙互鎵╁睍鍒板瓙绌洪棿鑱氱被锛岃�屼笉鏄悳绱㈡暣涓暟鎹┖闂淬�傚綋瀛樺湪寰堝灞炴�у苟涓旀暟鎹█鐤忔椂锛岃繖鏄湁鐢ㄧ殑銆備负浜嗚揪鍒板叏灞�鏈�浼橈紝鍩轰簬鍒掑垎鐨勮仛绫诲彲鑳介渶瑕佺┓涓炬墍鏈夊彲鑳界殑鍒掑垎锛岃绠楅噺鏋佸ぇ銆傚疄闄呬笂锛屽ぇ澶氭暟搴旂敤閮介噰鐢ㄤ簡娴佽鐨勫惎鍙戝紡鏂规硶锛屽k-鍧囧�煎拰k-涓績绠楁硶锛屾笎杩戠殑鎻愰珮鑱氱被璐ㄩ噺锛岄�艰繎灞�閮ㄦ渶浼樿В銆傝繖浜涘惎鍙戝紡鑱氱被鏂规硶寰堥�傚悎鍙戠幇涓皬瑙勬ā鐨勬暟鎹簱涓皬瑙勬ā鐨勬暟鎹簱涓殑鐞冪姸绨囥�備负浜嗗彂鐜板叿鏈夊鏉傚舰鐘剁殑绨囧拰瀵硅秴澶у瀷鏁版嵁闆嗚繘琛岃仛绫伙紝闇�瑕佽繘涓�姝ユ墿灞曞熀浜庡垝鍒嗙殑鏂规硶銆俒2]
2銆佸眰娆℃柟娉�(hierarchical methods)
杩欑鏂规硶瀵圭粰瀹氱殑鏁版嵁闆嗚繘琛屽眰娆′技鐨勫垎瑙ｏ紝鐩村埌鏌愮鏉′欢婊¤冻涓烘銆傚叿浣撳張鍙垎涓衡�滆嚜搴曞悜涓娾�濆拰鈥滆嚜椤跺悜涓嬧�濅袱绉嶆柟妗堛�備緥濡傚湪鈥滆嚜搴曞悜涓娾�濇柟妗堜腑锛屽垵濮嬫椂姣忎竴涓暟鎹邯褰曢兘缁勬垚涓�涓崟鐙殑缁勶紝鍦ㄦ帴涓嬫潵鐨勮凯浠ｄ腑锛屽畠鎶婇偅浜涚浉浜掗偦杩戠殑缁勫悎骞舵垚涓�涓粍锛岀洿鍒版墍鏈夌殑璁板綍缁勬垚涓�涓垎缁勬垨鑰呮煇涓潯浠舵弧瓒充负姝€�備唬琛ㄧ畻娉曟湁锛欱IRCH绠楁硶銆丆URE绠楁硶銆丆HAMELEON绠楁硶绛夛紱
灞傛鑱氱被鏂规硶鍙互鏄熀浜庤窛绂荤殑鎴栧熀浜庡瘑搴︽垨杩為�氭�х殑銆傚眰娆¤仛绫绘柟娉曠殑涓�浜涙墿灞曚篃鑰冭檻浜嗗瓙绌洪棿鑱氱被銆傚眰娆℃柟娉曠殑缂洪櫡鍦ㄤ簬锛屼竴鏃︿竴涓楠わ紙鍚堝苟鎴栧垎瑁傦級瀹屾垚锛屽畠灏变笉鑳借鎾ら攢銆傝繖涓弗鏍艰瀹氭槸鏈夌敤鐨勶紝鍥犱负涓嶇敤鎷呭績涓嶅悓閫夋嫨鐨勭粍鍚堟暟鐩紝瀹冨皢浜х敓杈冨皬鐨勮绠楀紑閿�銆傜劧鑰岃繖绉嶆妧鏈笉鑳芥洿姝ｉ敊璇殑鍐冲畾銆傚凡缁忔彁鍑轰簡涓�浜涙彁楂樺眰娆¤仛绫昏川閲忕殑鏂规硶銆偮燵2]聽
3銆佸熀浜庡瘑搴︾殑鏂规硶(density-based methods)
鍩轰簬瀵嗗害鐨勬柟娉曚笌鍏跺畠鏂规硶鐨勪竴涓牴鏈尯鍒槸锛氬畠涓嶆槸鍩轰簬鍚勭鍚勬牱鐨勮窛绂荤殑锛岃�屾槸鍩轰簬瀵嗗害鐨勩�傝繖鏍峰氨鑳藉厠鏈嶅熀浜庤窛绂荤殑绠楁硶鍙兘鍙戠幇鈥滅被鍦嗗舰鈥濈殑鑱氱被鐨勭己鐐广�傝繖涓柟娉曠殑鎸囧鎬濇兂灏辨槸锛屽彧瑕佷竴涓尯鍩熶腑鐨勭偣鐨勫瘑搴﹀ぇ杩囨煇涓�闃�鍊�锛屽氨鎶婂畠鍔犲埌涓庝箣鐩歌繎鐨勮仛绫讳腑鍘汇�備唬琛ㄧ畻娉曟湁锛欴BSCAN绠楁硶銆丱PTICS绠楁硶銆丏ENCLUE绠楁硶绛夛紱
4銆佸熀浜庣綉鏍肩殑鏂规硶(grid-based methods)
杩欑鏂规硶棣栧厛灏嗘暟鎹┖闂村垝鍒嗘垚涓烘湁闄愪釜鍗曞厓锛坈ell锛夌殑缃戞牸缁撴瀯,鎵�鏈夌殑澶勭悊閮芥槸浠ュ崟涓殑鍗曞厓涓哄璞＄殑銆傝繖涔堝鐞嗙殑涓�涓獊鍑虹殑浼樼偣灏辨槸澶勭悊閫熷害寰堝揩锛岄�氬父杩欐槸涓庣洰鏍囨暟鎹簱涓褰曠殑涓暟鏃犲叧鐨勶紝瀹冨彧涓庢妸鏁版嵁绌洪棿鍒嗕负澶氬皯涓崟鍏冩湁鍏炽�備唬琛ㄧ畻娉曟湁锛歋TING绠楁硶銆丆LIQUE绠楁硶銆乄AVE-CLUSTER绠楁硶锛�
寰堝绌洪棿鏁版嵁鎸栨帢闂锛屼娇鐢ㄧ綉鏍奸�氬父閮芥槸涓�绉嶆湁鏁堢殑鏂规硶銆傚洜姝わ紝鍩轰簬缃戞牸鐨勬柟娉曞彲浠ュ拰鍏朵粬鑱氱被鏂规硶闆嗘垚銆偮燵2]聽
5銆佸熀浜庢ā鍨嬬殑鏂规硶(model-based methods)
鍩轰簬妯″瀷鐨勬柟娉曠粰姣忎竴涓仛绫诲亣瀹氫竴涓ā鍨嬶紝鐒跺悗鍘诲鎵捐兘澶熷緢濂界殑婊¤冻杩欎釜妯″瀷鐨勬暟鎹泦銆傝繖鏍蜂竴涓ā鍨嬪彲鑳芥槸鏁版嵁鐐瑰湪绌洪棿涓殑瀵嗗害鍒嗗竷鍑芥暟鎴栬�呭叾瀹冦�傚畠鐨勪竴涓綔鍦ㄧ殑鍋囧畾灏辨槸锛氱洰鏍囨暟鎹泦鏄敱涓�绯诲垪鐨�姒傜巼鍒嗗竷鎵�鍐冲畾鐨勩�傞�氬父鏈変袱绉嶅皾璇曟柟鍚戯細缁熻鐨勬柟妗堝拰绁炵粡缃戠粶鐨勬柟妗堛��
褰撶劧鑱氱被鏂规硶杩樻湁锛�浼犻�掗棴鍖�娉曪紝甯冨皵鐭╅樀娉曪紝鐩存帴鑱氱被娉曪紝鐩稿叧鎬у垎鏋�鑱氱被锛屽熀浜庣粺璁＄殑鑱氱被鏂规硶绛夈��
鐮旂┒鎯呭喌
浼犵粺鐨勮仛绫诲凡缁忔瘮杈冩垚鍔熺殑瑙ｅ喅浜嗕綆缁存暟鎹殑鑱氱被闂銆備絾鏄敱浜庡疄闄呭簲鐢ㄤ腑鏁版嵁鐨勫鏉傛�э紝鍦ㄥ鐞嗚澶氶棶棰樻椂锛岀幇鏈夌殑绠楁硶缁忓父澶辨晥锛岀壒鍒槸瀵逛簬楂樼淮鏁版嵁鍜屽ぇ鍨嬫暟鎹殑鎯呭喌銆傚洜涓轰紶缁熻仛绫绘柟娉曞湪楂樼淮鏁版嵁闆嗕腑杩涜鑱氱被鏃讹紝涓昏閬囧埌涓や釜闂銆傗憼楂樼淮鏁版嵁闆嗕腑瀛樺湪澶ч噺鏃犲叧鐨勫睘鎬т娇寰楀湪鎵�鏈夌淮涓瓨鍦ㄧ皣鐨勫彲鑳芥�у嚑涔庝负闆讹紱鈶�楂樼淮绌洪棿涓暟鎹緝浣庣淮绌洪棿涓暟鎹垎甯冭绋�鐤忥紝鍏朵腑鏁版嵁闂磋窛绂诲嚑涔庣浉绛夋槸鏅亶鐜拌薄锛岃�屼紶缁熻仛绫绘柟娉曟槸鍩轰簬璺濈杩涜鑱氱被鐨勶紝鍥犳鍦ㄩ珮缁寸┖闂翠腑鏃犳硶鍩轰簬璺濈鏉ユ瀯寤虹皣銆�
楂樼淮鑱氱被鍒嗘瀽宸叉垚涓鸿仛绫诲垎鏋愮殑涓�涓噸瑕佺爺绌舵柟鍚戙�傚悓鏃�楂樼淮鏁版嵁鑱氱被涔熸槸鑱氱被鎶�鏈殑闅剧偣銆傞殢鐫�鎶�鏈殑杩涙浣垮緱鏁版嵁鏀堕泦鍙樺緱瓒婃潵瓒婂鏄擄紝瀵艰嚧鏁版嵁搴撹妯¤秺鏉ヨ秺澶с�佸鏉傛�ц秺鏉ヨ秺楂橈紝濡傚悇绉嶇被鍨嬬殑璐告槗浜ゆ槗鏁版嵁銆乄eb聽鏂囨。銆佸熀鍥犺〃杈炬暟鎹瓑锛屽畠浠殑缁村害锛�灞炴��锛夐�氬父鍙互杈惧埌鎴愮櫨涓婂崈缁达紝鐢氳嚦鏇撮珮銆備絾鏄紝鍙椻�滅淮搴︽晥搴斺�濈殑褰卞搷锛岃澶氬湪浣庣淮鏁版嵁绌洪棿琛ㄧ幇鑹ソ鐨勮仛绫绘柟娉曡繍鐢ㄥ湪楂樼淮绌洪棿涓婂線寰�鏃犳硶鑾峰緱濂界殑鑱氱被鏁堟灉銆�楂樼淮鏁版嵁鑱氱被鍒嗘瀽鏄仛绫诲垎鏋愪腑涓�涓潪甯告椿璺冪殑棰嗗煙锛屽悓鏃跺畠涔熸槸涓�涓叿鏈夋寫鎴樻�х殑宸ヤ綔銆�楂樼淮鏁版嵁鑱氱被鍒嗘瀽鍦ㄥ競鍦哄垎鏋愩�佷俊鎭畨鍏ㄣ�侀噾铻嶃�佸ū涔愩�佸弽鎭愮瓑鏂归潰閮芥湁寰堝箍娉涚殑搴旂敤銆�
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路聽1.鏁版嵁鑱氱被锛庣淮鍩虹櫨绉慬寮曠敤鏃ユ湡2013-04-20]
路聽2.Jiawei Han锛庢暟鎹寲鎺樻蹇典笌鎶�鏈細鏈烘宸ヤ笟鍑虹増绀撅紝2012
鏃堕棿搴忓垪
鏃堕棿搴忓垪
聽缂栬緫
鏈瘝鏉＄敱鈥滅鏅腑鍥解�濈櫨绉戠瀛﹁瘝鏉＄紪鍐欎笌搴旂敤宸ヤ綔椤圭洰瀹℃牳 銆�
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涓枃鍚�
鏃堕棿搴忓垪
澶栨枃鍚�
time series
绫诲瀷
搴忓垪
鏋勬垚瑕佺礌
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鎻忚堪鐜拌薄鐨勫彂灞曠姸鎬佸拰缁撴灉
鐩綍
00001.聽1聽鏃堕棿搴忓垪
00002.聽2聽浣滅敤
00003.聽3聽绉嶇被
00001.聽4聽鐗瑰緛
00002.聽5聽缂栧埗鍘熷垯
00003.聽6聽鍙橀噺鐗瑰緛
00001.聽7聽鍒嗘瀽鏂规硶
00002.聽8聽鍒嗘瀽妯″瀷
00003.聽9聽棰勬祴
鏃堕棿搴忓垪
缂栬緫
鏋勬垚瑕佺礌锛氶暱鏈熻秼鍔匡紝瀛ｈ妭鍙樺姩锛屽惊鐜彉鍔紝涓嶈鍒欏彉鍔�
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浣滅敤
缂栬緫
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缂栬緫
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鏃剁偣搴忓垪鐨勪富瑕佺壒鐐规湁锛�
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(浜�)鐩稿鏁版椂闂村簭鍒�
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鐗瑰緛
缂栬緫
1銆佹椂闂村簭鍒楀垎鏋愭硶鏄牴鎹繃鍘荤殑鍙樺寲瓒嬪娍棰勬祴鏈潵鐨勫彂灞�,瀹冪殑鍓嶆彁鏄亣瀹氫簨鐗╃殑杩囧幓寤剁画鍒版湭鏉ャ��
鏃堕棿搴忓垪鍒嗘瀽锛屾鏄牴鎹瑙備簨鐗╁彂灞曠殑杩炵画瑙勫緥鎬�,杩愮敤杩囧幓鐨勫巻鍙叉暟鎹�,閫氳繃缁熻鍒嗘瀽,杩涗竴姝ユ帹娴嬫湭鏉ョ殑鍙戝睍瓒嬪娍銆備簨鐗╃殑杩囧幓浼氬欢缁埌鏈潵杩欎釜鍋囪鍓嶆彁鍖呭惈涓ゅ眰鍚箟:涓�鏄笉浼氬彂鐢熺獊鐒剁殑璺宠穬鍙樺寲,鏄互鐩稿灏忕殑姝ヤ紣鍓嶈繘;浜屾槸杩囧幓鍜屽綋鍓嶇殑鐜拌薄鍙兘琛ㄦ槑鐜板湪鍜屽皢鏉ユ椿鍔ㄧ殑鍙戝睍鍙樺寲瓒嬪悜銆傝繖灏卞喅瀹氫簡鍦ㄤ竴鑸儏鍐典笅,鏃堕棿搴忓垪鍒嗘瀽娉曞浜庣煭銆佽繎鏈熼娴嬫瘮杈冩樉钁�,浣嗗寤朵几鍒版洿杩滅殑灏嗘潵,灏变細鍑虹幇寰堝ぇ鐨勫眬闄愭��,瀵艰嚧棰勬祴鍊煎亸绂诲疄闄呰緝澶ц�屼娇鍐崇瓥澶辫銆�
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(1)瓒嬪娍鎬�:鏌愪釜鍙橀噺闅忕潃鏃堕棿杩涘睍鎴栬嚜鍙橀噺鍙樺寲,鍛堢幇涓�绉嶆瘮杈冪紦鎱㈣�岄暱鏈熺殑鎸佺画涓婂崌銆佷笅闄嶃�佸仠鐣欑殑鍚屾�ц川鍙樺姩瓒嬪悜,浣嗗彉鍔ㄥ箙搴﹀彲鑳戒笉鐩哥瓑銆�
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缂栧埗鍘熷垯
缂栬緫
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(涓�)鏃舵湡闀跨煭鏈�濂戒竴鑷�
(浜�)鎬讳綋鑼冨洿搴旇涓�鑷�
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(鍥�)璁＄畻鏂规硶搴旇缁熶竴
(浜�)璁＄畻浠锋牸鍜岃閲忓崟浣嶅彲姣�
鍙橀噺鐗瑰緛
缂栬緫
闈炲钩绋虫�э紙nonstationarity锛屼篃璇戜綔涓嶅钩绋虫�э紝闈炵ǔ瀹氭�э級锛氬嵆鏃堕棿搴忓垪鍙橀噺鏃犳硶鍛堢幇鍑轰竴涓暱鏈熻秼鍔垮苟鏈�缁堣秼浜庝竴涓父鏁版垨鏄竴涓嚎鎬у嚱鏁般��
娉㈠姩骞呭害闅忔椂闂村彉鍖栵紙Time锛峷arying Volatility锛夛細鍗充竴涓椂闂村簭鍒楀彉閲忕殑鏂瑰樊闅忔椂闂寸殑鍙樺寲鑰屽彉鍖栬繖涓や釜鐗瑰緛浣垮緱鏈夋晥鍒嗘瀽鏃堕棿搴忓垪鍙橀噺鍗佸垎鍥伴毦銆�
骞崇ǔ鍨嬫椂闂存暟鍒楋紙Stationary Time Series锛夌郴鎸囦竴涓椂闂存暟鍒楀叾缁熻鐗规�у皢涓嶉殢鏃堕棿涔嬪彉鍖栬�屾敼鍙樿�呫��
鍒嗘瀽鏂规硶
缂栬緫
锛堜竴锛夋寚鏍囧垎鏋愭硶
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锛堜簩锛夋瀯鎴愬洜绱犲垎鏋愭硶
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鍒嗘瀽妯″瀷
缂栬緫
鏃堕棿鏁板垪鐨勭粍鍚堟ā鍨�
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2聽涔樻硶妯″瀷锛歒=T路S路C路I(甯哥敤妯″瀷) (Y,T聽璁￠噺鍗曚綅鐩稿悓鐨勬�婚噺鎸囨爣)(S,C,I聽瀵瑰師鏁板垪鎸囨爣澧炲姞鎴栧噺灏戠殑鐧惧垎姣�)
棰勬祴
缂栬緫
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鏃堕棿搴忓垪棰勬祴灏辨槸鍒╃敤缁熻鎶�鏈笌鏂规硶锛屼粠棰勬祴鎸囨爣鐨勬椂闂村簭鍒椾腑鎵惧嚭婕斿彉妯″紡锛屽缓绔嬫暟瀛︽ā鍨嬶紝瀵归娴嬫寚鏍囩殑鏈潵鍙戝睍瓒嬪娍鍋氬嚭瀹氶噺浼拌[1]聽銆�
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路聽1.杈涙不杩�,椤炬槑.鍩轰簬鏈�灏忎簩涔樻敮鎸佸悜閲忔満鐨勫鏉傞噾铻嶆椂闂村簭鍒楅娴媅J].娓呭崕澶у瀛︽姤锛堣嚜鐒剁瀛︾増锛�,2008,48(7):1147-1149.鈥�锛庝竾鏂规暟鎹簱[寮曠敤鏃ユ湡2017-09-09]
璇嶆潯鏍囩锛�
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鎺ㄨ崘绯荤粺
鎺ㄨ崘绯荤粺
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鎺ㄨ崘绯荤粺鏄埄鐢╗1]聽鐢靛瓙鍟嗗姟缃戠珯鍚戝鎴锋彁渚涘晢鍝佷俊鎭拰寤鸿锛屽府鍔╃敤鎴峰喅瀹氬簲璇ヨ喘涔颁粈涔堜骇鍝侊紝妯℃嫙閿�鍞汉鍛樺府鍔╁鎴峰畬鎴愯喘涔拌繃绋嬨�備釜鎬у寲鎺ㄨ崘鏄牴鎹敤鎴风殑鍏磋叮鐗圭偣鍜岃喘涔拌涓猴紝鍚戠敤鎴锋帹鑽愮敤鎴锋劅鍏磋叮鐨勪俊鎭拰鍟嗗搧銆�
闅忕潃鐢靛瓙鍟嗗姟瑙勬ā鐨勪笉鏂墿澶э紝鍟嗗搧涓暟鍜岀绫诲揩閫熷闀匡紝椤惧闇�瑕佽姳璐瑰ぇ閲忕殑鏃堕棿鎵嶈兘鎵惧埌鑷繁鎯充拱鐨勫晢鍝併�傝繖绉嶆祻瑙堝ぇ閲忔棤鍏崇殑淇℃伅鍜屼骇鍝佽繃绋嬫棤鐤戜細浣挎饭娌″湪淇℃伅杩囪浇闂涓殑娑堣垂鑰呬笉鏂祦澶便��
涓轰簡瑙ｅ喅杩欎簺闂锛�涓�у寲鎺ㄨ崘绯荤粺搴旇繍鑰岀敓銆備釜鎬у寲鎺ㄨ崘绯荤粺鏄缓绔嬪湪娴烽噺鏁版嵁鎸栨帢鍩虹涓婄殑涓�绉嶉珮绾у晢鍔℃櫤鑳藉钩鍙帮紝浠ュ府鍔╃數瀛愬晢鍔＄綉绔欎负鍏堕【瀹㈣喘鐗╂彁渚涘畬鍏ㄤ釜鎬у寲鐨勫喅绛栨敮鎸佸拰淇℃伅鏈嶅姟銆�
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Recommender system
搴旂敤
鐢靛瓙鍟嗗姟
瀹氫箟
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路聽3.鎺ㄨ崘绯荤粺锛庣煡缃慬寮曠敤鏃ユ湡2017-04-01]
路聽4.淇℃伅杩囨护锛庣煡缃慬寮曠敤鏃ユ湡2017-04-01]
鍥炲綊鍒嗘瀽
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1. Linear Regression绾挎�у洖褰�
瀹冩槸鏈�涓轰汉鐔熺煡鐨勫缓妯℃妧鏈箣涓�銆傜嚎鎬у洖褰掗�氬父鏄汉浠湪瀛︿範棰勬祴妯″瀷鏃堕閫夌殑鎶�鏈箣涓�銆傚湪杩欑鎶�鏈腑锛屽洜鍙橀噺鏄繛缁殑锛岃嚜鍙橀噺鍙互鏄繛缁殑涔熷彲浠ユ槸绂绘暎鐨勶紝鍥炲綊绾跨殑鎬ц川鏄嚎鎬х殑銆�
绾挎�у洖褰掍娇鐢ㄦ渶浣崇殑鎷熷悎鐩寸嚎锛堜篃灏辨槸鍥炲綊绾匡級鍦ㄥ洜鍙橀噺锛圷锛夊拰涓�涓垨澶氫釜鑷彉閲忥紙X锛変箣闂村缓绔嬩竴绉嶅叧绯汇��
澶氬厓绾挎�у洖褰掑彲琛ㄧず涓篩=a+b1*X +b2*X2+ e锛屽叾涓璦琛ㄧず鎴窛锛宐琛ㄧず鐩寸嚎鐨勬枩鐜囷紝e鏄宸」銆傚鍏冪嚎鎬у洖褰掑彲浠ユ牴鎹粰瀹氱殑棰勬祴鍙橀噺锛坰锛夋潵棰勬祴鐩爣鍙橀噺鐨勫�笺��
2.Logistic Regression閫昏緫鍥炲綊
閫昏緫鍥炲綊鏄敤鏉ヨ绠椻�滀簨浠�=Success鈥濆拰鈥滀簨浠�=Failure鈥濈殑姒傜巼銆傚綋鍥犲彉閲忕殑绫诲瀷灞炰簬浜屽厓锛�1 / 0锛岀湡/鍋囷紝鏄�/鍚︼級鍙橀噺鏃讹紝鎴戜滑灏卞簲璇ヤ娇鐢ㄩ�昏緫鍥炲綊銆傝繖閲岋紝Y鐨勫�间负0鎴�1锛屽畠鍙互鐢ㄤ笅鏂圭▼琛ㄧず銆�
odds= p/ (1-p) = probability of event occurrence / probability of not event occurrence
ln(odds) = ln(p/(1-p))
logit(p) = ln(p/(1-p)) =b0+b1X1+b2X2+b3X3....+bkXk
涓婅堪寮忓瓙涓紝p琛ㄨ堪鍏锋湁鏌愪釜鐗瑰緛鐨勬鐜囥�備綘搴旇浼氶棶杩欐牱涓�涓棶棰橈細鈥滄垜浠负浠�涔堣鍦ㄥ叕寮忎腑浣跨敤瀵规暟log鍛紵鈥濄��
鍥犱负鍦ㄨ繖閲屾垜浠娇鐢ㄧ殑鏄殑浜岄」鍒嗗竷锛堝洜鍙橀噺锛夛紝鎴戜滑闇�瑕侀�夋嫨涓�涓浜庤繖涓垎甯冩渶浣崇殑杩炵粨鍑芥暟銆傚畠灏辨槸Logit鍑芥暟銆傚湪涓婅堪鏂圭▼涓紝閫氳繃瑙傛祴鏍锋湰鐨勬瀬澶т技鐒朵及璁″�兼潵閫夋嫨鍙傛暟锛岃�屼笉鏄渶灏忓寲骞虫柟鍜岃宸紙濡傚湪鏅�氬洖褰掍娇鐢ㄧ殑锛夈��
3. Polynomial Regression澶氶」寮忓洖褰�
瀵逛簬涓�涓洖褰掓柟绋嬶紝濡傛灉鑷彉閲忕殑鎸囨暟澶т簬1锛岄偅涔堝畠灏辨槸澶氶」寮忓洖褰掓柟绋嬨�傚涓嬫柟绋嬫墍绀猴細
y=a+b*x^2
鍦ㄨ繖绉嶅洖褰掓妧鏈腑锛屾渶浣虫嫙鍚堢嚎涓嶆槸鐩寸嚎銆傝�屾槸涓�涓敤浜庢嫙鍚堟暟鎹偣鐨勬洸绾裤��
4. Stepwise Regression閫愭鍥炲綊
鍦ㄥ鐞嗗涓嚜鍙橀噺鏃讹紝鎴戜滑鍙互浣跨敤杩欑褰㈠紡鐨勫洖褰掋�傚湪杩欑鎶�鏈腑锛岃嚜鍙橀噺鐨勯�夋嫨鏄湪涓�涓嚜鍔ㄧ殑杩囩▼涓畬鎴愮殑锛屽叾涓寘鎷潪浜轰负鎿嶄綔銆�
杩欎竴澹妇鏄�氳繃瑙傚療缁熻鐨勫�硷紝濡俁-square锛宼-stats鍜孉IC鎸囨爣锛屾潵璇嗗埆閲嶈鐨勫彉閲忋�傞�愭鍥炲綊閫氳繃鍚屾椂娣诲姞/鍒犻櫎鍩轰簬鎸囧畾鏍囧噯鐨勫崗鍙橀噺鏉ユ嫙鍚堟ā鍨嬨�備笅闈㈠垪鍑轰簡涓�浜涙渶甯哥敤鐨勯�愭鍥炲綊鏂规硶锛�
鏍囧噯閫愭鍥炲綊娉曞仛涓や欢浜嬫儏銆傚嵆澧炲姞鍜屽垹闄ゆ瘡涓楠ゆ墍闇�鐨勯娴嬨��
鍚戝墠閫夋嫨娉曚粠妯″瀷涓渶鏄捐憲鐨勯娴嬪紑濮嬶紝鐒跺悗涓烘瘡涓�姝ユ坊鍔犲彉閲忋��
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杩欑寤烘ā鎶�鏈殑鐩殑鏄娇鐢ㄦ渶灏戠殑棰勬祴鍙橀噺鏁版潵鏈�澶у寲棰勬祴鑳藉姏銆傝繖涔熸槸澶勭悊楂樼淮鏁版嵁闆嗙殑鏂规硶涔嬩竴銆俒2]
5. Ridge Regression宀洖褰�
褰撴暟鎹箣闂村瓨鍦ㄥ閲嶅叡绾挎�э紙鑷彉閲忛珮搴︾浉鍏筹級鏃讹紝灏遍渶瑕佷娇鐢ㄥ箔鍥炲綊鍒嗘瀽銆傚湪瀛樺湪澶氶噸鍏辩嚎鎬ф椂锛屽敖绠℃渶灏忎簩涔樻硶锛圤LS锛夋祴寰楃殑浼拌鍊间笉瀛樺湪鍋忓樊锛屽畠浠殑鏂瑰樊涔熶細寰堝ぇ锛屼粠鑰屼娇寰楄娴嬪�间笌鐪熷疄鍊肩浉宸敋杩溿�傚箔鍥炲綊閫氳繃缁欏洖褰掍及璁″�兼坊鍔犱竴涓亸宸�硷紝鏉ラ檷浣庢爣鍑嗚宸��
鍦ㄧ嚎鎬х瓑寮忎腑锛岄娴嬭宸彲浠ュ垝鍒嗕负聽2聽涓垎閲忥紝涓�涓槸鍋忓樊閫犳垚鐨勶紝涓�涓槸鏂瑰樊閫犳垚鐨勩�傞娴嬭宸彲鑳戒細鐢辫繖涓よ�呮垨涓よ�呬腑鐨勪换浣曚竴涓�犳垚銆傚湪杩欓噷锛屾垜浠皢璁ㄨ鐢辨柟宸墍閫犳垚鐨勮宸�偮燵2]聽
宀洖褰掗�氳繃鏀剁缉鍙傛暟位锛坙ambda锛夎В鍐冲閲嶅叡绾挎�ч棶棰樸�傝鐪嬩笅闈㈢殑绛夊紡锛�
L2=argmin||y=x尾||
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鍦ㄨ繖涓叕寮忎腑锛屾湁涓や釜缁勬垚閮ㄥ垎銆傜涓�涓槸鏈�灏忎簩涔橀」锛屽彟涓�涓槸尾-骞虫柟鐨勎诲�嶏紝鍏朵腑尾鏄浉鍏崇郴鏁板悜閲忥紝涓庢敹缂╁弬鏁颁竴璧锋坊鍔犲埌鏈�灏忎簩涔橀」涓互寰楀埌涓�涓潪甯镐綆鐨勬柟宸��
6. Lasso Regression濂楃储鍥炲綊
瀹冪被浼间簬宀洖褰掞紝Lasso聽锛圠east Absolute Shrinkage and Selection Operator锛変篃浼氬氨鍥炲綊绯绘暟鍚戦噺缁欏嚭鎯╃綒鍊奸」銆傛澶栵紝瀹冭兘澶熷噺灏戝彉鍖栫▼搴﹀苟鎻愰珮绾挎�у洖褰掓ā鍨嬬殑绮惧害銆傜湅鐪嬩笅闈㈢殑鍏紡锛�
L1=agrmin||y-x尾||
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Lasso聽鍥炲綊涓嶳idge鍥炲綊鏈変竴鐐逛笉鍚岋紝瀹冧娇鐢ㄧ殑鎯╃綒鍑芥暟鏄疞1鑼冩暟锛岃�屼笉鏄疞2鑼冩暟銆傝繖瀵艰嚧鎯╃綒锛堟垨绛変簬绾︽潫浼拌鐨勭粷瀵瑰�间箣鍜岋級鍊间娇涓�浜涘弬鏁颁及璁＄粨鏋滅瓑浜庨浂銆備娇鐢ㄦ儵缃氬�艰秺澶э紝杩涗竴姝ヤ及璁′細浣垮緱缂╁皬鍊艰秺瓒嬭繎浜庨浂銆傝繖灏嗗鑷存垜浠浠庣粰瀹氱殑n涓彉閲忎腑閫夋嫨鍙橀噺銆�
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浼樼偣
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绠楁硶
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