Freaks City

Problem A: Digit Frequency Analysis Given a functon (f(x)) that counts the frequency of the most common digit in number (x), compute (\sum_{i=l}^{r}f(i)) for large ranges (up to (10^{18})). Approach: Represent digit counts as a state vector (S = {c_0, \dots, c_9}) Transform into frequency-of-counts representation (S' = {a_0, \dots, a_{18}}) Us ...

Probabilistic Path Enumeration in Directed Acyclic Graphs Given a directed acyclic graph (DAG) consisting of $V$ vertices and $E$ edges, each edge $e_k$ possesses an independent activation probability $p_k$. The objective is to compute the expected number of functional paths originating from vertex $0$ and terminating at vertex $V-1$. A path is ...