Maximum Subarray: Difference between revisions
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== Parameters == | == Parameters == | ||
n: length of array | $n$: length of array | ||
d: dimensionality of array | $d$: dimensionality of array | ||
== Table of Algorithms == | == Table of Algorithms == | ||
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| [[Brute Force (1D Maximum Subarray Maximum Subarray Problem)|Brute Force]] || 1977 || $O(n^{3})$ || $O({1})$ | | [[Brute Force (1D Maximum Subarray Maximum Subarray Problem)|Brute Force]] || 1977 || $O(n^{3})$ || $O({1})$ || Exact || Deterministic || | ||
|- | |- | ||
| [[Grenander (1D Maximum Subarray Maximum Subarray Problem)|Grenander]] || 1977 || $O(n^{2})$ || $O(n)$ || Exact || Deterministic || | | [[Grenander (1D Maximum Subarray Maximum Subarray Problem)|Grenander]] || 1977 || $O(n^{2})$ || $O(n)$ || Exact || Deterministic || | ||
|- | |- | ||
| [[Faster Brute Force (via x(L:U) = x(L:U-1)+x(U)) (1D Maximum Subarray Maximum Subarray Problem)|Faster Brute Force (via x(L:U) = x(L:U-1)+x(U))]] || 1977 || $O(n^{2})$ || $O({1})$ | | [[Faster Brute Force (via x(L:U) = x(L:U-1)+x(U)) (1D Maximum Subarray Maximum Subarray Problem)|Faster Brute Force (via x(L:U) = x(L:U-1)+x(U))]] || 1977 || $O(n^{2})$ || $O({1})$ || Exact || Deterministic || [https://dl.acm.org/doi/pdf/10.1145/358234.381162 Time] | ||
|- | |- | ||
| [[Shamos (1D Maximum Subarray Maximum Subarray Problem)|Shamos]] || 1978 || $O( | | [[Shamos (1D Maximum Subarray Maximum Subarray Problem)|Shamos]] || 1978 || $O(n \log n)$ || $O(\log n)$ || Exact || Deterministic || | ||
|- | |- | ||
| [[Kadane's Algorithm (1D Maximum Subarray Maximum Subarray Problem)|Kadane's Algorithm]] || 1982 || $O(n)$ || $O({1})$ auxiliary || Exact || Deterministic || | | [[Kadane's Algorithm (1D Maximum Subarray Maximum Subarray Problem)|Kadane's Algorithm]] || 1982 || $O(n)$ || $O({1})$ auxiliary || Exact || Deterministic || | ||
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| [[Perumalla and Deo (1D Maximum Subarray Maximum Subarray Problem)|Perumalla and Deo]] || 1995 || $O(log n)$ || $O(n)$ auxiliary || Exact || Parallel || [https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.1291&rep=rep1&type=pdf Time] | | [[Perumalla and Deo (1D Maximum Subarray Maximum Subarray Problem)|Perumalla and Deo]] || 1995 || $O(\log n)$ || $O(n)$ auxiliary || Exact || Parallel || [https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.1291&rep=rep1&type=pdf Time] | ||
|- | |- | ||
| [[Gries (1D Maximum Subarray Maximum Subarray Problem)|Gries]] || 1982 || $O(n)$ || $O({1})$ auxiliary || Exact || Deterministic || [https://www.sciencedirect.com/science/article/pii/0167642383900151?via%3Dihub Time] | | [[Gries (1D Maximum Subarray Maximum Subarray Problem)|Gries]] || 1982 || $O(n)$ || $O({1})$ auxiliary || Exact || Deterministic || [https://www.sciencedirect.com/science/article/pii/0167642383900151?via%3Dihub Time] | ||
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| [[Bird (1D Maximum Subarray Maximum Subarray Problem)|Bird]] || 1989 || $O(n)$ || $O({1})$ auxiliary || Exact || Deterministic || [https://dl.acm.org/doi/10.1093/comjnl/32.2.122 Time] | | [[Bird (1D Maximum Subarray Maximum Subarray Problem)|Bird]] || 1989 || $O(n)$ || $O({1})$ auxiliary || Exact || Deterministic || [https://dl.acm.org/doi/10.1093/comjnl/32.2.122 Time] | ||
|- | |- | ||
| [[Ferreira, Camargo, Song (1D Maximum Subarray Maximum Subarray Problem)|Ferreira, Camargo, Song]] || 2014 || $O(log n)$ || $O(n)$ auxiliary || Exact || Parallel || [https://ieeexplore.ieee.org/document/6972008 Time] | | [[Ferreira, Camargo, Song (1D Maximum Subarray Maximum Subarray Problem)|Ferreira, Camargo, Song]] || 2014 || $O(\log n)$ || $O(n)$ auxiliary || Exact || Parallel || [https://ieeexplore.ieee.org/document/6972008 Time] | ||
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Latest revision as of 08:23, 10 April 2023
Description
Given a $d$-dimensional array $M$ with $n^d$ real-valued entries, find the $d$-dimensional subarray of $M$ which maximizes the sum of the elements it contains.
Related Problems
Subproblem: 1D Maximum Subarray, 2D Maximum Subarray, Maximum Square Subarray
Related: 2D Maximum Subarray, Maximum Square Subarray
Parameters
$n$: length of array
$d$: dimensionality of array
Table of Algorithms
| Name | Year | Time | Space | Approximation Factor | Model | Reference |
|---|---|---|---|---|---|---|
| Brute Force | 1977 | $O(n^{3})$ | $O({1})$ | Exact | Deterministic | |
| Grenander | 1977 | $O(n^{2})$ | $O(n)$ | Exact | Deterministic | |
| Faster Brute Force (via x(L:U) = x(L:U-1)+x(U)) | 1977 | $O(n^{2})$ | $O({1})$ | Exact | Deterministic | Time |
| Shamos | 1978 | $O(n \log n)$ | $O(\log n)$ | Exact | Deterministic | |
| Kadane's Algorithm | 1982 | $O(n)$ | $O({1})$ auxiliary | Exact | Deterministic | |
| Perumalla and Deo | 1995 | $O(\log n)$ | $O(n)$ auxiliary | Exact | Parallel | Time |
| Gries | 1982 | $O(n)$ | $O({1})$ auxiliary | Exact | Deterministic | Time |
| Bird | 1989 | $O(n)$ | $O({1})$ auxiliary | Exact | Deterministic | Time |
| Ferreira, Camargo, Song | 2014 | $O(\log n)$ | $O(n)$ auxiliary | Exact | Parallel | Time |
Reductions TO Problem
| Problem | Implication | Year | Citation | Reduction |
|---|---|---|---|---|
| Distance Product | if: to-time: $O(n^{3-\epsilon})$ for some $\epsilon > {0}$ then: from-time: $O(n^{3-\epsilon})$ |
1998 | https://dl.acm.org/doi/abs/10.5555/314613.314823 | link |
| Negative Triangle Detection | 1998 | https://dl.acm.org/doi/abs/10.5555/314613.314823 | link |
Reductions FROM Problem
| Problem | Implication | Year | Citation | Reduction |
|---|---|---|---|---|
| Negative Triangle Detection | 2018 | https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 5.4 | link | |
| Max-Weight k-Clique | if: to-time: $O(n^{d+\lfloor d/{2}\rfloor-\epsilon})$ for $d$-dimensional hypercube arrays then: from-time: $O(n^{k-\epsilon})$ on $n$ vertex graphs for $k=d+\lfloor d/{2}\rfloor$ |
2016 | https://arxiv.org/pdf/1602.05837.pdf | link |