Given an unsorted array can we make a sorting algorithm with O(n) time complexity and O(1) space complexity

DeepSeek V3.1

Vote: No

Sorting an unsorted array with O(n) time and O(1) space complexity is impossible for general data. The lower bound for comparison-based sorting is Ω(n log n), meaning any such algorithm must perform at least that many comparisons in the worst case. Non-comparison sorts like Counting Sort or Radix Sort can achieve O(n) time but require O(k) or O(n) extra space, violating the O(1) space constraint. Thus, no algorithm can sort arbitrary data with both O(n) time and O(1) space.

GPT-5.4

Vote: No

For a general unsorted array of arbitrary values, comparison-based sorting has a lower bound of O(n log n) time. So you cannot have a general-purpose sorting algorithm that guarantees O(n) time with only O(1) extra space. There are special-case linear-time sorts like counting sort or radix sort, but they require extra assumptions about the data, such as bounded integer ranges or fixed-length keys. They also typically use extra memory beyond O(1). So in the usual general case, the answer is no.

Claude Haiku 4.5

Vote: No

No, we cannot create a sorting algorithm that achieves both O(n) time complexity and O(1) space complexity for a general unsorted array. The fundamental issue is the information-theoretic lower bound for comparison-based sorting. Any comparison-based sorting algorithm requires at least Ω(n log n) comparisons in the worst case to sort n elements, which translates to Ω(n log n) time complexity. This is a proven mathematical limit, not just a limitation of current algorithms. While O(n) sorting algorithms do exist (like counting sort, radix sort, and bucket sort), they all require additional space complexity beyond O(1). For example, counting sort requires O(k) extra space where k is the range of input values, and radix sort requires O(n + k) space. To achieve O(1) space with general comparison-based sorting, you are constrained to in-place algorithms like quicksort, heapsort, or insertion sort, all of which have at best O(n log n) time complexity for the general case. Therefore, for a general unsorted array without constraints on the data, O(n) time and O(1) space simultaneously is not achievable.

Grok 4.1 Fast

Vote: No

Sorting an unsorted array of arbitrary comparable elements requires Ω(n log n) time in the worst case, as proven by the decision tree lower bound for comparison-based sorting algorithms. This holds because distinguishing among n! possible permutations requires at least log₂(n!) ≈ n log n bits of information, and each comparison yields at most 1 bit. Non-comparison-based sorts like radix or counting sort achieve O(n) time only under restrictive assumptions (e.g., fixed-size integers with bounded range), but even then, they typically require O(n) or O(k) extra space, violating O(1) space. No known algorithm sorts general arrays in O(n) time with O(1) space. In-place algorithms like heapsort or quicksort achieve O(1) space but take O(n log n) time. Thus, it's impossible.

Claude Opus 4.6

Vote: No

In the general case of comparison-based sorting, the theoretical lower bound is O(n log n) time complexity. No comparison-based sorting algorithm can do better than this, regardless of space constraints. Non-comparison-based sorts like counting sort or radix sort can achieve O(n) time complexity, but they require O(k) or O(n) additional space, not O(1). For example, counting sort needs an auxiliary array proportional to the range of values, and radix sort similarly needs extra space for the buckets. There is no known general-purpose sorting algorithm that achieves both O(n) time and O(1) space simultaneously. The question asks about an unsorted array in general, without specifying constraints on the data (like a bounded range of integers), so we must consider the general case. Even if we consider specialized algorithms, achieving O(n) time with O(1) space for arbitrary data remains impossible under standard computational models. The information-theoretic lower bound of O(n log n) for comparison-based sorting cannot be bypassed with constant space alone.

Kimi K2.5

Vote: No

No, this is impossible for general-purpose sorting. Comparison-based sorting algorithms have a proven lower bound of Ω(n log n) time complexity, making O(n) time impossible for comparison sorts. While non-comparison sorts like Counting Sort or Radix Sort can achieve O(n) time complexity, they require O(k) or O(n) auxiliary space where k is the range of input values. The O(1) space constraint rules out these approaches since we cannot allocate additional storage proportional to the input size or value range. The only exception would be highly constrained cases (e.g., a permutation of 1..n), but for an arbitrary unsorted array, achieving both O(n) time and O(1) space simultaneously violates established computational complexity bounds.

Gemini 2.5 Flash

Vote: No

For a general unsorted array, it is not possible to create a sorting algorithm with both O(n) time complexity and O(1) space complexity. First, consider comparison-based sorting algorithms. The theoretical lower bound for such algorithms is O(n log n). Algorithms like Heap Sort achieve O(n log n) time complexity and O(1) auxiliary space complexity (in-place sorting). Therefore, an O(n) time complexity cannot be achieved by comparison-based methods while maintaining O(1) space. Second, consider non-comparison-based sorting algorithms, such as Counting Sort or Radix Sort. These algorithms can achieve O(n) time complexity under certain conditions. However, they typically require auxiliary space that depends on the range of input values (k for Counting Sort) or the number of digits/buckets (for Radix Sort/Bucket Sort). This auxiliary space is often O(k) or O(n), which is not O(1) unless the range of values (k) is a constant, independent of the input size 'n' and the maximum possible value an element can take. For a general unsorted array, the range of values is usually not a constant. In summary, algorithms that achieve O(n) time usually require O(n) or O(k) space, and algorithms that achieve O(1) space usually have a time complexity of at least O(n log n) (for comparison-based sorts). There are very specific cases (e.g., sorting an array of only 0s, 1s, and 2s) where O(n) time and O(1) space are possible, but these are not general sorting algorithms for arbitrary unsorted arrays.

Gemini 3.1 Pro

Vote: No

For a general unsorted array, it is impossible to achieve a sorting algorithm with both O(n) time complexity and O(1) space complexity. First, any comparison-based sorting algorithm has a mathematically proven lower time bound of Ω(n log n). Therefore, no comparison sort can run in O(n) time. Second, non-comparison-based sorting algorithms (like Counting Sort, Radix Sort, or Bucket Sort) can achieve O(n) time complexity under certain conditions. However, they inherently require extra auxiliary memory to map or count elements, leading to O(n) or O(k) space complexity, which violates the strictly O(1) space requirement. While highly restricted arrays (e.g., arrays with only 0s, 1s, and 2s) can be sorted in O(n) time and O(1) space using algorithms like the Dutch National Flag approach, a general sorting algorithm cannot meet both constraints simultaneously.

Final Summary

The AI Roundtable reached a rare state of absolute mathematical harmony, with all eight models concluding that a general-purpose O(n) time and O(1) space sorting algorithm is impossible. From Claude Opus 4.6 to Gemini 2.5 Flash, the group highlighted the inescapable tension between the Ω(n log n) lower bound for comparison-based sorts and the high memory requirements of linear-time alternatives like Radix Sort.

No agreement after 1 rounds of discussion. Final vote: No: 8

Strongest Arguments

• No: The information-theoretic lower bound for comparison-based sorting is Ω(n log n) because distinguishing among n! permutations requires at least log₂(n!) ≈ n log n bits of information; meanwhile, non-comparison sorts that achieve O(n) time require auxiliary space proportional to the data range or input size, violating the O(1) constraint.