kb/data/en.wikipedia.org/wiki/3SUM-2.md

---
title: "3SUM"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/3SUM"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:01:54.553179+00:00"
instance: "kb-cron"
---

If there is a solution for 3SUM: 
  
    
      
        z
        =
        x
        +
        y
      
    
    {\displaystyle z=x+y}
  
, then: 
  
    
      
        T
        [
        h
        (
        z
        )
        ]
        =
        T
        [
        h
        (
        x
        )
        ]
        +
        T
        [
        h
        (
        y
        )
        ]
      
    
    {\displaystyle T[h(z)]=T[h(x)]+T[h(y)]}
  
 and 
  
    
      
        h
        (
        z
        )
        =
        h
        (
        x
        )
        +
        h
        (
        y
        )
      
    
    {\displaystyle h(z)=h(x)+h(y)}
  
, so this solution will be found by the Conv3SUM solver on T.
Conversely, if a Conv3SUM is found on T, then obviously it corresponds to a 3SUM solution on S since T is just a permutation of S.
This idealized solution doesn't work, because any hash function might map several distinct elements of S to the same cell of T. The trick is to create an array ⁠
  
    
      
        
          T
          
            ∗
          
        
      
    
    {\displaystyle T^{*}}
  
⁠ by selecting a single random element from each cell of T, and run Conv3SUM on ⁠
  
    
      
        
          T
          
            ∗
          
        
      
    
    {\displaystyle T^{*}}
  
⁠. If a solution is found, then it is a correct solution for 3SUM on S. If no solution is found, then create a different random ⁠
  
    
      
        
          T
          
            ∗
          
        
      
    
    {\displaystyle T^{*}}
  
⁠ and try again. Suppose there are at most R elements in each cell of T. Then the probability of finding a solution (if a solution exists) is the probability that the random selection will select the correct element from each cell, which is 
  
    
      
        (
        1
        
          /
        
        R
        
          )
          
            3
          
        
      
    
    {\displaystyle (1/R)^{3}}
  
. By running Conv3SUM 
  
    
      
        
          R
          
            3
          
        
      
    
    {\displaystyle R^{3}}
  
 times, the solution will be found with a high probability.
Unfortunately, we do not have linear perfect hashing, so we have to use an almost linear hash function, i.e. a function h such that:

  
    
      
        h
        (
        x
        +
        y
        )
        =
        h
        (
        x
        )
        +
        h
        (
        y
        )
      
    
    {\displaystyle h(x+y)=h(x)+h(y)}
  
 or

  
    
      
        h
        (
        x
        +
        y
        )
        =
        h
        (
        x
        )
        +
        h
        (
        y
        )
        +
        1
      
    
    {\displaystyle h(x+y)=h(x)+h(y)+1}
  

This requires to duplicate the elements of S when copying them into T, i.e., put every element 
  
    
      
        x
        ∈
        S
      
    
    {\displaystyle x\in S}
  
 both in 
  
    
      
        T
        [
        h
        (
        x
        )
        ]
      
    
    {\displaystyle T[h(x)]}
  
 (as before) and in 
  
    
      
        T
        [
        h
        (
        x
        )
        ]
        −
        1
      
    
    {\displaystyle T[h(x)]-1}
  
. So each cell will have 2R elements, and we will have to run Conv3SUM 
  
    
      
        (
        2
        R
        
          )
          
            3
          
        
      
    
    {\displaystyle (2R)^{3}}
  
 times.

== 3SUM-hardness ==
A problem is called 3SUM-hard if solving it in subquadratic time implies a subquadratic-time algorithm for 3SUM.  The concept of 3SUM-hardness was introduced by Gajentaan & Overmars (1995). They proved that a large class of problems in computational geometry are 3SUM-hard, including the following ones. (The authors acknowledge that many of these problems are contributed by other researchers.)

Given a set of lines in the plane, are there three that meet in a point?
Given a set of non-intersecting axis-parallel line segments, is there a line that separates them into two non-empty subsets?
Given a set of infinite strips in the plane, do they fully cover a given rectangle?
Given a set of triangles in the plane, compute their measure.
Given a set of triangles in the plane, does their union have a hole?
A number of visibility and motion planning problems, e.g.,
Given a set of horizontal triangles in space, can a particular triangle be seen from a particular point?
Given a set of non-intersecting axis-parallel line segment obstacles in the plane, can a given rod be moved by translations and rotations between a start and finish positions without colliding with the obstacles?
By now there are a multitude of other problems that fall into this category. An example is the decision version of X + Y sorting: given sets of numbers X and Y of n elements each, are there n² distinct x + y for x ∈ X, y ∈ Y?

== See also ==
Subset sum problem

== Notes ==

== References ==
Kane, Daniel M.; Lovett, Shachar; Moran, Shay (2018), "Near-optimal linear decision trees for k-SUM and related problems", Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 554–563, arXiv:1705.01720, doi:10.1145/3188745.3188770, ISBN 9781450355599, S2CID 30368541
Chan, Timothy M. (2020), "More logarithmic-factor speedups for 3SUM, (median,+)-convolution, and some geometric 3SUM-hard problems", ACM Transactions on Algorithms, 16 (1) 7: 1–23, doi:10.1145/3363541, MR 4060405
Grønlund, Allan; Pettie, Seth (2018), "Threesomes, degenerates, and love triangles", Journal of the ACM, 65 (4) 22: 1–25, arXiv:1404.0799, doi:10.1145/3185378, MR 3795516
Freund, Ari (2017), "Improved Subquadratic 3SUM", Algorithmica, 44 (2): 440–458, doi:10.1007/s00453-015-0079-6, S2CID 253979651.
Gold, Omer; Sharir, Micha (2017), "Improved Bounds for 3SUM, k-SUM, and linear degeneracy", in Pruhs, Kirk; Sohler, Christian (eds.), 25th Annual European Symposium on Algorithms, ESA 2017, September 4–6, 2017, Vienna, Austria, LIPIcs, vol. 87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, pp. 42:1–42:13, doi:10.4230/LIPICS.ESA.2017.42, ISBN 978-3-95977-049-1, S2CID 691387
Baran, Ilya; Demaine, Erik D.; Pătraşcu, Mihai (2008), "Subquadratic algorithms for 3SUM", Algorithmica, 50 (4): 584–596, doi:10.1007/s00453-007-9036-3, S2CID 9855995.
Demaine, Erik D.; Mitchell, Joseph S. B.; O'Rourke, Joseph (July 2005), "Problem 11: 3SUM Hard Problems", The Open Problems Project, retrieved 2008-09-02{{citation}}:  CS1 maint: deprecated archival service (link).
Erickson, Jeff (1999), "Lower bounds for linear satisfiability problems", Chicago Journal of Theoretical Computer Science, 1999, MIT Press.
Gajentaan, Anka; Overmars, Mark H. (1995), "On a class of O(n2) problems in computational geometry", Computational Geometry: Theory and Applications, 5 (3): 165–185, doi:10.1016/0925-7721(95)00022-2, hdl:1874/17058.
King, James (2004), A survey of 3SUM-hard problems (PDF).