LZ77压缩算法的时间与空间复杂度是什么？如何以最优时间与空间复杂度实现该算法

Hey there! Let's break down the time and space complexity of LZ77, plus actionable technical details to help you implement it with optimal efficiency—since you're targeting the best possible performance, I'll get into the specifics that matter most.

First, let's start with the basics, then move to optimized implementations:

• Naive Implementation: Runs in O(n²) time in the worst case. Here's why: for each character in the input (total n characters), you scan the entire sliding history window (size W) to find the longest matching substring. If W is close to n (e.g., no repeated patterns), this becomes O(n*W) ≈ O(n²).
• Optimized Implementation: Can be brought down to O(n) or O(n log W) time with smart data structures:
  • Using a suffix automaton built incrementally for the history window lets you find the longest matching substring in O(L) time per step (L is the lookahead buffer size), leading to an overall O(n) runtime.
  • Rolling hash (Rabin-Karp) computes hash values for all substrings in the history window and lookahead buffer in constant time per step. Comparing hashes lets you quickly find potential matches, with a small overhead for verifying actual substrings to avoid hash collisions—this brings runtime to O(n) on average.
  • A hash table mapping characters to linked lists of their positions lets you start checking matches from the most recent occurrence first, cutting down on unnecessary comparisons. This typically results in O(n log W) time in the worst case, but often performs better in practice.

The space complexity is dominated by the sliding window components:

• Baseline: O(W + L), where W is the size of the history window and L is the lookahead buffer size. Since L is usually much smaller than W (common values are W=32KB, L=256B), this simplifies to O(W).
• Optimized Implementations: Most efficient data structures add only linear overhead relative to W. For example, a suffix automaton uses O(W) space, a rolling hash table uses O(W) space for storing hash values, so the overall space remains O(W)—no extra exponential costs here.

Here are concrete steps to keep both time and space tight:

• Use a circular buffer for the history window: Instead of shifting elements every time the window moves (which is O(W) per step), overwrite the oldest element in a fixed-size array. This keeps window updates at O(1) time and avoids unnecessary memory copies.
• Early termination for no matches: If the first character in the lookahead buffer doesn't exist in the history window, immediately output a (0, 0, c) token without searching further—this saves a lot of wasted cycles on non-repeating data.
• Limit match length checks: Once you find a match of length equal to the lookahead buffer size, stop searching—you can't get a longer match than that. Similarly, if the remaining input is shorter than the current longest match, terminate early.
• Avoid redundant comparisons: When using position lists from a hash table, start checking from the most recent occurrence of the character first. If you find a partial match, extend it as far as possible before moving to older positions—this minimizes the number of character-by-character comparisons.

内容的提问来源于stack exchange，提问作者Mahammad Ayman