hello-algo/en/docs/chapter_searching/replace_linear_by_hashing.md

Hash Optimization Strategy

In algorithm problems, we often reduce the time complexity of algorithms by replacing linear search with hash-based search. Let's use an algorithm problem to deepen our understanding.

!!! question

Given an integer array `nums` and a target element `target`, search for two elements in the array whose "sum" equals `target`, and return their array indices. Any solution will do.

Linear Search: Trading Time for Space

Consider directly traversing all possible combinations. As shown in the figure below, we open a two-layer loop and judge in each round whether the sum of two integers equals target. If so, return their indices.

The code is shown below:

[file]{two_sum}-[class]{}-[func]{two_sum_brute_force}

This method has a time complexity of O(n^2) and a space complexity of O(1), which is very time-consuming with large data volumes.

Hash-Based Search: Trading Space for Time

Consider using a hash table where key-value pairs are array elements and element indices respectively. Loop through the array, performing the steps shown in the figure below in each round:

1. Check if the number target - nums[i] is in the hash table. If so, directly return the indices of these two elements.
2. Add the key-value pair nums[i] and index i to the hash table.

=== "<1>"

=== "<2>"

=== "<3>"

The implementation code is shown below, requiring only a single loop:

[file]{two_sum}-[class]{}-[func]{two_sum_hash_table}

This method reduces the time complexity from O(n^2) to O(n) through hash-based search, greatly improving runtime efficiency.

Since an additional hash table needs to be maintained, the space complexity is O(n). Nevertheless, this method achieves a more balanced overall time-space efficiency, making it the optimal solution for this problem.