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The secret of determinant computing: How to solve it cleverly using polynomial circuits?
In the calculation complexity theory, arithmetic circuits are regarded as standard model for calculating polynomials.Basically, the function of an arithmetic circuit is to receive variables or numbers as inputs, and then perform addition or multiplication operations.This model provides a formal way to understand the complexity of computational polynomials.So, how to effectively calculate a given polynomial?This has become one of the core issues of the research.
Arithmetic circuit is a directed acyclic graph with the inlet of each input gate zero and marked as a variable or field element.Other gates are marked as addition gates or multiplication gates.Each circuit has two complexity measures: size and depth.The size of the circuit refers to the number of gates in it, and the depth of the circuit refers to the length of the longest directed path.
The arithmetic circuit calculates the polynomial in a natural way, the input gate calculates its marked polynomial, the addition gate calculates the sum of the polynomials of its children nodes, and the multiplication gate calculates the product of the polynomials of the children nodes.
Upper boundary analysis
In the study of polynomial computational complexity, some clever circuits and algorithms have been found.A famous example is Strassen's matrix multiplication algorithm.Usually calculating the product of two n × n matrices requires a circuit of about n³ size, but Strassen proves that it can be used to calculate using a circuit of about n².807 size.
Computing the determinant of the n × n matrix is also an interesting story.A pure method requires circuits of about n!, but we know that determinants can be calculated using polynomial-sized circuits, and the depths of these circuits are linear.But Berkowitz proposes an improvement that the size of the circuit is still polynomial, but the depth is limited to O(log²(n)).
However, for a permanent n × n matrix, the best known circuit size is about 2^n, which is the depth three circuit provided by Ryser's theorem.
Next Realm Challenge
The knowledge about proof of the lower bound is very limited, especially for polynomials of small degrees.For example, calculating very high levels of polynomials requires large circuits, and our main goal is to prove the lower bound for polynomials of small degrees.A major open problem is to find clear examples of a circuit with a small degree of polynomial but requiring a superpolynomial size.
Although counting arguments tell us that some polynomials of small degrees may also require circuits of superpolynomial size, these results usually fail to deepen our understanding of the computational process.
For example, the lower bound so far can only reach the scale of Ω(n log d), which is mainly reflected in the work of Strassen and Baur and Strassen.
Algebra P and NP problems
The most noticeable open problem in the theory of computational complexity is the P vs. NP problem.Valiant's algebraic analogy problem VP vs. VNP is one of them.VP is an analogy of the polynomial degree principle, while VNP can be regarded as a problem similar to NP.Valiant proves that a permanent polynomial is a complete polynomial of the VNP class. Therefore, if you want to prove that VP is not equal to VNP, you need to prove that there is no circuit with the polynomial size in the permanent polynomial.
The significance of decreasing depth
In our understanding of polynomial calculations, Valiant and other scholars’ research provides important references.They show that if a polynomial has a circuit of size s, its depth can also be reduced to O(log(r) log(s)), which provides reference guidance for solving other similar problems.
This result not only extends Berkowitz's circuit method, but also helps us better understand the calculation of polynomials.
In this rapidly changing era, can we find new ways to gain insight into the structure and complexity of circuit computing to meet the challenges of future computing needs?
