Continuing the theme from last lecture, let M₁, … , M_n be a series of matrices, and r₀, … , r_n be a series of dimensions such that M_i is an r_i-1 * r_i matrix.

Our goal is to compute the multiplication of all M:

best(i, j) = min_k best_k(i, j)
best_k(i, j) = best(i, k) + best(k, j) + r_i-1 * r_k * r_j

Using these rules, we can carefully fill a lower triangular table (being careful of the order of filling):

Matrix Cost/Order Algorithm:

For i = 1 to n
    B[i, i] = 0
EndFor

For gap = 1 to (n-1)
    for i = 1 to n – gap
        j = i + gap
        cost = min_k {B[i, k] + B[k+1, j] + r_i-1 * r_k * r_j}
        B[i, j] = cost
    EndFor
End For

return B[1,n]

• Need to remember choices!
• Whenever we assign B[i, j] we know the best k
• Use another array Mem[i, j] = best k found when doing B[i, j]
• Mem(1, n) tells where to split M₁, … , M_n

Dividing a polygon into triangles:

This was once a very popular optimization technique for 3d rendering (and especially games) before graphics hardware supported complex shapes–triangles were fast to draw:

• Find the minimum weight of triangulation
• weight = sum of the lengths of new edges
• Polygon w/ vertices v₁, … , v_n– assume counterclockwise.
• D[i, j] = distance from v_i to v_j, unless outside the polygon, in which case the distance is infinite
• C(i, j) = minimum cost for triangulating polygon v₁, … , v_n
• C_k(1, n) = best answer using vertex k for triangle with v₁ and v_n
• C(1, n) = min C_k(1, n)
• C_k(1, n) = C(k, n) + C(1, k) + D[k, n] + D[1, k]

Polygon Triangulating Algorithm:

For i = 1 to n
    C[i, i+1] = -D[i, i+1]
EndFor

For gap = 2 to n -1
    for i = 1 to n – gap
        j = i + gap
        C[i, j] = min_k {C[i, k] + C[k, j] + D[i, k] + D[k, j]}
    EndFor
EndFor

return C[1, n]

This entry was posted on Friday, February 11th, 2005 at 1:23 pm and is tagged with nbsp nbsp nbsp nbsp nbsp, optimization technique, triangular table, dynamic programming, polygon, vertices, mink, vertex, graphics hardware, rk, matrices, triangles, multiplication, ck, gap, algorithm, shapes, triangle, array, matrix. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback.