Count White Balls After K Steps
Problem Description
You are given a ring of N positions, indexed from 0 to N-1. Each position i initially contains one ball, whose color is either white or black. Additionally, each position i may contain a flip gate, indicated by gate[i] (where 1 means a gate is present and 0 means no gate).
In one synchronous step, every ball moves one position clockwise. Whenever a ball crosses a position containing a flip gate, its color toggles. A ball starting at index i will be located at (i + K) mod N after K steps, and its final color depends solely on whether it crosses an even or odd number of gates during those K moves.
For each query, you are provided with an integer K. Your task is to determine the total number of white balls after exactly K steps. Note that K may be significantly larger than N, making a step-by-step simulation approach too slow.
Design an algorithm that efficiently preprocesses the ring and answers each query, ideally in O(1) time per query after the initial preprocessing phase.
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Core dilemma
The central engineering conflict is efficiently calculating the cumulative effect of state changes (ball color toggles) over a potentially vast number of steps (`K`) in a cyclic system, without resorting to slow step-by-step simulation, while ensuring `O(1)` query time after preprocessing. This requires identifying and leveraging periodicity or mathematical properties of the system's evolution.
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Count White Balls After K Steps
Problem Description
You are given a ring of N positions, indexed from 0 to N-1. Each position i initially contains one ball, whose color is either white or black. Additionally, each position i may contain a flip gate, indicated by gate[i] (where 1 means a gate is present and 0 means no gate).
In one synchronous step, every ball moves one position clockwise. Whenever a ball crosses a position containing a flip gate, its color toggles. A ball starting at index i will be located at (i + K) mod N after K steps, and its final color depends solely on whether it crosses an even or odd number of gates during those K moves.
For each query, you are provided with an integer K. Your task is to determine the total number of white balls after exactly K steps. Note that K may be significantly larger than N, making a step-by-step simulation approach too slow.
Design an algorithm that efficiently preprocesses the ring and answers each query, ideally in O(1) time per query after the initial preprocessing phase.