What is the time complexity of the iterative constructive method for solving a system of congruences?

Conditions

• The moduli are pairwise coprime.
• Naive iteration is used to find the multiplier for each step.

Reasoning, step by step

1. Identify the number of congruences k and their moduli $m_1$, $m_2$, ..., $m_k$.
2. Recognize that solving the first congruence takes $O(1)$ time.
3. For each subsequent congruence i (from 2 to k), the search takes at most $m_i$ steps.
4. Sum the maximum steps for all stages to get the total complexity O(sum_{$i=2$}^$k m_i$).

Example

The video explicitly writes '$O(m_2 + m_3 + m_4)$' on the board and states, 'From a programming perspective, this defines the algorithm's efficiency. The time complexity is determined by the sum of the moduli... since each stage requires at most $m_i$ iterations.'

Common misconceptions

• Assuming the complexity is O($m_1 * m_2$ * ... * $m_k$).
• Forgetting that the first modulus does not contribute to the summation in the same way as the subsequent ones in this specific iterative formulation.

Watch the explanation

Connected concepts

Explore next

• Why is the search bounded by the modulus at each stage?
• How does the remainder trick improve practical computation?
• What happens to the complexity if the moduli are not coprime?