冲刺NOIP系列之算法整理一、高精度算法1.高精度加法procedure add(a,b:hp,var c:hp);var i:longint;beginfillchar(c,sizeof(c),0);c[0]:=1;if a[0]>b[0] thenc[0]:=a[0] else c[0]:=b[0];for i:=1 to a[0] do inc(c[i],a[i]);for i:=1 to b[0] do inc(c[i],b[i]);for i:=1 to c[0] dobeginc[i+1]:=c[i+1]+c[i] div 10;c[i]:=c[i] mod 10;end;while c[c[0]+1]>0 dobegininc(c[0]);c[c[0]+1]:=c[c[0]+1]+c[c[0]] div 10;c[c[0]]:=c[c[0]] mod 10;end;end;2.高精度减法procedure minus(a,b:hp;var c:hp);var i:longint;beginfillchar(c,sizeof(c),0); c[0]:=a[0];for i:=1 to c[0] do c[i]:=a[i]-b[i];for i:=1 to c[0] doif c[i]<0 thenbegindec(c[i+1]);c[i]:=c[i]+10;end;while (c[0]>1)and(c[c[0]]=0) do dec(c[0]);end;3.高精度乘法(单精、高精)procedure mulnum(a:hp;x:longint;var c:hp);var i:longint;beginfillchar(c,sizeof(c),0);c[0]:=a[0];for i:=1 to c[0] do c[i]:=a[i]*x;for i:=1 to c[0] dobeginc[i+1]:=c[i+1]+c[i] div 10;c[i]:=c[i] mod 10;end;while c[c[0]+1]>0 dobegininc(c[0]);c[c[0]+1]:=c[c[0]+1]+c[c[0]] div 10;c[c[0]]:=c[c[0]] div 10;end;end;procedure mul(a,b:hp;var c:hp);var i:longint;beginfillchar(c,sizeof(c),0);c[0]:=a[0]+b[0]-1;for i:=1 to a[0] dofor j:=1 to b[0] doinc(c[i+j-1],a[i]*b[i]);for i:=1 to c[0] dobeginc[i+1]:=c[i+1]+c[i] div 10;c[i]:=c[i] mod 10;end;while c[c[0]+1]>0 dobeginc[c[0]+1]:=c[c[0]+1]+c[c[0]] div 10;c[c[0]]:=c[c[0]] mod 10;end;end;4.高精度除法procedure divnum(a:hp;x:longint;var c:hp;var d:longint); var i:longint;begind:=0; fillchar(c,sizeof(c),0); c[0]:=a[0];for i:=c[0] downto 1 dobegind:=d*10+a[i];c[i]:=d div x;d:=d mod x;end;while (c[0]>1)and(c[c[0]]=0) do dec(c[0]);end;5.高精度比较function compare(a,b:hp):longint;beginwhile a[a[0]]=0 do dec(a[0]);while b[b[0]]=0 do dec(b[0]);while a[a[0]+1]>0 do inc(a[0]);while b[b[0]+1]>0 do inc(b[0]);if a[0]>b[0] then exit(1);if a[0]<b[0] then exit(-1);for i:=a[0] downto 1 dobeginif a[i]>b[i] then exit(1);if a[i]<b[i] then exit(-1);end;exit(0);end;二、数学算法1.欧几里得(最大公约数)function gcd(a,b:longint):longint;beginif b=0 then exit(a)else gcd:=gcd(b,a mod b);end;2.同余方程vara,b,x,y,k,n,m2,m1,a2,a1,g,t,c:longint;flag:boolean;function extended_gcd(a,b:longint;var x,y:longint):longint; vart:longint;beginif b=0 thenbeginx:=1; y:=0;exit(a);end;extended_gcd:=extended_gcd(b,a mod b,x,y);t:=x;x:=y;y:=t - (a div b)*y;end;beginread(n);read(m1,a1);while n>1 dobegindec(n);read(m2,a2);g:=extended_gcd(m1,m2,x,y);c:=a2-a1;if c mod g<>0 then begin flag:=true;break;end;k:=m2 div g;t:=(c div g *x mod k+k) mod k;a1:=a1+t*m1;m1:=m1 div g *m2;end;if flag=false then write(a1) else write(-1);close(input);close(output);end.3.快速幂function soso(a,b:longint):longint;var total:longint;begintotal:=1;while b>0 dobeginif odd(b) then total:=(total*a) mod 10000;b:=b shr 1;a:=(a*a) mod 10000;end;exit(total);end;三、图论1.spfavara,b,e:array[1..1000] of longint;vis:array[1..2000] of boolean;q,d,f:array[1..2001] of longint;n,m,i,s,t:longint;procedure qsort(l,r:longint);var i,j,x,y:longint;beginx:=a[(l+r) shr 1];repeatwhile a[i]<x do inc(i);while a[j]>x do dec(j);if not(i>j) thenbeginy:=a[i]; a[i]:=a[j]; a[j]:=y;y:=b[i]; b[i]:=b[j]; b[j]:=y;y:=e[i]; e[i]:=e[j]; e[j]:=y;inc(i); dec(j);end;until i>j;if i<r then qsort(i,r);if l<j then qsort(l,j);end;procedure spfa(s:longint);var i,k,l,t:longint;beginfillchar(vis,sizeof(vis),0);for i:=1 to n dod[i]:=maxlongint;d[s]:=0;l:=0; t:=1; q[1]:=s; vis[s]:=true; repeatl:=l mod 10000 +1;k:=q[l];for i:=f[k] to f[k+1]-1 doif d[k]+e[i]<d[b[i]] thenbegind[b[i]]:=d[k]+e[i];if not vis[b[i]] then begint:=t mod 10000 +1;q[t]:=b[i];vis[b[i]]:=true;end;end;vis[k]:=false;until l=t;end;beginreadln(n,m);for i:=1 to m doreadln(a[i],b[i],e[i]);for i:=1 to m doif f[a[i]]=0 then f[a[i]]:=i;f[n+1]:=m+1;for i:=n downto 1 doif f[i]=0 then f[i]:=f[i+1];readln(s,t);spfa(s);writeln(d[t]);end.2.dijkstravar dist:Array[1..1001,1..1001] of longint;n,m,i,j:longint;aa,bb,zz:longint;v:array[1..1001] of boolean;procedure dijkstra(s:longint);var i,j,min,k:longint;beginv[s]:=true;dist[s,s]:=0;for i:=1 to n-1 dobeginmin:=10000000;for j:=1 to n doif (dist[s,j]<min)and(not v[j]) thenbegin k:=j; min:=dist[s,j]; end;v[k]:=true;for j:=1 to n doif dist[k,j]>0 thenif (dist[s,k]+dist[k,j]<dist[s,j])or(dist[s,j]=0) thendist[s,j]:=dist[s,k]+dist[k,j];end;end;beginreadln(n,m);for i:=1 to m dobeginreadln(aa,bb,zz);dist[aa,bb]:=zz;end;dijkstra(1);for i:=2 to n do writeln(dist[1,i]);end.3. kruskalvar f:array[1..105] of longint;a,b,w:array[1..10005] of longint;n,i,j,k,x,y,total:longint;procedure kuaipai(st,ed:longint);var i,j,mid:longint;beginif st>=ed then exit;i:=st-1; j:=ed+1;mid:=w[(st+ed) >> 1];while i<j dobeginrepeat inc(i) until w[i]>=mid;repeat dec(j) until w[j]<=mid;if i<j thenbeginw[i]:=w[i] xor w[j];w[j]:=w[i] xor w[j];w[i]:=w[i] xor w[j];a[i]:=a[i] xor a[j];a[j]:=a[i] xor a[j];a[i]:=a[i] xor a[j];b[i]:=b[i] xor b[j];b[j]:=b[i] xor b[j];b[i]:=b[i] xor b[j];end;end;kuaipai(st,j); kuaipai(j+1,ed);end;function find(v:longint):longint;beginif f[v]=v then exit(v)elsebeginf[v]:=find(f[v]);find:=f[v];end;end;beginreadln(n);k:=0;for i:=1 to n dofor j:=1 to n dobeginread(x);if j>i then continue;inc(k);a[k]:=i;b[k]:=j;w[k]:=x;end;kuaipai(1,k);for i:=1 to n do f[i]:=i;total:=0;for i:=1 to k dobeginx:=find(a[i]); y:=find(b[i]);if x<>y thenbegininc(total,w[i]);f[x]:=y;end;end;writeln(total);end.4. 并查集beginfor i:=1 to n do f[i]:=i;function find(v:longint):longint;beginif f[v]=v then exit(v)elsebeginf[v]:=find(f[v]);exit(f[v]);end;end;for i:=1 to n do find(i);end;4.trajan①(强连通分量)varv,f :array[1..100]of boolean; dfn,low :array[1..100]of integer;a :array[0..100,0..100]of integer; i,j,n,m,x,y,deep,d:integer;count:longint;stack :array[1..100]of integer; function min(x,y:longint):integer;beginif x>y then exit(y) else exit(x);end;procedure print(x:integer);beginwhile stack[deep]<>x dobeginwrite(stack[deep],' ');f[stack[deep]]:=false;dec(deep);end;writeln(stack[deep]);f[stack[deep]]:=false;dec(deep);end;procedure dfs(x:integer);vari:integer;begininc(d); dfn[x]:=d;low[x]:=d; inc(deep);stack[deep]:=x;f[x]:=true;for i:=1 to a[x,0] doif not v[a[x,i]] thenbeginv[a[x,i]]:=true;dfs(a[x,i]);low[x]:=min(low[a[x,i]],low[x]);endelseif f[a[x,i]] thenbegin inc(count); low[x]:=min(low[x],dfn[a[x,i]]);end;if dfn[x]=low[x] then print(x);end;begincount:=0;readln(n,m);fillchar(a,sizeof(a),0);for i:=1 to m dobeginreadln(x,y);inc(a[x,0]);a[x,a[x,0]]:=y;end;for i:=1 to n doif not v[i] thenbeginv[i]:=true;dfs(i);end;writeln(count);end.②(求LCA)var n,m,q:longint;f,colour:array[1..1000]of longint;lca:array[1..1000,1..1000]of longint;g:array[1..1000,1..1000]of boolean;function find(x:longint):longint;beginif f[x]=x then exit(x);find:=find(f[x]);end;procedure tarjan(u:longint);var i,j:longint;beginf[u]:=u;colour[u]:=1;for i:=1 to n doif (g[u,i])and(colour[i]=0) then begin tarjan(i);f[i]:=u;end;for i:=1 to n doif (colour[i]=2)then begin lca[u,i]:=find(i);lca[i,u]:=lca[u,i];end; colour[u]:=2;end;procedure init;var i,j,x,y:longint;beginread(n,m,q);for i:=1 to m dobeginread(x,y);g[x,y]:=true;g[y,x]:=true;end;for i:=1 to n do f[i]:=i;end;procedure main;var i,j:longint;beginif f[i]=i then tarjan(i);end;procedure print;var i,j,x,y:longint;beginfor i:=1 to q dobeginread(x,y);write(lca[x,y]);end;end;begininit;main;print;end.四、基本算法策略1.KMPvar s:ansistring;m,ans:longint;next,l,r,temp:array[0..10000]of longint; procedure get_next(t:ansistring);var i,j,k,l:longint;beginl:=length(t);k:=0; next[1]:=0;for i:=2 to l dobeginwhile (k>0)and(t[i]<>t[k+1]) do k:=next[k];if t[i]=t[k+1] then inc(k);next[i]:=k;end;end;procedure kmp1(s,t:ansistring);var i,j,l1,l2,k:longint;beginfillchar(next,sizeof(next),0);get_next(t);k:=0;l1:=length(s);l2:=length(t);beginwhile (k>0)and(s[i]<>t[k+1]) do k:=next[k];if s[i]=t[k+1] then inc(k);l[i]:=k;if k=l2 then exit;end;end;procedure uget_next(t:ansistring);var i,j,l,k:longint;beginfillchar(next,sizeof(next),0);l:=length(t);k:=l+1;next[l]:=k;for i:=l-1 downto 1 dobeginwhile (k<l+1)and(t[i]<>t[k-1]) do k:=next[k];if t[i]=t[k-1] then dec(k);next[i]:=k;end;end;procedure kmp2(s,t:ansistring);var i,j,l1,l2,k:longint;beginuget_next(t);l1:=length(s);l2:=length(t);k:=l2+1;for i:=l1 downto 1 dobeginwhile (k<l2+1)and(t[k-1]<>s[i]) do k:=next[k];if s[i]=t[k-1] then dec(k);r[i]:=l2+1-k;if k=1 then exit;end;end;procedure check(s,ss:ansistring);var i,j,l1,l2:longint;beginl1:=length(s);l2:=length(ss);for i:=1 to l1 doif l[i]<l[i-1] then l[i]:=l[i-1];for i:=1 to l1-1 doif l[i]+r[i+1] >=l2 then begin inc(ans);break;end; end;procedure main;var i,j,n:longint;ss:ansistring;beginreadln(s);readln(m);for i:=1 to m dobeginreadln(ss);if length(ss)<2 then continue;kmp1(s,ss);kmp2(s,ss);check(s,ss);end;end;beginmain;write(ans);end.2.RMQvar fmax,fmin:array[0..20,1..51000] of longint;n,i,j,ma,mi,l,r,q,t:longint;function max(x,y:longint):longint;beginif x>y then exit(x) else exit(y);end;function min(x,y:longint):longint;beginif x>Y then exit(y) else exit(x);end;beginreadln(n,q);t:=trunc(ln(n)/ln(2));for i:=1 to n dobeginreadln(fmax[0,i]);fmin[0,i]:=fmax[0,i];end;for i:=1 to t dofor j:=1 to n dobeginfmax[i,j]:=max(fmax[i-1,j],fmax[i-1,j+(1 << (i-1))]);fmin[i,j]:=min(fmin[i-1,j],fmin[i-1,j+(1 << (i-1))]);end;for i:=1 to q dobeginreadln(l,r);t:=trunc(ln(r-l+1)/ln(2));ma:=max(fmax[t,l],fmax[t,r-(1 << t)+1]);mi:=min(fmin[t,l],fmin[t,r-(1 << t)+1]);writeln(ma,' ',mi);end;end.3. 单调队列vartemp,ans:int64;n,p,q,i,j:longint;a:array[0..400000] of longint;b,r,l:array[0..400000] of longint;beginreadln(n);for i:=1 to n do read(a[i]);p:=1;q:=0;for i:=1 to n+1 dobeginwhile (p<=q) and (a[i]<a[b[q]]) dobeginr[b[q]]:=i;dec(q);end;inc(q);b[q]:=i;end;fillchar(b,sizeof(b),0);p:=1;q:=0;for i:=n downto 0 dobeginwhile (p<=q) and (a[i]<a[b[q]]) dobeginl[b[q]]:=i;dec(q);end;inc(q);b[q]:=i;end;for i:=1 to n dobegintemp:=(r[i]-l[i]-1)*a[i];if temp>ans then ans:=temp;end;writeln(ans); end.。
