Freaks City

Given (n,c,k), compute (\sum_{i=0}^n f_{ic}^k) modulo (p=10^9+9), where (f_i) is the (i)-th Fibonacci number. Constraitns: (n,c\in[1,10^{18}]), (k\le 10^5), (T\le 200), time limit 3 seconds. Use the closed-form expression for Fibonacci numbers: (f_i = \frac{1}{\sqrt5}\left(\alpha^i - \beta^i\right)), where (\alpha = \frac{1+\sqrt5}{2}), (\beta ...

The ZKW segment tree is a non-recursive data structure that performs bottom-up updates and queries. It is often used as a replacement for Fenwick trees when range updates and range queries are needed. The key concept is to use two "shrinking" pointers (l and r) that move upward from the leaf level. To range updates with lazy tags, we ...

Core Concept of Dynamic Programming Dynamic Programming (DP) is an algorithmic technique used when a problem exhibits overlapping subproblems and optimal substructure. Unlike greedy algorithms—which make locally optimal choices without considering previous states—DP builds solutions incrementally, where each state is derived from one or more pr ...