Matrix Multiplication (Matrix Product)
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Description
Matrix Multiplication or Matrix Product is a binary operation that produces a matrix from two matrices with entries in a field; or; more generally; in a ring or even a semiring.
Related Problems
Subproblem: Boolean Matrix Multiplication, Matrix Product Verification
Related: Boolean Matrix Multiplication (Combinatorial), Matrix Product Verification, Distance Product, $(\min, \leq)$ Product
Parameters
$n$: dimension of square matrix
Table of Algorithms
| Name | Year | Time | Space | Approximation Factor | Model | Reference |
|---|---|---|---|---|---|---|
| Naive algorithm | 1940 | $O(n^{3})$ | $O({1})$ auxiliary | Exact | Deterministic | |
| Strassen's algorithm | 1969 | $O(n^{(log7/log2)}) ~ O(n^{2.{80}7})$ | $O(n^{2})$ | Exact | Deterministic | Time & Space |
| Pan's algorithm | 1978 | $O(n^{(log({143640})/log({70}))}) ~ O(n^{2.{79}5})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| Romani's algorithm | 1981 | $O(n^{2.{5166}5})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| Coppersmith–Winograd algorithm | 1981 | $O(n^{2.{49554}8})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| Strassen's algorithm | 1986 | $O(n^{(log54/log5)}) ~ O(n^{({2.4785})})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| Coppersmith–Winograd algorithm | 1990 | $O(n^{2.{375}5})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| Vassilevska Williams | 2014 | $O(n^{2.{37287}3})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| François Le Gall | 2014 | $O(n^{2.{372863}9})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| Bini's algorithm | 1979 | $O(n^{2.{779}9})$ | $O(n^{2})$ | $O(n logn)$ error | Deterministic | Time |
| Schonhage's algorithm | 1980 | $O(n^{({3}*\log {52}/l \og {110})}) ~ O(n^{2.{521}8})$ | $O(n^{2})$ | ? | Deterministic | Time |
| Output-Sensitive Quantum BMM | 2018 | O*( \min \{n^{1/3} L^{17/{3}0}, n^{1.5} L^{1/4}\}) | Exact | Quantum | Time | |
| 2018 | $O(n^{3} / log^{2.25} n)$ | Exact | Deterministic | Time | ||
| O'Neil | 1973 | $O(n^{3})$ | Exact | Deterministic | Time | |
| Method of Four Russians | 1970 | $O(n^{3}/(log n)$^{2}) | Exact | Deterministic | Time | |
| Bansal, Williams | 2009 | $O(n^{3} * (log log n)$^{2} / log^{2.25} n) | Exact | Randomized | Time | |
| Bansal, Williams | 2009 | $O(n^{3} * (log log n)$^{2} / (w * (log n)^{7}/{6})) | Exact | Randomized | Time | |
| Chan | 2015 | $O(n^{3} * (log log n)$^{3} / log^{3} n) | Exact | Deterministic | Time | |
| Chan | 2015 | $O(n^{3} * (log w)$^{3} / (w * log^{2} n)) | Exact | Deterministic | Time | |
| Yu | 2015 | $O(n^{3}*poly(log log n)$/log^{4} n) | Exact | Deterministic | Time |
Time Complexity Graph
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