> For the complete documentation index, see [llms.txt](https://liuzhenglaichn.gitbook.io/algorithm/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://liuzhenglaichn.gitbook.io/algorithm/prefix-state-map.md).

# Prefix State Map

I haven't found an official name of this kind of problem. I just named it as "Prefix State Map".

This kind of problem is usually:

* about finding the longest/shortest subarray which satisfies some condition.
* the state of the subarray can be deduced using the the states of the prefix array. For example, to get the state of `A[i..j]`, you can use the state of two prefix array, `A[0..(i-1)]` and `A[0..j]`.

## Algorithm

Let `m` be a map from the state `state[i]` of a prefix subarray `A[0..i]` to the index of the first/last occurrence of the state `state[i]`.

For the current index `i` and its state `state[i]`, assume we want to get a subarray which ends at `i` and has a `target_state` and has the longest/shortest length.

Then we can use `state[i]` and `target_state` to compute a `prefix_state`. `j = m[prefix_state]` is the corresponding index of the prefix array. `A[(j+1) .. i]` is the longest/shortest optimal subarray.

## Pseudo Code

```cpp
int ans = 0; // the length of the optimal subarray
int state = 0; // assume the initial state is 0
unordered_map<int, int> m{{0, -1}}; // let -1 be the index of state `0`.
for (int i = 0; i < A.size(); ++i) {
    state = nextState(state, A[i]); // update state using A[i]
    int prefixState = getPrefixState(state); // based on problem description, we can compute a prefix state using the current state.
    ans = max(ans, i - m[prefixState]); // Use `min` if we are looking for the shortest subarray.
    m[state] = min(m[state], i); // Use `max` if we are looking for the shortest subarray.
}
```

## Problems

* [325. Maximum Size Subarray Sum Equals k (Medium)](https://leetcode.com/problems/maximum-size-subarray-sum-equals-k/)
* [525. Contiguous Array (Medium)](https://leetcode.com/problems/contiguous-array/)
* [974. Subarray Sums Divisible by K (Medium)](https://leetcode.com/problems/subarray-sums-divisible-by-k/)
* [1371. Find the Longest Substring Containing Vowels in Even Counts (Medium)](https://leetcode.com/problems/find-the-longest-substring-containing-vowels-in-even-counts/)
* [1542. Find Longest Awesome Substring (Hard)](https://leetcode.com/problems/find-longest-awesome-substring/)
* [1590. Make Sum Divisible by P (Medium)](https://leetcode.com/problems/make-sum-divisible-by-p)


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter:

```
GET https://liuzhenglaichn.gitbook.io/algorithm/prefix-state-map.md?ask=<question>&goal=<endgoal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.