3SUM (3SUM)
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Description
Given a set $S$ of integers, determine whether there is a subset of $S$ of size 3 that sums to 0.
Related Problems
Generalizations: Real 3SUM
Subproblem: 3SUM', All-Integers 3SUM
Related: All-Integers 3SUM
Parameters
S: the set of integers
n: the number of integers in the set
Table of Algorithms
| Name | Year | Time | Space | Approximation Factor | Model | Reference |
|---|---|---|---|---|---|---|
| Textbook SortandBinarySearch | - | $O(n^{2} log n)$ | $O(n)$ | Exact | Deterministic | |
| Textbook SortandTwoSidedTraversal | - | $O(n^{2})$ | $O(n)$ | Exact | Deterministic | |
| Baran, Demaine, Patrascu | 2008 | $O(n^{2}/max(w/(log w)$^{2}, (log n)^{2}/(log log n)^{2})) | Exact | Randomized | Time | |
| Baran, Demaine, Patrascu | 2008 | $O(n^{2}/(w^{2}/(log w)$^{2})) | Exact | Randomized | Time | |
| Baran, Demaine, Patrascu | 2008 | $O(n^{2}/MB)$ | Exact | Randomized | Time | |
| Baran, Demaine, Patrascu | 2008 | $O(n^{2}*(log M)$^{2}/MB) | Exact | Randomized | Time | |
| Gronlund, Pettie | 2014 | $O(n^{2}/((log n)$/(log log n))^{2}/{3}) | Exact | Deterministic | Time | |
| Gronlund, Pettie | 2014 | $O(n^{2}*(log log n)$^{2}/(log n)) | Exact | Randomized | Time | |
| Freund | 2017 | $O(n^{2}*(log log n)$/(log n)) | Exact | Deterministic | Time | |
| Chan | 2018 | $O(n^{2}*(log log n)$^{$O({1})$}/(log n)^{2}) | Exact | Deterministic | Time |
Reductions TO Problem
| Problem | Implication | Year | Citation | Reduction |
|---|---|---|---|---|
| 3SUM' | if: to-time $N^{2-\epsilon}$ for some $\epsilon > {0}$ then: from-time: $N^{2-\epsilon'}$ for some $\epsilon' > {0}$ |
1995 | https://doi-org.ezproxy.canberra.edu.au/10.1016/0925-7721(95)00022-2 | link |
| 3 Points on Line | if: to-time $N^{2-\epsilon}$ for some $\epsilon > {0}$ then: from-time: $N^{2-\epsilon'}$ for some $\epsilon' > {0}$ |
1995 | https://doi-org.ezproxy.canberra.edu.au/10.1016/0925-7721(95)00022-2 | link |
| Local Alignment | if: to-time $N^{2-\delta-\epsilon} for two strings of size $n$ and alphabet of size $n^{1-\delta}$ for some $\espilon > {0}$,$\delta \in ({0},{1})$ then: from-time: $n^{2-\epsilon'}$ for some $\epsilon' > {0}$ |
2014 | https://link-springer-com.ezproxy.canberra.edu.au/chapter/10.1007/978-3-662-43948-7_4 | link |
| All-Integers 3SUM | if: to-time: $T(n)$ then: from-time: $O(T(n))$ |
link |
Reductions FROM Problem
| Problem | Implication | Year | Citation | Reduction |
|---|---|---|---|---|
| 3SUM' | if: to-time $N^{2-\epsilon}$ for some $\epsilon > {0}$ then: from-time: $N^{2-\epsilon'}$ for some $\epsilon' > {0}$ |
1995 | https://doi-org.ezproxy.canberra.edu.au/10.1016/0925-7721(95)00022-2 | link |
| All-Integers 3SUM | if: to-time: $O(n^{2-\epsilon})$ for some $\epsilon > {0}$ then: from-time: $O(n^{1.5} + n^{2-\epsilon / 2})$ |
2018 | https://dl-acm-org.ezproxy.canberra.edu.au/doi/pdf/10.1145/3186893, Theorem 8.1 | link |
References/Citation
https://link-springer-com.ezproxy.canberra.edu.au/article/10.1007/s00453-007-9036-3