Testwiki:Reference desk/Archives/Computing/2020 August 22

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August 22

any reason that array based sparse matrix will be slower than linkedlist sparse matrix?

Hi there, Is there a reason that a sparse matrix represented by an array of a linked list will exhibit a faster performance than a sparse array representation of a matrix? --Exx8 (talk) 18:40, 22 August 2020 (UTC)

This depends both on the specifics of in particular the sparse array representation, for which there are many options, and the specific matrix operations being performed.  --Lambiam 21:29, 22 August 2020 (UTC)

sparse matrix multiplication.--Exx8 (talk) 07:08, 23 August 2020 (UTC)

Let $R_i(M)$ denote the set of column indices $j$ such that $M_{i,j}\ne 0$, and let similarly $C_i(M)$ denote the set of row indices $i$ such that $M_{i,j}\ne 0$. (So $j\in C_i$ iff $i\in R_j$.) If $C=AB$, then $C_{i,j}={\mathrm{\Sigma }}_k{A}_{i,k}{B}_{k,j}$. Random access of elements in sparse representation, as seemingly required, is expensive, but note that $k$ can be restricted to the intersection of $R_i(A)$ and $C_j(B)$. If $A$ is represented as a collection of sparse row vectors and $B$ as a collection of sparse column vectors,, it is cheap to find the values of $k$ that contribute to the sum.

One can use linked lists to represent the sparse row and column vectors; if doubly linked, insertion is easier, but for matrix multiplication that is not an important issue.  --Lambiam 10:24, 23 August 2020 (UTC)