### Abstract

In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is to determine whether the sets intersect. For this problem, we give a protocol that communicates a total of O(k log^{(r)} k) bits over r rounds and errs with very small probability. Here we can take r = log * k to obtain a O(k) total communication log * k-round protocol with exponentially small error probability, improving on the O(k)-bits O(log k)-round constant error probability protocol of Hastad and Wigderson from 1997. In the exists-equal problem, the players receive vectors x, y ∈ [t]n and the goal is to determine whether there exists a coordinate i such that xi = yi. Namely, the exists-equal problem is the OR of n equality problems. Observe that existsequal is an instance of sparse set disjointness with k = n, hence the protocol above applies here as well, giving an O(n log(r) n) upper bound. Our main technical contribution in this paper is a matching lower bound: we show that when t = Ω(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size Ω(n log(r) n). Our lower bound holds even for super-constant r ≤ log * n, showing that any O(n) bits exists-equal protocol should have log* n - O(1) rounds. Note that the protocol we give errs only with less than polynomially small probability and provides guarantees on the total communication for the harder set disjointness problem, whereas our lower bound holds even for constant error probability protocols and for the easier exists-equal problem with guarantees on the maxcommunication. Hence our upper and lower bounds match in a strong sense. Our lower bound on the constant round protocols for existsequal shows that solving the OR of n instances of the equality problems requires strictly more than n times the cost of a single instance. To our knowledge this is the first example of such a super-linear increase in complexity.

Original language | English |
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Title of host publication | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

Pages | 678-687 |

Number of pages | 10 |

DOIs | |

Publication status | Published - 2013 |

Event | 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 - Berkeley, CA, United States Duration: Oct 27 2013 → Oct 29 2013 |

### Other

Other | 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 |
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Country | United States |

City | Berkeley, CA |

Period | 10/27/13 → 10/29/13 |

### Fingerprint

### Keywords

- Communication complexity
- Direct-sum
- Isoperimetric inequality
- Roundelimination

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS*(pp. 678-687). [6686204] https://doi.org/10.1109/FOCS.2013.78

**On the communication complexity of sparse set disjointness and exists-equal problems.** / Saǧlam, Mert; Tardos, G.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS.*, 6686204, pp. 678-687, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013, Berkeley, CA, United States, 10/27/13. https://doi.org/10.1109/FOCS.2013.78

}

TY - GEN

T1 - On the communication complexity of sparse set disjointness and exists-equal problems

AU - Saǧlam, Mert

AU - Tardos, G.

PY - 2013

Y1 - 2013

N2 - In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is to determine whether the sets intersect. For this problem, we give a protocol that communicates a total of O(k log(r) k) bits over r rounds and errs with very small probability. Here we can take r = log * k to obtain a O(k) total communication log * k-round protocol with exponentially small error probability, improving on the O(k)-bits O(log k)-round constant error probability protocol of Hastad and Wigderson from 1997. In the exists-equal problem, the players receive vectors x, y ∈ [t]n and the goal is to determine whether there exists a coordinate i such that xi = yi. Namely, the exists-equal problem is the OR of n equality problems. Observe that existsequal is an instance of sparse set disjointness with k = n, hence the protocol above applies here as well, giving an O(n log(r) n) upper bound. Our main technical contribution in this paper is a matching lower bound: we show that when t = Ω(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size Ω(n log(r) n). Our lower bound holds even for super-constant r ≤ log * n, showing that any O(n) bits exists-equal protocol should have log* n - O(1) rounds. Note that the protocol we give errs only with less than polynomially small probability and provides guarantees on the total communication for the harder set disjointness problem, whereas our lower bound holds even for constant error probability protocols and for the easier exists-equal problem with guarantees on the maxcommunication. Hence our upper and lower bounds match in a strong sense. Our lower bound on the constant round protocols for existsequal shows that solving the OR of n instances of the equality problems requires strictly more than n times the cost of a single instance. To our knowledge this is the first example of such a super-linear increase in complexity.

AB - In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is to determine whether the sets intersect. For this problem, we give a protocol that communicates a total of O(k log(r) k) bits over r rounds and errs with very small probability. Here we can take r = log * k to obtain a O(k) total communication log * k-round protocol with exponentially small error probability, improving on the O(k)-bits O(log k)-round constant error probability protocol of Hastad and Wigderson from 1997. In the exists-equal problem, the players receive vectors x, y ∈ [t]n and the goal is to determine whether there exists a coordinate i such that xi = yi. Namely, the exists-equal problem is the OR of n equality problems. Observe that existsequal is an instance of sparse set disjointness with k = n, hence the protocol above applies here as well, giving an O(n log(r) n) upper bound. Our main technical contribution in this paper is a matching lower bound: we show that when t = Ω(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size Ω(n log(r) n). Our lower bound holds even for super-constant r ≤ log * n, showing that any O(n) bits exists-equal protocol should have log* n - O(1) rounds. Note that the protocol we give errs only with less than polynomially small probability and provides guarantees on the total communication for the harder set disjointness problem, whereas our lower bound holds even for constant error probability protocols and for the easier exists-equal problem with guarantees on the maxcommunication. Hence our upper and lower bounds match in a strong sense. Our lower bound on the constant round protocols for existsequal shows that solving the OR of n instances of the equality problems requires strictly more than n times the cost of a single instance. To our knowledge this is the first example of such a super-linear increase in complexity.

KW - Communication complexity

KW - Direct-sum

KW - Isoperimetric inequality

KW - Roundelimination

UR - http://www.scopus.com/inward/record.url?scp=84893470398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893470398&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2013.78

DO - 10.1109/FOCS.2013.78

M3 - Conference contribution

AN - SCOPUS:84893470398

SN - 9780769551357

SP - 678

EP - 687

BT - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

ER -