insight - Algorithms and Complexity

### Analyzing Iterative Algorithms Using Combinatorial Diagrams

Iterative algorithms can be analyzed using a combinatorial diagram basis, which reveals that the asymptotic behavior of these algorithms is dominated by tree-shaped diagrams that represent asymptotically independent Gaussian random variables.

### A Faster Fully Polynomial-Time Randomized Approximation Scheme (FPRAS) for Counting the Number of Strings Accepted by a Non-Deterministic Finite Automaton (NFA)

The authors present a faster fully polynomial-time randomized approximation scheme (FPRAS) for the #NFA problem, which is to determine the size of the set of words of a given length accepted by a non-deterministic finite automaton (NFA). The new FPRAS significantly improves the time complexity compared to the previous state-of-the-art FPRAS.

### Efficient Pseudopolynomial-Time Algorithm for the Knapsack Problem Leveraging Rectangular Monotone Min-Plus Convolution and Balancing

We present a randomized algorithm for the Knapsack problem that runs in time e O(n + t√pmax), where n is the number of items, t is the knapsack capacity, and pmax is the maximum item profit. This improves upon the previous best known e O(n + t · pmax)-time algorithm.

### Efficient Algorithm for Computing Power Series Composition in Near-Linear Time

We present a simple and efficient algorithm that computes the composition of two power series in near-linear time complexity, improving upon the previous best algorithms.