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Due to Observation 2, when the algorithm is about to compute the entry in the second row and j-th column of H u, u ′ (i.e. the entry corresponding to T u v 2 and S u ′ v j ′ ), all required values for computing x∗, α, and β, are already stored in H, and therefore these values may be computed in O(d u ) time.
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To compute the entries of S, visit all vertices w ∈ V in a bottom-up traversal.
Next, we provide a strategy to compute the entries of S in accordance with this definition.
To compute the entries of Best referred to the index (i + 1), we only need to know the entries referred to the index i.
We can thus compute the entries of S u and Q ux, respectively, in polynomial time by solving maximum weighted matching problems with non-negative weights.
The matrix B is computed by applying the Inside VMT algorithm (Table 2) with a simple modification: in the base-cases of the recurrence (line 2 in Procedure Compute-Inside-Sub-Matrix, Table 2), the condition for computing the entry B i, j is that h-1(i) ≤ h-1 j) h-1 jrather i ≤ j.
This is done by shifting the integration point locations when computing the entries of the mass matrix, while ensuring the mass conservation.
This assumes, however, that all the entries of A for pairs with distance l - 1 are known before computing the entries of the pairs with distance l.
Note that in the case v = w we have that S v, v) is the weight of an optimal orientation of the subtree rooted at v. Next, we describe how our algorithm computes the entries of S in accordance with this definition.
Computing column j in matrices EDT α and ED, the algorithm performs O matrix-vector min-plus multiplications (Equations 15 and 16), each taking O n 2 λ 2 log | D | 2 (n ) time using the algorithm in Section "The EDDC algorithm based on efficient D-discrete min-plus matrix-vector multiplication", and computes the entry-wise minimum of | Σ| i-length vectors (Equation 16) in O (| Σ| i) time.
Let us now compute the stencil entries for our inpainting task from (3) resp.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com