Triangular division of Convex polygons (Hnoi ' 97)

First, the question description

Given a convex polygon with n (n<50) vertices (numbered from 1 to n), the weights of each vertex are known. Ask how to divide this convex polygon into N-2 disjoint triangles, making the sum of the weights of the vertices of these triangles a minimum.

Input file: Number of vertices in first row n

Weights for n vertices in the second row (from 1 to N)

Output format: Minimum and value

The way the triangles are composed

Input Example: 5

122 123 245 231

Output example: the minimum is:12214884

The formation of 3 triangle:

3 4 5, 1 5 3, 1 2 3

Second, analysis of test questions

This is a typical dynamic programming problem. Set F[i,j] (I<J) represents the maximum product from vertex I to vertex J convex polygon triangulation, we can get the following dynamic transfer equation:

F[i,j]=min{f[i,k]+f[k,j]+s[i]*s[j]*s[k]} (I<K<J)

Target status: F[1,n]

But we can find that because of the sum of the product here, when the input data is larger than the length of the shape or even the scope of the solid, so we also need high-precision calculation, but this is the basic skills of everyone, the program is not written, please complete the reader by themselves.

III. Reference Procedures

Var s:array[1..50] of Integer;

F:ARRAY[1..50,1..50] of Comp;

D:ARRAY[1..50,1..50] of Byte;

N:integer;

Procedure Init; (Input data)

Var I:integer;

Begin

READLN (N);

For I:=1 to N do Read (S[i]);

End;

Procedure Dynamic; (Dynamic planning)

Var I,j,k:integer;

Begin

For I:=1 to N do

For j:=i+1 to N do f[i,j]:=maxlongint; (Assign initial value)

For i:=n-2 Downto 1 do

For j:=i+2 to N do

For k:=i+1 to J-1 do

If (F[i,j]>f[i,k]+f[k,j]+s[i]*s[j]*s[k]) then

Begin

F[I,J]:=F[I,K]+F[K,J]+S[I]*S[J]*S[K];

D[i,j]:=k; (Record parent node)

End;

End;

Procedure Print (I,j:integer); (Output each triangle)

Begin

If J=i+1 then Exit;

Write (', ', I, ', J, ', d[i,j]);

Out (I,d[i,j]);

Out (D[I,J],J);

End;

Procedure out; (Output information)

Begin

Assign (Output, ' Output.Txt '); Rewrite (Output);

Writeln (' The minimum is: ', f[1,n]:0:0);

Writeln (' The formation of ', N-2, ' triangle: ');

Write (1, "', N, ' d[1,n]);

Out (1,d[1,n]);

Out (D[1,n],n);

Close (Output);

End;

Begin (main program)

Init;

Dynamic;

Out;

End.