Partition arguments in multiparty communication complexity

Consider the “Number in Hand” multiparty communication complexity model, where k players holding inputs x₁,…,x_k∈{0,1}ⁿ communicate to compute the value f(x₁,…,x_k) of a function f known to all of them. The main lower bound technique for the communication complexity of such problems is that of partition arguments: partition the k players into two disjoint sets of players and find a lower bound for the induced two-party communication complexity problem.In this paper, we study the power of partition arguments. Our two main results are very different in nature:

(i)

For randomized communication complexity, we show that partition arguments may yield bounds that are exponentially far from the true communication complexity. Specifically, we prove that there exists a 3-argument function f whose communication complexity is Ω(n), while partition arguments can only yield an Ω(logn) lower bound. The same holds for nondeterministiccommunication complexity.

(ii)

For deterministic communication complexity, we prove that finding significant gaps between the true communication complexity and the best lower bound that can be obtained via partition arguments, would imply progress on a generalized version of the “log-rank conjecture” in communication complexity. We also observe that, in the case of computing relations (search problems), very large gaps do exist.

We conclude with two results on the multiparty “fooling set technique”, another method for obtaining communication complexity lower bounds.     

An information statistics approach to data stream and communication complexity

We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on communication complexity, we show that it also admits a direct sum theorem. Direct sum decomposition reduces our task to that of proving conditional information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations.Our paradigm leads to two main results:(1) An improved lower bound for the multi-party set-disjointness problem in the general communication complexity model, and a nearly optimal lower bound in the one-way communication model. As a consequence, we show that for any real k>2, approximating the kth frequency moment in the data stream model requires essentially Ω(n1−2/k) space; this resolves a conjecture of Alon et al. (J. Comput. System Sci. 58(1) (1999) 137).(2) A lower bound for the L_p approximation problem in the general communication model; this solves an open problem of Saks and Sun (in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 360-369). As a consequence, we show that for p>2, approximating the L_p norm to within a factor of n^ε in the data stream model with constant number of passes requires Ω(n1−4ε−2/p) space.     

Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations

The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the Ω(n²) lower bound and the O(n3+?) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2′) the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered “cross-sections” of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1′) and (2′) have recently been shown to have a nearly tight bound: both complexities are O(n2+?) with lower bound Ω(n²).In this paper, we determine the combinatorial complexities of these four problems for some very simple input configurations. In particular, we study point configurations with just two degrees of freedom (DOFs), exploring both the Voronoi diagrams and the corresponding Minkowski sums. We consider the traditional versions of these problems to have 4 DOFs. We show that even for these simple configurations the combinatorial complexities have upper bounds of either O(n²) or O(n2+?) and lower bounds of Ω(n²).     

Refined memorization for vertex cover

Memorization is a technique which allows to speed up exponential recursive algorithms at the cost of an exponential space complexity. This technique already leads to the currently fastest algorithm for fixed-parameter vertex cover, whose time complexity is O(k^1.2832k^1.5+kn), where n is the number of nodes and k is the size of the vertex cover. Via a refined use of memorization, we obtain an O(k^1.2759k^1.5+kn) algorithm for the same problem. We moreover show how to further reduce the complexity to O(k^1.2745k⁴+kn).     

Communication complexity towards lower bounds on circuit depth

Karchmer, Raz, and Wigderson (1995) discuss the circuit depth complexity of n-bit Boolean functions constructed by composing up to d = log n/log log n levels of k = log n-bit Boolean functions. Any such function is in AC¹ . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires depth. This would separate AC¹ from NC¹. They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit depth lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk—O(d ²(k log k)^1/2) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds. Received: July 30, 1999.     

Higher-order Voronoi diagrams on triangulated surfaces

We study the complexity of higher-order Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the order-j Voronoi diagrams of m sites, for j=1,…,k, is O(k²n²+k²m+knm), which is asymptotically tight in the worst case.     

Switching functions whose monotone complexity is nearly quadratic

A sequence of monotone switching functions h_n:{0,1}ⁿ→ {0,1}ⁿ is constructed, such that the monotone complexity of h_n grows faster than Ω(n² log^?2n). Previously the best lower bounds of this nature were several Ω(n^{$\text{3}\text{2}$} bounds due to Pratt, Paterson, Mehlhorn and Galil and Savage.     

Randomized parallel list ranking for distributed memory multiprocessors

We present a randomized parallel list ranking algorithm for distributed memory multiprocessors, using a BSP type model. We first describe a simple version which requires, with high probability, log(3p)+log ln(n)=Õ(logp+log logn) communication rounds (h-relations withh=Õ(n/p)) andÕ(n/p)) local computation. We then outline an improved version that requires high probability, onlyr?(4k+6) log(2/3p)+8=Õ(k logp) communication rounds wherek=min{i?0 |ln(i+1)n?(2/3p)²ⁱ⁺¹}. Notekn) is an extremely small number. Forn andp?4, the value ofk is at most 2. Hence, for a given number of processors,p, the number of communication rounds required is, for all practical purposes, independent ofn. Forn?1, 500,000 and 4?p?2048, the number of communication rounds in our algorithm is bounded, with high probability, by 78, but the actual number of communication rounds observed so far is 25 in the worst case. Forn?10010100 and 4?p?2048, the number of communication rounds in our algorithm is bounded, with high probability, by 118; and we conjecture that the actual number of communication rounds required will not exceed 50. Our algorithm has a considerably smaller member of communication rounds than the list ranking algorithm used in Reid-Miller’s empirical study of parallel list ranking on the Cray C-90.⁽¹⁾ To our knowledge, Reid-Miller’s algorithm⁽¹⁾ was the fastest list ranking implementation so far. Therefore, we expect that our result will have considerable practical relevance.     

The communication complexity of the Hamming distance problem

We investigate the randomized and quantum communication complexity of the Hamming Distance problem, which is to determine if the Hamming distance between two n-bit strings is no less than a threshold d. We prove a quantum lower bound of Ω(d) qubits in the general interactive model with shared prior entanglement. We also construct a classical protocol of O(dlogd) bits in the restricted Simultaneous Message Passing model with public random coins, improving previous protocols of O(d²) bits [A.C.-C. Yao, On the power of quantum fingerprinting, in: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 77-81], and O(dlogn) bits [D. Gavinsky, J. Kempe, R. de Wolf, Quantum communication cannot simulate a public coin, quant-ph/0411051, 2004].     

A quadratic edge-finding filtering algorithm for cumulative resource constraints

The cumulative scheduling constraint, which enforces the sharing of a finite resource by several tasks, is widely used in constraint-based scheduling applications. Propagation of the cumulative constraint can be performed by several different filtering algorithms, often used in combination. One of the most important and successful of these filtering algorithms is edge-finding. Recent work by Vilím has resulted in a ?? (kn log n) algorithm for cumulative edge-finding (where n is the number of tasks and k is the number of distinct capacity requirements), as well as a new related filter, timetable edge-finding, with a complexity of ??(n ²). We present a sound ??(n ²) filtering algorithm for standard cumulative edge-finding, orthogonal to the work of Vilím; we also show how this algorithm’s filtering may be improved by incorporating some reasoning from extended edge-finding, with no increase in complexity. The complexity of the new algorithm does not strictly dominate previous edge-finders for small k, and it sometimes requires more iterations to reach the same fixpoint; nevertheless, results from Project Scheduling Problem Library benchmarks show that in practice this algorithm consistently outperforms earlier edge-finding filters, and remains competitive with timetable edge-finding, despite the latter algorithm’s generally stronger filtering.     

Communication Complexity of Gossiping by Packets

This paper considers the problem of gossiping with packets of limited size in a network with a cost function. We show that the problem of determining the minimum cost necessary to perform gossiping among a given set of participants with packets of limited size is NP-hard. We also give an approximate (with respect to the cost) gossiping algorithm. The ratio between the cost of an optimal algorithm and the approximate one is less than 1 + 2(k− 1)/n, wherenis the number of nodes participating in the gossiping process andk≤n− 1 is the upper bound on the number of individual blocks of information that a packet can carry. We also analyze the communication time and communication complexity, i.e., the product of the communication cost and time, of gossiping algorithms.     

Chromatic distribution of k-nearest neighbors of a line segment in a planar colored point set

Let P be a set of n colored points distributed arbitrarily in R². The chromatic distribution of the k-nearest neighbors of a query line segment ? is to report the number of points of each color among the k-nearest points of the query line segment. While solving this problem, we have encountered another interesting problem, namely the semicircular range counting query. Here a set of n points is given. The objective is to report the number of points inside a given semicircular range. We propose a simple algorithm for this problem with preprocessing time and space complexity O(n³), and the query time complexity O(logn). Finally, we propose the algorithm for reporting the chromatic distribution of k nearest neighbors of a query line segment. Using our proposed technique for semicircular range counting query, it runs in O(log²n) time.     

Multimedia Indexing with the SMART system

The storage and retrieval of multimedia data is a crucial problem in multimedia information systems due to the huge storage requirements. It is necessary to provide an efficient methodology for the indexing of multimedia data for rapid retrieval. The aim of this paper is to introduce a methodology to represent, simplify, store, retrieve and reconstruct an image from a repository. An algebraic representation of the spatio-temporal relations present in a document is constructed from an equivalent graph representation and used to index the document. We use this representation to simplify and later reconstruct the complete index. This methodology has been tested by implementation of a prototype system called Simplified Modeling to Access and ReTrieve multimedia information (SMART). Experimental results show that the complexity of an index of a 2D document is O (n^*(n−1)/k) with k≥2 as opposed to the O (n^*(n−1)/2) known so far. Since k depends on the number of objects in an image more complex documents have lower overall complexity.     

Reporting intersections among thick objects

Let E be a set of n objects in fixed dimension d. We assume that each element of E has diameter smaller than D and has volume larger than V. We give a new divide and conquer algorithm that reports all the intersecting pairs in O(nlogn+(D^d/V)(n+k)) time and using O(n) space, where k is the number of intersecting pairs. It makes use of simple data structures and primitive operations, which explains why it performs very well in practice. Its restriction to unit balls in low dimensions is optimal in terms of time complexity, space complexity and algebraic degree.     

Graph Complexity and Slice Functions

Abstract. A graph-theoretic approach to study the complexity of Boolean functions was initiated by Pudlák, Rödl, and Savický [PRS] by defining models of computation on graphs. These models generalize well-known models of Boolean complexity such as circuits, branching programs, and two-party communication complexity. A Boolean function f is called a 2-slice function if it evaluates to zero on inputs with less than two 1's and evaluates to one on inputs with more than two 1's. On inputs with exactly two 1's f may be nontrivially defined. There is a natural correspondence between 2-slice functions and graphs. Using the framework of graph complexity, we show that sufficiently strong superlinear monotone lower bounds for the very special class of {2-slice functions} would imply superpolynomial lower bounds over a complete basis for certain functions derived from them. We prove, for instance, that a lower bound of n ^1+Ω(1) on the (monotone) formula size of an explicit 2-slice function f on n variables would imply a 2 ^Ω(?) lower bound on the formula size over a complete basis of another explicit function g on l variables, where l=Θ( log n) . We also consider lower bound questions for depth-3 bipartite graph complexity. We prove a weak lower bound on this measure using algebraic methods. For instance, our result gives a lower bound of Ω(( log n) ³ / ( log log n) ⁵ ) for bipartite graphs arising from Hadamard matrices, such as the Paley-type bipartite graphs. Lower bounds for depth-3 bipartite graph complexity are motivated by two significant applications: (i) a lower bound of n ^Ω(1) on the depth-3 complexity of an explicit n -vertex bipartite graph would yield superlinear size lower bounds on log-depth Boolean circuits for an explicit function, and (ii) a lower bound of $\exp((\log \log n)^{\omega(1)})$ would give an explicit language outside the class Σ ₂ ^cc of the two-party communication complexity as defined by Babai, Frankl, and Simon [BFS]. Our lower bound proof is based on sign-representing polynomials for DNFs and lower bounds on ranks of ±1 matrices even after being subjected to sign-preserving changes to their entries. For the former, we use a result of Nisan and Szegedy [NS] and an idea from a recent result of Klivans and Servedio [KS]. For the latter, we use a recent remarkable lower bound due to Forster [F1].     

On the Quantum Query Complexity of Local Search in Two and Three Dimensions

The quantum query complexity of searching for local optima has been a subject of much interest in the recent literature. For the d-dimensional grid graphs, the complexity has been determined asymptotically for all fixed d≥5, but the lower dimensional cases present special difficulties, and considerable gaps exist in our knowledge. In the present paper we present near-optimal lower bounds, showing that the quantum query complexity for the 2-dimensional grid [n]² is Ω(n ^1/2?δ ), and that for the 3-dimensional grid [n]³ is Ω(n ^1?δ ), for any fixed δ>0.A general lower bound approach for this problem, initiated by Aaronson (based on Ambainis’ adversary method for quantum lower bounds), uses random walks with low collision probabilities. This approach encounters obstacles in deriving tight lower bounds in low dimensions due to the lack of degrees of freedom in such spaces. We solve this problem by the novel construction and analysis of random walks with non-uniform step lengths. The proof employs in a nontrivial way sophisticated results of Sárközy and Szemerédi, Bose and Chowla, and Halász from combinatorial number theory, as well as less familiar probability tools like Esseen’s Inequality.     

Threshold quantum secret sharing between multiparty and multiparty using Greenberger–Horne–Zeilinger state

We propose a (t, m)?(s, n) threshold quantum secret sharing protocol between multiparty (m members in group 1) and multiparty (n members in group 2) using a sequence of Greenberger–Horne–Zeilinger (GHZ) states, which is useful and efficient when the parties of communication are not all present. In the protocol, Alice prepares a sequence of GHZ states in one of the eight different states and sends the last two particles to the first agent while other members encode their information on the sequence via unitary transformations. Finally the last member in group 2 measures the qubits. It is shown that this scheme is safe.     

A faster parameterized algorithm for set packing

We present an efficient parameterized algorithm for the (k,t)-set packing problem, in which we are looking for a collection of k disjoint sets whose union consists of t elements. The complexity of the algorithm is O(²O(t)nNlogN). For the special case of sets of bounded size, this improves the O(k(ck)n) algorithm of Jia et al. [J. Algorithms 50 (1) (2004) 106].