基本入門(mén)的C/C++算法總結(jié)

C C++,算法實(shí)例

一、數(shù)論算法

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1．求兩數(shù)的最大公約數(shù)
function gcd(a,b:integer):integer;
begin
if b=0 then gcd:=a
else gcd:=gcd (b,a mod b);
end ;

2．求兩數(shù)的最小公倍數(shù)
function lcm(a,b:integer):integer;
begin
if a<b then swap(a,b);
lcm:=a;
while lcm mod b>0 do inc(lcm,a);
end;

3．素?cái)?shù)的求法
A.小范圍內(nèi)判斷一個(gè)數(shù)是否為質(zhì)數(shù)：
function prime (n: integer): Boolean;
var I: integer;
begin
for I:=2 to trunc(sqrt(n)) do
if n mod I=0 then begin
prime:=false; exit;
end;
prime:=true;
end;

B.判斷l(xiāng)ongint范圍內(nèi)的數(shù)是否為素?cái)?shù)（包含求50000以?xún)?nèi)的素?cái)?shù)表）：
procedure getprime;
var
i,j:longint;
p:array[1..50000] of boolean;
begin
fillchar(p,sizeof(p),true);
p[1]:=false;
i:=2;
while i<50000 do begin
if p[i] then begin
j:=i*2;
while j<50000 do begin
p[j]:=false;
inc(j,i);
end;
end;
inc(i);
end;
l:=0;
for i:=1 to 50000 do
if p[i] then begin
inc(l);pr[l]:=i;
end;
end;{getprime}

function prime(x:longint):integer;
var i:integer;
begin
prime:=false;
for i:=1 to l do
if pr[i]>=x then break
else if x mod pr[i]=0 then exit;
prime:=true;
end;{prime}


二、圖論算法

1．最小生成樹(shù)

A.Prim算法：

procedure prim(v0:integer);
var
lowcost,closest:array[1..maxn] of integer;
i,j,k,min:integer;
begin
for i:=1 to n do begin
lowcost[i]:=cost[v0,i];
closest[i]:=v0;
end;
for i:=1 to n-1 do begin
{尋找離生成樹(shù)最近的未加入頂點(diǎn)k}
min:=maxlongint;
for j:=1 to n do
if (lowcost[j]<min) and (lowcost[j]<>0) then begin
min:=lowcost[j];
k:=j;
end;
lowcost[k]:=0; {將頂點(diǎn)k加入生成樹(shù)}
{生成樹(shù)中增加一條新的邊k到closest[k]}
{修正各點(diǎn)的lowcost和closest值}
for j:=1 to n do
if cost[k,j]<lwocost[j] then begin
lowcost[j]:=cost[k,j];
closest[j]:=k;
end;
end;
end;{prim}

B.Kruskal算法：(貪心)

按權(quán)值遞增順序刪去圖中的邊，若不形成回路則將此邊加入最小生成樹(shù)。
function find(v:integer):integer; {返回頂點(diǎn)v所在的集合}
var i:integer;
begin
i:=1;
while (i<=n) and (not v in vset[i]) do inc(i);
if i<=n then find:=i else find:=0;
end;

procedure kruskal;
var
tot,i,j:integer;
begin
for i:=1 to n do vset[i]:=[i];{初始化定義n個(gè)集合，第I個(gè)集合包含一個(gè)元素I}
p:=n-1; q:=1; tot:=0; {p為尚待加入的邊數(shù)，q為邊集指針}
sort;
{對(duì)所有邊按權(quán)值遞增排序，存于e[I]中，e[I].v1與e[I].v2為邊I所連接的兩個(gè)頂點(diǎn)的序號(hào)，e[I].len為第I條邊的長(zhǎng)度}
while p>0 do begin
i:=find(e[q].v1);j:=find(e[q].v2);
if i<>j then begin
inc(tot,e[q].len);
vset[i]:=vset[i]+vset[j];vset[j]:=[];
dec(p);
end;
inc(q);
end;
writeln(tot);
end;

2.最短路徑

A.標(biāo)號(hào)法求解單源點(diǎn)最短路徑：
var
a:array[1..maxn,1..maxn] of integer;
b:array[1..maxn] of integer; {b[i]指頂點(diǎn)i到源點(diǎn)的最短路徑}
mark:array[1..maxn] of boolean;

procedure bhf;
var
best,best_j:integer;
begin
fillchar(mark,sizeof(mark),false);
mark[1]:=true; b[1]:=0;{1為源點(diǎn)}
repeat
best:=0;
for i:=1 to n do
If mark[i] then {對(duì)每一個(gè)已計(jì)算出最短路徑的點(diǎn)}
for j:=1 to n do
if (not mark[j]) and (a[i,j]>0) then
if (best=0) or (b[i]+a[i,j]<best) then begin
best:=b[i]+a[i,j]; best_j:=j;
end;
if best>0 then begin
b[best_j]:=best；mark[best_j]:=true;
end;
until best=0;
end;{bhf}

B.Floyed算法求解所有頂點(diǎn)對(duì)之間的最短路徑：
procedure floyed;
begin
for I:=1 to n do
for j:=1 to n do
if a[I,j]>0 then p[I,j]:=I else p[I,j]:=0; {p[I,j]表示I到j(luò)的最短路徑上j的前驅(qū)結(jié)點(diǎn)}
for k:=1 to n do {枚舉中間結(jié)點(diǎn)}
for i:=1 to n do
for j:=1 to n do
if a[i,k]+a[j,k]<a[i,j] then begin
a[i,j]:=a[i,k]+a[k,j];
p[I,j]:=p[k,j];
end;
end;

C. Dijkstra 算法：

var
a:array[1..maxn,1..maxn] of integer;
b,pre:array[1..maxn] of integer; {pre[i]指最短路徑上I的前驅(qū)結(jié)點(diǎn)}
mark:array[1..maxn] of boolean;
procedure dijkstra(v0:integer);
begin
fillchar(mark,sizeof(mark),false);
for i:=1 to n do begin
d[i]:=a[v0,i];
if d[i]<>0 then pre[i]:=v0 else pre[i]:=0;
end;
mark[v0]:=true;
repeat {每循環(huán)一次加入一個(gè)離1集合最近的結(jié)點(diǎn)并調(diào)整其他結(jié)點(diǎn)的參數(shù)}
min:=maxint; u:=0; {u記錄離1集合最近的結(jié)點(diǎn)}
for i:=1 to n do
if (not mark[i]) and (d[i]<min) then begin
u:=i; min:=d[i];
end;
if u<>0 then begin
mark[u]:=true;
for i:=1 to n do
if (not mark[i]) and (a[u,i]+d[u]<d[i]) then begin
d[i]:=a[u,i]+d[u];
pre[i]:=u;
end;
end;
until u=0;
end;

3.計(jì)算圖的傳遞閉包

Procedure Longlink;
Var
T:array[1..maxn,1..maxn] of boolean;
Begin
Fillchar(t,sizeof(t),false);
For k:=1 to n do
For I:=1 to n do
For j:=1 to n do T[I,j]:=t[I,j] or (t[I,k] and t[k,j]);
End;


4．無(wú)向圖的連通分量

A.深度優(yōu)先
procedure dfs ( now,color: integer);
begin
for i:=1 to n do
if a[now,i] and c[i]=0 then begin {對(duì)結(jié)點(diǎn)I染色}
c[i]:=color;
dfs(I,color);
end;
end;

B 寬度優(yōu)先（種子染色法）


5．關(guān)鍵路徑

幾個(gè)定義： 頂點(diǎn)1為源點(diǎn)，n為匯點(diǎn)。
a. 頂點(diǎn)事件最早發(fā)生時(shí)間Ve[j], Ve [j] =max{ Ve [j] + w[I,j] },其中Ve (1)= 0;
b. 頂點(diǎn)事件最晚發(fā)生時(shí)間 Vl[j], Vl [j] =min{ Vl[j] – w[I,j] },其中 Vl(n)= Ve(n);
c. 邊活動(dòng)最早開(kāi)始時(shí)間 Ee[I], 若邊I由