scholarly journals On the Distributed Complexity of Large-Scale Graph Computations

2021 ◽  
Vol 8 (2) ◽  
pp. 1-28
Author(s):  
Gopal Pandurangan ◽  
Peter Robinson ◽  
Michele Scquizzato
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Distributed Algorithm ◽  
Message Passing ◽  
Large Scale ◽  
Communication Complexity ◽  
Minimal Amount ◽  
Theoretic Approach ◽  
Graph Problems ◽  
Round Complexity

Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower bounds for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds.

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

Lower Bounds on OBDD Proofs with Several Orders

2021 ◽  
Vol 22 (4) ◽  
pp. 1-30
Author(s):  
Sam Buss ◽  
Dmitry Itsykson ◽  
Alexander Knop ◽  
Artur Riazanov ◽  
Dmitry Sokolov
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Multiparty Communication ◽  
Constant Degree ◽  
Polynomial Size ◽  
Explicit Constant ◽  
Open Question ◽  
System 1 ◽  
Exponential Size

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

A LOWER BOUND FOR ORDER-PRESERVING BROADCAST IN THE POSTAL MODEL

1993 ◽  
Vol 03 (04) ◽  
pp. 313-320 ◽  
Author(s):  
PHILIP D. MACKENZIE
Keyword(s):  
Communication Network ◽  
Lower Bound ◽  
Lower Bounds ◽  
Message Passing ◽  
Upper Bounds ◽  
Communication Latency ◽  
Multiple Messages ◽  
Postal Model ◽  
Parameter Ranges

In the postal model of message passing systems, the actual communication network between processors is abstracted by a single communication latency factor, which measures the inverse ratio of the time it takes for a processor to send a message and the time that passes until the recipient receives the message. In this paper we examine the problem of broadcasting multiple messages in an order-preserving fashion in the postal model. We prove lower bounds for all parameter ranges and show that these lower bounds are within a factor of seven of the best upper bounds. In some cases, our lower bounds show significant asymptotic improvements over the previous best lower bounds.

scholarly journals A new lower bound for the smallest singular value

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2711-2723
Author(s):  
Ksenija Doroslovacki ◽  
Ljiljana Cvetkovic ◽  
Ernest Sanca
Keyword(s):  
Lower Bound ◽  
Boundary Value Problems ◽  
Lower Bounds ◽  
Large Scale ◽  
Block Structure ◽  
Upper Bounds ◽  
Singular Value ◽  
Ecological Modelling ◽  
Infinity Norm ◽  
Smallest Singular Value

The aim of this paper is to obtain new lower bounds for the smallest singular value for some special subclasses of nonsingularH-matrices. This is done in two steps: first, unifying principle for deriving new upper bounds for the norm 1 of the inverse of an arbitrary nonsingular H-matrix is presented, and then, it is combined with some well-known upper bounds for the infinity norm of the inverse. The importance and efficiency of the results are illustrated by an example from ecological modelling, as well as on a type of large-scale matrices posessing a block structure, arising in boundary value problems.

scholarly journals Lower Bounds for Dynamic Algebraic Problems

1998 ◽  
Vol 5 (11) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Johan P. Hansen ◽  
Peter Bro Miltersen
Keyword(s):  
Fourier Transform ◽  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Symmetric Functions ◽  
Matrix Multiplication ◽  
Worst Case ◽  
Dynamic Matrix ◽  
Wide Range ◽  
On Line

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form "change input xi to value v" or "what is the value of output yi?". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent<br />algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal  Omega(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an <br />Omega(sqrt(n)) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and<br />Tate's O(sqrt(n log n)) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Omega(sqrt(n)) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting<br />of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an ((log n)2= log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields.

On the role of shared entanglement

2008 ◽  
Vol 8 (1&2) ◽  
pp. 82-95
Author(s):  
D. Gavinsky
Keyword(s):  
Lower Bounds ◽  
Message Passing ◽  
Communication Complexity ◽  
Optimal Solution ◽  
Famous Theorem ◽  
Trade Offs ◽  
Wide Range ◽  
Bounded Error ◽  
Strong Direct Product Theorem ◽  
Apparent Similarity

Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former. In particular, there is no known ``entanglement analogue'' for the famous theorem by Newman, saying that the number of shared random bits required for solving any communication problem can be at most logarithmic in the input length (i.e., using more than $\asO[]{\log n}$ shared random bits would not reduce the complexity of an optimal solution). In this paper we prove that the same is not true for entanglement. We establish a wide range of tight (up to a polylogarithmic factor) entanglement vs.\ communication trade-offs for relational problems. The low end is:\ for any $t>2$, reducing shared entanglement from $log^tn$ to $\aso[]{log^{t-2}n}$ qubits can increase the communication required for solving a problem almost exponentially, from $\asO[]{log^tn}$ to $\asOm[]{\sqrt n}$. The high end is:\ for any $\eps>0$, reducing shared entanglement from $n^{1-\eps}\log n$ to $\aso[]{n^{1-\eps}/\log n}$ can increase the required communication from $\asO[]{n^{1-\eps}\log n}$ to $\asOm[]{n^{1-\eps/2}/\log n}$. The upper bounds are demonstrated via protocols which are \e{exact} and work in the \e{simultaneous message passing model}, while the lower bounds hold for \e{bounded-error protocols}, even in the more powerful \e{model of 1-way communication}. Our protocols use shared EPR pairs while the lower bounds apply to any sort of prior entanglement. We base the lower bounds on a strong direct product theorem for communication complexity of a certain class of relational problems. We believe that the theorem might have applications outside the scope of this work.

scholarly journals Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

2021 ◽  
Vol 13 (4) ◽  
pp. 1-37
Author(s):  
Valentine Kabanets ◽  
Sajin Koroth ◽  
Zhenjian Lu ◽  
Dimitrios Myrisiotis ◽  
Igor C. Oliveira
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Communication Complexity ◽  
Inner Product ◽  
Threshold Function ◽  
Seed Length ◽  
Average Case ◽  
Linear Threshold ◽  
Product Function

The class FORMULA[s]∘G consists of Boolean functions computable by size- s De Morgan formulas whose leaves are any Boolean functions from a class G. We give lower bounds and (SAT, Learning, and pseudorandom generators ( PRG s )) algorithms for FORMULA[n 1.99 ]∘G, for classes G of functions with low communication complexity . Let R (k) G be the maximum k -party number-on-forehead randomized communication complexity of a function in G. Among other results, we show the following: • The Generalized Inner Product function GIP k n cannot be computed in FORMULA[s]° G on more than 1/2+ε fraction of inputs for s=o(n 2 /k⋅4 k ⋅R (k) (G)⋅log⁡(n/ε)⋅log⁡(1/ε)) 2 ). This significantly extends the lower bounds against bipartite formulas obtained by [62]. As a corollary, we get an average-case lower bound for GIP k n against FORMULA[n 1.99 ]∘PTF k −1 , i.e., sub-quadratic-size De Morgan formulas with degree-k-1) PTF ( polynomial threshold function ) gates at the bottom. Previously, it was open whether a super-linear lower bound holds for AND of PTFs. • There is a PRG of seed length n/2+O(s⋅R (2) (G)⋅log⁡(s/ε)⋅log⁡(1/ε)) that ε-fools FORMULA[s]∘G. For the special case of FORMULA[s]∘LTF, i.e., size- s formulas with LTF ( linear threshold function ) gates at the bottom, we get the better seed length O(n 1/2 ⋅s 1/4 ⋅log⁡(n)⋅log⁡(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n halfspaces in the regime where ε≤1/n, complementing a recent result of [45]. • There exists a randomized 2 n-t #SAT algorithm for FORMULA[s]∘G, where t=Ω(n\√s⋅log 2 ⁡(s)⋅R (2) (G))/1/2. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n 1.99 ]∘LTF. • The Minimum Circuit Size Problem is not in FORMULA[n 1.99 ]∘XOR; thereby making progress on hardness magnification, in connection with results from [14, 46]. On the algorithmic side, we show that the concept class FORMULA[n 1.99 ]∘XOR can be PAC-learned in time 2 O(n/log n) .

scholarly journals Robust Bell inequalities from communication complexity

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 72 ◽  
Author(s):  
Sophie Laplante ◽  
Mathieu Laurière ◽  
Alexandre Nolin ◽  
Jérémie Roland ◽  
Gabriel Senno
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Bell Inequality ◽  
Bell Inequalities ◽  
Local Distribution ◽  
Classical Communication ◽  
Quantum Communication Complexity ◽  
Complexity Lower Bounds ◽  
Quantum Distributions

The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.

scholarly journals Better protocol for XOR game using communication protocol and nonlocal boxes

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1261-1276
Author(s):  
Ryuhei Mori
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Communication Protocol ◽  
Constant Factor ◽  
Efficient Communication ◽  
Information Causality ◽  
Special Case

Buhrman showed that an efficient communication protocol implies a reliable XOR game protocol. This idea rederives Linial and Shraibman’s lower bound of randomized and quantum communication complexities, which was derived by using factorization norms, with worse constant factor in much more intuitive way. In this work, we improve and generalize Buhrman’s idea, and obtain a class of lower bounds for randomized communication complexity including an exact Linial and Shraibman’s lower bound as a special case. In the proof, we explicitly construct a protocol for XOR game from a randomized communication protocol by using a concept of nonlocal boxes and Paw lowski et al.’s elegant protocol, which was used for showing the violation of information causality in superquantum theories.

Sign in / Sign up

Export Citation Format

Share Document