pascal 基本算法

基本算法模块对于NOIP，基础是相当重要的，在3个小时之内做完4道题，那么就要求我们有相当快的速度。

特别是对于一些简单的、常用的算法模块，一定要要熟练掌握并灵活运用。

由于NOIP是一个比较基础的比赛，因此基本算法的掌握尤为重要，所以要求能够把这些基本的模块快速、准确的移植到不同的程序中，才能在稳中取胜。

基本算法模块中最重要的是基本程序框架，也就是说，要养成适合于自己的程序风格，这样对于程序编写的速度与程序的准确度都有较大的提高。

模块目录一、排序1．选择排序2．插入排序3．冒泡排序4．快速排序5．堆排序6．归并排序7．线性时间排序二、高精度1．高精度比较2．高精度加法3．高精度减法4．单精度乘法5．高精度乘法6．单精度除法7．高精度除法8．进制转换三、数论1．欧几里德算法2．扩展欧几里德3．求最小公倍数4．求解线形同余方程5．素数的判断6．素数的生成四、排列组合1．排列生成算法2．组合生成算法3．排列按序生成法4．排列字典序生成法五、图论1．图的读入2．深度优先搜索3．广度优先搜索4．强连同分量5．拓扑排序6．最小生成树7．最短路径六、背包问题1．装满背包2．一维价值最大背包3．二位价值最大背包一、排序算法vara:array[1..maxn]of longint;——排序对象1．选择排序——Select_sortprocedure select_sort;beginfor i:=1to n-1dofor j:=i+1to n doif a[i]>a[j]thenbegin temp:=a[i];a[i]:=a[j];a[j]:=temp;end;end;2．插入排序——Insert_sortprocedure insert_sort;beginfor i:=2to n dobeginkey:=a[i];j:=i-1;while(key<a[j])and(j>0)do begin a[j+1]:=a[j];dec(j);end;a[j+1]:=key;end;end;3．冒泡排序——Bubble_sortprocedure bubble_sort;beginfor i:=1to n-1dofor j:=n downto i+1doif a[j]<a[j-1]thenbegin temp:=a[j];a[j]:=a[j-1];a[j-1]:=temp;end;end;4．快速排序——Quick_sortprocedure qsort(s,t:longint);vari,j,x:longint;begini:=s;j:=t;x:=a[(i+j)div2];repeatwhile a[i]<x do inc(i);{找左边比他大的}while a[j]>x do dec(j);{找右边比他小的}if i<=j then{交换}begintemp:=a[i];a[i]:=a[j];a[j]:=temp;inc(i);dec(j);end;until i>j;if s<j then qsort(s,j);if i<t then qsort(i,t);end;5．堆排序——Heap_sortprocedure heap(i,n:longint);{将第i个元素向下筛}varj,x:longint;beginj:=i*2;x:=a[i];while j<=n dobeginif(j<n)and(a[j]<a[j+1])then inc(j);if x<a[j]then begina[i]:=a[j];i:=j;j:=i*2;endelse j:=n+1;end;a[i]:=x;end;procedure heap_sort;beginfor i:=n div2downto1do heap(i,n);for i:=n downto2dobegintemp:=a[i];a[i]:=a[1];a[1]:=temp;heap(1,i-1);end;end;6．归并排序——Merge_sortprocedure mergesort(s,t:longint);varm,i,j,k:longint;beginif t-s=0then exit;m:=(s+t)div2;mergesort(s,m);mergesort(m+1,t);for i:=1to m-s+1do b[i]:=a[s+i-1];for j:=m+1to t do c[j-m]:=a[j];i:=1;j:=1;b[m-s+2]:=max;c[t-m+1]:=max;for k:=s to t doif b[i]<c[j]then begin a[k]:=b[i];inc(i);endelse begin a[k]:=c[j];inc(j);end;end;7．线性时间排序——基数排序、计数排序、桶排序二、高精度算法——High_precisionconstmaxcount=进制位maxlen=记录高精度数组大小typebignum=array[0..maxlen]of longint;0为位数1．高精度比较function compare(a,b:bignum):longint;beginwhile a[a[0]]=0do dec(a[0]);{检查位数是否正确}while b[b[0]]=0do dec(b[0]);while a[a[0]+1]>0do inc(a[0]);while b[b[0]+1]>0do inc(b[0]);if a[0]>b[0]then exit(1);if a[0]<b[0]then exit(-1);for i:=a[0]downto1dobeginif a[i]>b[i]then exit(1);if a[i]<b[i]then exit(-1);end;exit(0);end;2．高精度加法procedure add(a,b:bignum;var c:bignum);vari:longint;beginfillchar(c,sizeof(c),0);c[0]:=1;if a[0]>b[0]then c[0]:=a[0]else c[0]:=b[0];for i:=1to a[0]do inc(c[i],a[i]);for i:=1to b[0]do inc(c[i],b[i]);for i:=1to c[0]dobegininc(c[i+1],c[i]div maxcount);c[i]:=c[i]mod10;end;while c[c[0]+1]>0dobegininc(c[0]);inc(c[c[0]+1],c[c[0]]div maxcount);c[c[0]]:=c[c[0]]mod maxcount;end;end;3．高精度减法procedure minus(a,b:bignum;var c:bignum);vari:longint;beginfillchar(c,sizeof(c),0);c[0]:=a[0];for i:=1to c[0]do c[i]:=a[i]-b[i];for i:=1to c[0]doif c[i]<0thenbegindec(c[i+1]);inc(c[i],maxcount);end;while(c[0]>1)and(c[c[0]]=0)do dec(c[0]);end;4．单精度乘法procedure mulnum(a:bignum;x:longint,var c:bignum);vari:longint;beginfillchar(c,sizeof(c),0);c[0]:=a[0];for i:=1to c[0]do c[i]:=a[i]*x;for i:=1to c[0]dobegininc(c[i+1],c[i]div maxcount);c[i]:=c[i]mod10;end;while c[c[0]+1]>0dobegininc(c[0]);inc(c[c[0]+1],c[c[0]]div maxcount);c[c[0]]:=c[c[0]]mod maxcount;end;end;5．高精度乘法procedure mul(a,b:bignum;var c:bignum);vari,j:longint;beginfillchar(c,sizeof(c),0);c[0]:=a[0]+b[0]-1;for i:=1to a[0]dofor j:=1to b[0]doinc(c[i+j-1],a[i]*b[j]);for i:=1to c[0]dobegininc(c[i+1],c[i]div maxcount);c[i]:=c[i]mod10;end;while c[c[0]+1]>0dobegininc(c[0]);inc(c[c[0]+1],c[c[0]]div maxcount);c[c[0]]:=c[c[0]]mod maxcount;end;end;6．单精度除法function divnum(a:bignum;x:longint;var c:bignum):longint;vari,temp:longint;beginfillchar(c,sizeof(c),0);c[0]:=a[0];temp:=0;for i:=a[0]downto1dobegintemp:=temp*maxcount+a[i];c[i]:=temp div x;temp:=temp mod x;end;while(c[o]>1)and(c[c[0]]=0)do dec(c[0]);exit(temp);end;7．高精度除法procedure div(a,b:bignum;var c,d:bignum);vari:longint;beginfillchar(c,sizeof(c),0);c[0]:=a[0]-b[0]+1;fillchar(d,sizeof(d),0);d[0]:=1;for i:=c[0]downto1dobeginc[i]:=maxcount;repeatdec(c[i]);mul(c,b,temp);until compare(a,temp)>=0;end;while(c[o]>1)and(c[c[0]]=0)do dec(c[0]);minus(a,temp,d);end;8．进制转换10进制数用bignum记，maxcount=10k进制数用string记constrepchar:array[0..35]ofstring=(‘0’,‘1’,’2’,……,’a’,’b’,……,’z’);——数码对应的字符repnum:array[48..122]of longint=(0,1,2……,34,35);——字符的ASCCI码对应的数码k进制转十进制：procedure change_to_ten(s:string;k:longint):bignum;vari,l:longint;temp:bignum;beginl:=length(s);temp[0]:=1;temp[1]:=repnum[ord(s[l])];for i:=1to l-1dobegininc(temp[1],repnum[ord(s[l-i])]);mulnum(temp,k);end;exit(temp);end;十进制转k进制：procedure change_to_k(num:bignum;k:longint):string;vari,temp:longint;s:string;beginif(num[0]=1)and(num[1]=0)then exit(‘0’);while not((num[0]=1)and(num[1]=0))dobegintemp:=divnum(num,k,num);s:=repchar[temp]+s;end;exit(s);end;三、数论算法1．求最大公约数——gcd（欧几里德算法）递归(recursion)：function gcd(a,b:longint):longint;beginif b=0then exit(a);exit(gcd(b,a mod b));end;非递归(iterative)：function gcd(a,b:longint):longint;vart:longint;beginwhile b<>0dobegint:=a;a:=b;b:=t mod b;end;exit(a);end;2．扩展欧几里德算法function extended_euclid(a,b:longint;var x,y:longint):longint;varp,q:longint;beginif b=0thenbeginx:=1;y:=0;exit(a);end;p:=extended_euclid(b,a mod b,x,y);q:=x;x:=y;y:=q-a div b*y;exit(p);end;3．求最小公倍数k:=a*b div gcd(a,b);4．求解线性同余方程typeansnode=recordansnum:longint;——解的个数num:array[1..maxnnum]of longint;——解end;procedure modular_linear_equation(a,b,n:longint;var ans:ansnode);vard,i,x,y,temp:longint;begind:=extended_euclid(a,n,x,y);if b mod d<>0then ans.ansnum:=0else beginans.ansnum:=d;temp:=(x*(b div d))mod n;for i:=1to d do ans.num[i]:=(temp+i*(n div d))mod n;end;end;5．素数的判断function prime_bool(x:longint):boolean;vari:longint;beginfor i:=2to trunc(sqrt(x))doif x mod i=0then exit(false);exit(true);end;6．素数的生成maxnum=生成质数的范围maxprime=对应范围中的质数个数varprime:array[0..maxprime]of longint;——存储质数bool:array[1..maxnnum]of boolean;——存储每个数是不是质数procedure prime_make;vari,j:longint;beginfillchar(bool,sizeof(bool),0);i:=2;while i<=maxnnum dobeginif not p[i]thenbeginj:=2*i;while i<=maxnnum dobeginp[j]:=true;inc(j,i);end;inc(prime[0]);prime[prime[0]]:=i;end;inc(i);end;end;四、排列组合算法1．排列生成算法——m的n排列vara:array[0..maxn]of longint;——排列方案b:array[0..maxm]of boolean;——每个数是否被用过递归(recursion)：procedure make_permutation_recursion(t:longint)vari:longint;beginif t=n+1thenbeginwrite(a[1]);for i:=2to n do write(‘‘,a[i]);writeln;exit;end;for i:=1to m doif not b[i]thenbeginb[i]:=true;a[t]:=i;make(t+1);b[i]:=false;end;end;非递归(iterative)：procedure make_permutation_iterative(m,n:longint);vari,j:longint;begini:=1;a[1]:=0;repeatj:=a[i]+1;while(j<=m)and(b[j])do inc(j);if j<=mthen begina[i]:=j;b[j]:=true;if i=nthen beginwrite(a[1]);for j:=2to n do write(‘‘,a[j]);writeln;b[a[n]]:=false;endelse begininc(i);a[i]:=0;end;endelse begindec(i);b[a[i]]:=false;end;until i=0;end;2．组合生成算法——m的n组合procedure make_combination(t:longint)vari:longint;beginif t=n+1thenbeginwrite(a[1]);for i:=2to n do write(‘‘,a[i]);writeln;exit;end;for i:=a[t-1]to m doif not b[i]thenbeginb[i]:=true;a[t]:=i;make(t+1);b[i]:=false;end;end;3．排列按序生成法constpower:array[1..maxn]of longint=(…);power[i]为i的阶乘typepermutationnode=array[1..maxn]of longint;——排列方案求n的全排的字典序：function get_number(n:longint;a:permutationnode):longint;varb:array[1..maxn]of longint;i,j,s:longint;beginfor i:=1to n do b[i]:=i-1;s:=0;for i:=1to n-1dobegininc(s,b[a[i]]*power[n-i]);for j:=a[i]+1to n do dec(b[j]);end;exit(s+1);end;求字典序为m的n的全排：function get_permutation(m,n:longint;):permutationnode;varuse:array[1..maxn]of boolean;a:array[0..maxn]of longint;temp:permutationnode;begindec(m);for i:=1to n-1dobegina[i]:=m mod(i+1);m:=m div(i+1);end;a[0]:=0;for i:=1to n dobeginj:=0;for k:=1to a[n-i]+1dobegininc(j);while use[j]do inc(j);end;temp[i]:=j;use[j]:=true;end;exit(temp);end;4．排列字典序生成法——求n的某个全排的下m个字典序排列procedure make_next(n,m:longint;a:permutationnode):permutationnode;vari,j,k,t,temp:longint;beginfor t:=1to m dobegini:=n;while(i>1)and(a[i]<a[i-1])do dec(i);j:=n;while a[j]<a[i-1]do dec(j);temp:=a[i-1];a[i-1]:=a[j];a[j]:=temp;for k:=i to(i+n)div2dobegintemp:=a[k];a[k]:=a[n+i-k];a[n+i-k]:=temp;end;end;exit(a);end;五、图论算法1．图的读入以点为基础读入（没有特殊说明，一般以此方法读入）：varvetex:array[1..maxn,0..maxn]of longint;——邻接表，记录与那些点相连map:array[1..maxn,1..maxn]of longint;——邻接矩阵，记录点点之间的距离procedure initgraph;vari,u,v,c:longint;beginreadln(n,e);for i:=1to e dobeginreadln(u,v,c);inc(vetex[u,0]);vetex[u,vetex[u,0]]:=v;map[u,v]:=c;end;end；以边为基础读入：typenode=recordu,v,w:longint;——u为起点，v为终点，w为权end;varvetex:array[1..maxe]of node;——记录边procedure initgraph;vari:longint;beginreadln(n,e);for i:=1to e dowith vetex[i]do readln(u,v,w);end;2．深度优先搜索——DFSvartime:longint;——时间flag:array[1..maxn]of boolean;——是否标记procedure DFS(t:longint);vari:longint;begininc(time);gettime[t]:=time;flag[t]:=true;for i:=1to vetex[t,0]doif not flag[vetex[t,i]]thenbeginfrom[vetex[t,i]]:=t;dep[vetex[t,i]]:=dep[t]+1;DFS(vetex[t,i]);end;inc(time);finishtime[t]:=time;end;3．广度优先搜索——BFSprocedure BFS(t:longint);vartime,open,closed,i,v:longint;flag:array[1..maxn]of boolean;x0:array[1..maxn]of longint;beginfillchar(flag,sizeof(flag),0);open:=0;closed:=1;x0[1]:=t;dep[t]:=0;time:=1;flag[t]:=true;flagtime[t]:=1;repeatinc(open);v:=x0[open];inc(time);finishtime[v]:=time;for i:=1to vetex[v,0]doif not flag[vetex[v,i]]thenbegininc(closed);x0[closed]:=vetex[v,i];flag[vetex[v,i]]:=true;dep[vetex[v,i]]:=dep[v]+1;inc(time);gettime[vetex[v,i]]:=time;end;until open>=closed;end;4．强连通分量varconnected:array[1..maxn,0..maxn]of longint;——connect[i,0]为此分量包含的节点数total:longint;——强连同分量的个数procedure strongly_connected;vari,time:longint;flag:array[1..maxn]of boolean;sign:array[1..maxn]of longint;procedure sort1(t:longint);vari:longint;beginflag[t]:=true;for i:=1to n doif(map[t,i]<>0)and(not flag[i])thensort1(i);inc(time);sign[time]:=t;end;procedure sort2(t:longint);vari:longint;beginflag[t]:=true;for i:=1to n doif(not flag[i])and(map[i,t]<>0)thensort2(i);inc(connected[total,0]);connected[total,conneted[total,0]]:=t;end;beginfillchar(flag,sizeof(flag),0);for i:=1to n doif not flag[i]thensort1(i);for i:=n downto1doif not flag[sign[i]]thenbegininc(total);sort(sign[i]);end;end;5．拓扑排序procedure topological_sort;vari,open,closed:longint;flag:array[1..maxn]of boolean;beginopen:=0;closed:=0;for i:=1to n doif inv[i]=0thenbegininc(closed);flag[i]:=true;AOV[closed]:=i;end;if closed=0then exit{error};repeatinc(open);v:=AOV[open];for i:=1to vetex[v,0]doif not flag[vetex[v,i]]thenbegindec(inv[vetex[v,i]]);if inv[vetex[v,i]]=0thenbegininc(closed);AOV[closed]:=vetex[v,i];flag[vetex[v,i]]:=true;end;end;until open=closed;if closed<n then exit{error};end;6．最小生成树Prime:procedure prime_normal;vari,j,min,mj:longint;flag:array[1..maxn]of boolean;lowcost:array[1..maxn]of longint;beginfillchar(lowcost,sizeof(lowcost),$5F);lowcost[1]:=0;flag[1]:=true;for i:=1to v[1,0]dolowcost[v[1,i]]:=map[1,v[1,i]];for i:=2to n dobeginmin:=maxlongint;for j:=1to n doif(not flag[j])and(lowcost[j]<min)thenbeginmin:=lowcost[j];mj:=j;end;flag[mj]:=true;inc(totallen,min);for j:=1to v[mj,0]doif(not flag[v[mj,j]])and(lowcost[v[mj,j]]>map[mj,v[mj,j]])thenlowcost[v[mj,j]]:=map[mj,v[mj,j]];end;end;Kruskal——以边为基础读入:procedure kruskal;varset1,set2,vetex_pointer,last_set_num:longint;function find(x:longint):longint;beginif father[x]=x then find:=father[x]else begin father[x]:=find(father[x]);find:=father[x];end;end;beginqsort(1,e);——对vetex以w为关键字从小到大排序for i:=1to n do father[i]:=i;vetex_pointer:=1;last_set_num:=n;while(last_set_num>1)and(vetex_pointer<=e)dobeginset1:=find(vetex[vetex_pointer].u);set2:=find(vetex[vetex_pointer].v);if set1<>set2thenbegininc(totallen,vetex[vetex_pointer].w);dec(last_set_num);father[set1]:=set2;end;inc(vetex_pointer);end;writeln(totallen);end;7．最短路径Dijktra:procedure Dijkstra(s:longint);vari,j,min,mi:longint;beginfillchar(shortest,sizeof(shortest),$5F);shortest[s]:=0;for i:=1to n dobeginmin:=max;for j:=1to n doif(not flag[j])and(shortest[j]<min)thenbegin min:=shortest[j];mi:=j;end;flag[mi]:=true;for j:=1to vetex[mi,0]doif(not flag[vetex[mi,j]])and(shortest[vetex[mi,j]]>min+map[mi,vetex[mi,j]])thenshortest[vetex[mi,j]]:=min+map[mi,vetex[mi,j]];end;end;Floyd:procedure Floyd;vari,j,k:longint;beginfillchar(len,sizeof(len),$5F);for i:=1to n dobeginlen[i,i]:=0;for j:=1to vetex[i,0]dolen[i,vetex[i,j]]:=map[i,vetex[i,j]];end;for k:=1to n dofor i:=1to n dofor j:=1to n doif len[i,k]+len[k,j]<len[i,j]thenlen[i,j]:=len[i,k]+len[k,j];end;Bellman-ford——以边为基础读入:procedure Bellman-ford(s:longint);vari,j:longint;bool:boolean;beginfillchar(shortest,sizeof(shortest),$5F);shortest[s]:=0;bool:=true;for i:=1to n-1doif bool thenbeginbool:=false;for j:=1to e doif shortest[vetex[j].v]>shortest[vetex[j].u]+vetex[j].w thenbeginshortest[vetex[j].v]:=shortest[vetex[j].u]+vetex[j].w;bool:=true;end;end;for j:=1to e doif shortest[vetex[j].v]>shortest[vetex[j].u]+vetex[j].w thenexit(flase);exit(true);end;SPFA:procedure SPFA(s:longint);varu,i:longint;x0:array[1..maxn]of longint;beginfillchar(shortest,sizeof(shortest),$5f);fillchar(flag,sizeof(flag),0);open:=0;closed:=1;x0[1]:=s;shortest[s]:=0;flag[s]:=true;repeatinc(open);u:=x0[open];for i:=1to vetex[u,0]doif shortest[vetex[u,i]]<shortest[u]+map[u,vetex[u,i]]thenbeginshortest[vetex[u,i]]:=shortest[u]+map[u,vetex[u,i]];if not flag[vetex[u,i]]thenbegininc(closed);x0[closed]:=vetex[u,i];flag[vetex[u,i]]:=true;end;end;flag[u]:=false;until open>=closed;end;六、背包问题1．尽量把容量为w的箱子装满varf:array[0..maxw]of boolean;weight:array[1..maxn]of longint;function p1:longint;vari,j:longint;beginfillchar(f,sizeof(f),0);f[0]:=true;for i:=1to n dofor j:=w downto weight[i]dof[j]:=f[j]or f[j-weight[j]];i:=w;while not f[i]do dec(i);exit(i);end;2．在容量为w的箱子中装入物品使总价值最高varf:array[0..maxw]of longint;weight,value:array[1..maxn]of longint;function p2:longint;vari,j:longint;beginfillchar(f,sizeof(f),$FF);f[0]:=0;for i:=1to n dofor j:=w downto weight[i]doif f[j-weight[i]]<>-1thenf[j]:=max(f[j],f[j-weight[i]]+value[i]);j:=0;for i:=0to w do j:=max(j,f[i]);exit(j);end;3．在满足两个量的限制(w,h)条件下使总价值最高typenode=recordw,h:longint;value:longint;end;varf:array[0..maxw,0..maxh]of longint;num:array[1..maxn]of node;function p3:longint;vari,j:longint;beginfillchar(f,sizeof(f),$FF);f[0,0]:=0;for i:=1to n dofor j:=w downto num[i].w dofor k:=h downto num[i].h doif f[j-num[i].w,k-num[i].h]<>-1thenf[j,k]:=max(f[j,k],f[j-num[i].w,k-num[i].h]+num[i].value);i:=0;for j:=0to w dofor k:=0to h do i:=max(i,f[j,k]);exit(i);end;。