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en/docs/chapter_searching/replace_linear_by_hashing.md
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# Hash optimization strategies # Hash optimization strategies
In algorithm problems, **we often reduce the time complexity of algorithms by replacing linear search with hash search**. Let's use an algorithm problem to deepen understanding. In algorithm problems, **we often reduce an algorithm's time complexity by replacing a linear search with a hash-based search**. Let's use an algorithm problem to deepen the understanding.
!!! question !!! question
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## Linear search: trading time for space ## Linear search: trading time for space
Consider traversing all possible combinations directly. As shown in the figure below, we initiate a two-layer loop, and in each round, we determine whether the sum of the two integers equals `target`. If so, we return their indices. Consider traversing through all possible combinations directly. As shown in the figure below, we initiate a nested loop, and in each iteration, we determine whether the sum of the two integers equals `target`. If so, we return their indices.
![Linear search solution for two-sum problem](replace_linear_by_hashing.assets/two_sum_brute_force.png) ![Linear search solution for two-sum problem](replace_linear_by_hashing.assets/two_sum_brute_force.png)
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[file]{two_sum}-[class]{}-[func]{two_sum_brute_force} [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. This method has a time complexity of $O(n^2)$ and a space complexity of $O(1)$, which becomes very time-consuming with large data volumes.
## Hash search: trading space for time ## Hash search: trading space for time
Consider using a hash table, with key-value pairs being the array elements and their indices, respectively. Loop through the array, performing the steps shown in the figure below each round. Consider using a hash table, where the key-value pairs are the array elements and their indices, respectively. Loop through the array, performing the steps shown in the figure below during each iteration.
1. Check if the number `target - nums[i]` is in the hash table. If so, directly return the indices of these two elements. 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. 2. Add the key-value pair `nums[i]` and index `i` to the hash table.
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[file]{two_sum}-[class]{}-[func]{two_sum_hash_table} [file]{two_sum}-[class]{}-[func]{two_sum_hash_table}
``` ```
This method reduces the time complexity from $O(n^2)$ to $O(n)$ by using hash search, greatly improving the running efficiency. This method reduces the time complexity from $O(n^2)$ to $O(n)$ by using hash search, significantly improving the running efficiency.
As it requires maintaining an additional hash table, the space complexity is $O(n)$. **Nevertheless, this method has a more balanced time-space efficiency overall, making it the optimal solution for this problem**. As it requires maintaining an additional hash table, the space complexity is $O(n)$. **Nevertheless, this method has a more balanced time-space efficiency overall, making it the optimal solution for this problem**.