The communication complexity of enumeration, elimination, and selection

Ambainis, Andris; Buhrman, Harry; Gasarch, William; Kalyanasundaram, Bala; Torenvliet, Leen

doi:10.1109/ccc.2000.856734

Cited by 5 publications

(8 citation statements)

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“…We suspect that a random function f (x, y), on an input size sufficiently large compared to k and 1 δ , should be a suitable inner function. Our conjecture also appears somewhat related to the Enumeration and Elimination Conjectures of [1] (so far unresolved). These are another type of variant of the Direct Sum Conjecture of [8].…”

Section: Our Resultsmentioning

confidence: 69%

Multitask Efficiencies in the Decision Tree Model

Drucker¹

2009

2009 24th Annual IEEE Conference on Computational Complexity

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In Direct Sum problems [8], one tries to show that for a given computational model, the complexity of computing a collection F = {f 1 (x 1 ), . . . f l (x l )} of finite functions on independent inputs is approximately the sum of their individual complexities. In this paper, by contrast, we study the diversity of ways in which the joint computational complexity can behave when all the f i are evaluated on a common input. We focus on the deterministic decision tree model, with depth as the complexity measure; in this model we prove a result to the effect that the 'obvious' constraints on joint computational complexity are essentially the only ones.The proof uses an intriguing new type of cryptographic data structure called a 'mystery bin' which we construct using a small polynomial separation between deterministic and unambiguous query complexity shown by Savický. We also pose a variant of the Direct Sum Conjecture of [8] which, if proved for a single family of functions, could yield an analogous result for models such as the communication model.

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Section: Our Resultsmentioning

confidence: 69%

Multitask Efficiencies in the Decision Tree Model

Drucker¹

2009

2009 24th Annual IEEE Conference on Computational Complexity

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“…Hence, an ensemble of states, called an ǫ-approximate state t-design Υ (ǫ) t has been studied [21,[25][26][27][28][29][30][31][32][33]]. An ǫ-approximate state t-design is defined by [18,19].…”

Section: Random States and State T-designmentioning

confidence: 99%

“…We especially investigate the ensemble of thermal states in comparison with the unitarily invariant ensemble. To this end, we exploit the concept of a state t-design, an ensemble of states simulating, up to the or-der t, statistical moments of random states [18,19], and investigate whether or not a state t-design is approximately achievable in random global/local Hamiltonian systems at finite temperature. This provides an insight into the validity of the foundation of physics using random states or a state t-design when the system respects a local structure and is at finite temperature.…”

Section: Introductionmentioning

confidence: 99%

Thermal states of random quantum many-body systems

Nakata

Osborne

2014

Phys. Rev. A

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We study a distribution of thermal states given by random Hamiltonians with a local structure. We show that the ensemble of thermal states monotonically approaches the unitarily invariant ensemble with decreasing temperature if all particles interact according to a single random interaction and achieves a state $t$-design at temperature $O(1/\log(t))$. For the system where the random interactions are local, we show that the ensemble achieves a state $1$-design. We then provide numerical evidence indicating that the ensemble undergoes a phase transition at finite temperature.Comment: 5 pages, 4 figure

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“…Namely, in these cases no significant savings is possible. For general two-party deterministic protocols, this fundamental question (stated as a conjecture in [2]) remains open: Question 1. Is it true that, for all functions f : {0, 1} n × {0, 1} n → {0, 1}, only a minor saving is possible; formally,…”

Section: Introductionmentioning

confidence: 99%

On the Power of Multiplexing in Number-on-the-Forehead Protocols

Aharoni¹,

Kushilevitz²

2014

Preprint

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We study the direct-sum problem for k-party "Number On the Forehead" (NOF) deterministic communication complexity. We prove several positive results, showing that the complexity of computing a function f in this model, on ℓ instances, may be significantly cheaper than ℓ times the complexity of computing f on a single instance. Quite surprisingly, we show that this is the case for "most" (boolean, k-argument) functions. We then formalize two-types of sufficient conditions on a NOF protocol Q, for a single instance, each of which guarantees some communication complexity savings when appropriately extending Q to work on ℓ instances. One such condition refers to what each party needs to know about inputs of the other parties, and the other condition, additionally, refers to the communication pattern that the single-instance protocol Q uses. In both cases, the tool that we use is "multiplexing": we combine messages sent in parallel executions of protocols for a single instance, into a single message for the multi-instance (direct-sum) case, by xoring them with each other.

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The communication complexity of enumeration, elimination, and selection

Cited by 5 publications

References 50 publications

Multitask Efficiencies in the Decision Tree Model

Multitask Efficiencies in the Decision Tree Model

Thermal states of random quantum many-body systems

On the Power of Multiplexing in Number-on-the-Forehead Protocols

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