Figure 8. Sparse Matrix.

) Example organization of a sparse cloud into a flat array with supporting offset data, and demonstration of its use. (A) sparse cloud cells in pink/blue are supplemented with the set of padding cells (white with ▪) that ensure that any Forward/Backward calculation dependencies will refer to either a cloud or padding cell (to avoid conditionals in the DP inner loop). (B) Table of values required to compute offsets into flat array during DP recurrence computation: the row offset is the column index of the first cell in the row; the block offset is the index in the flat array of the first cell in the row. (C) Pseudocode for retrieving a value from the flat array given logical (i.e. implicit full matrix) row and column indices. The retrieval function is fast in practice, and circumvents the use of conditional logic. Note: This is a slight simplification of the actual implementation, which must support access to each of the $M,I,$ and $D$ values that correspond to the same logical row and column. (D) Representation of the flat array in memory. Note: the visualization has been simplified for clarity; in practice, each element in a block shown actually corresponds to a tuple of three values, one for each of the $M,I,$ and $D$ matrices. Similarly, each padding cell shown in the flat representation corresponds to a group of three identical padding values.