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Strassen's breakthrough: How can the calculation of matrix multiplication be greatly simplified?
In computational complexity theory, arithmetic circuits have become the standard model for computing polynomials. These circuits work by taking variables or numbers as input and then performing addition or multiplication operations, making them a formal way to understand the polynomial complexity of computations. However, the question of how to compute a particular polynomial most efficiently is still worth pondering.
An arithmetic circuit is a directed acyclic graph where each node with zero in-degree is called an input gate and is labeled as a variable or field element.
The size and depth of arithmetic circuits are two key complexity measures. The size of a circuit is the number of its gates, while its depth is the length of the longest directed path from input to output. For example, an arithmetic circuit can calculate polynomials through input gates and then perform addition and multiplication operations based on the calculated subnodes.
Upper and Lower Bounds
When exploring the complexity of computing polynomials, we can ask ourselves the question: How do we find the best way to compute a certain polynomial? This involves first building a circuit that can compute the given polynomial, which is called an upper bound. Then show that no other circuit can do better, and this is the lower bound.
While the two tasks of upper and lower bounds are conceptually closely related, proving lower bounds is usually more challenging because all possible circuits need to be analyzed simultaneously.
A notable example is Strathern's algorithm, which was shown to compute the product of two n×n matrices with a size of about n2.807. This represents a significant simplification over the traditional O(n3) approach. Strathern's innovations stemmed primarily from his clever method for multiplying 2×2 matrices, which laid the foundation for more efficient matrix multiplication.
Challenges of the Nether
While many clever circuits have been found to find upper bounds on polynomials, the task of proving lower bounds is extremely difficult. Especially for polynomials of small degree, the complexity of the problem can be illustrated if it can be shown that some polynomials require circuits of superpolynomial size. However, the main challenge is to find an explicit polynomial that can be proven to exceed the polynomial size requirement, which has become one of the key focuses of current research.
Lower bounds for polynomials such as x1d + ... + xnd are given by Strathern et al. proved it to be Ω(n log d).
The research results presented by Strathern not only lead us to a deeper understanding of arithmetic circuits, but also successfully focus attention on the complexity problems caused by the global circuit size required by polynomials. If such results can be further applied to a wider range of polynomials, it is expected to solve many unsolved problems.
Algebraic P and NP Problems
Another topic worth paying attention to is the P and NP problem in algebra. In this question, can one solve a problem with the same efficiency as confirming whether a solution to a given problem exists? This is an important theoretical challenge because it is not only about polynomial computation, but also involves the core issue of computational complexity as a whole.
The VP and VNP problem proposed by Valiant is a wonderful algebraic problem involving the computation and representation capabilities of polynomials.
In-depth study of VP and VNP problems may provide unique insights into the complexity of arithmetic computations. As research continues, we look forward to more breakthroughs in the future that will challenge the boundaries of traditional computing theory.
In this rapidly changing world of mathematics and computing, as theory advances and practical applications expand, the complexity of the calculation process should at least cause us to think deeply. Can future computing models be further optimized?
