Results 1  10
of
14
Mixnetworks with Restricted Routes
 Proceedings of Privacy Enhancing Technologies workshop (PET 2003). SpringerVerlag, LNCS 2760
, 2003
"... We present a mix network topology that is based on sparse expander graphs, with each mix only communicating with a few neighbouring others. We analyse the anonymity such networks provide, and compare it with fully connected mix networks and mix cascades. We prove that such a topology is efficient si ..."
Abstract

Cited by 41 (9 self)
 Add to MetaCart
We present a mix network topology that is based on sparse expander graphs, with each mix only communicating with a few neighbouring others. We analyse the anonymity such networks provide, and compare it with fully connected mix networks and mix cascades. We prove that such a topology is efficient since it only requires the route length of messages to be relatively small in comparison with the number of mixes to achieve maximal anonymity. Additionally mixes can resist intersection attacks while their batch size, that is directly linked to the latency of the network, remains constant. A worked example of a network is also presented to illustrate how these results can be applied to create secure mix networks in practise.
Correlation Clustering in General Weighted Graphs
 Theoretical Computer Science
, 2006
"... We consider the following general correlationclustering problem [1]: given a graph with real nonnegative edge weights and a 〈+〉/〈− 〉 edge labeling, partition the vertices into clusters to minimize the total weight of cut 〈+ 〉 edges and uncut 〈− 〉 edges. Thus, 〈+ 〉 edges with large weights (represen ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
We consider the following general correlationclustering problem [1]: given a graph with real nonnegative edge weights and a 〈+〉/〈− 〉 edge labeling, partition the vertices into clusters to minimize the total weight of cut 〈+ 〉 edges and uncut 〈− 〉 edges. Thus, 〈+ 〉 edges with large weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while 〈− 〉 edges with large weights encourage the endpoints to belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be the best possible by the problem definition. Correlation clustering was introduced by Bansal, Blum, and Chawla [1], motivated by both document clustering and agnostic learning. They proved NPhardness and gave constantfactor approximation algorithms for the special case in which the graph is complete (full information) and every edge has the same weight. We give an O(log n)approximation algorithm for the general case based on a linearprogramming rounding and the “regiongrowing ” technique. We also prove that this linear program has a gap of Ω(log n), and therefore our approximation is tight under this approach. We also give an O(r 3)approximation algorithm for Kr,rminorfree graphs. On the other hand, we show that the problem is equivalent to minimum multicut, and therefore APXhard and difficult to approximate better than Θ(logn). 1
Expander based dictionary data structures
, 2005
"... We consider dictionary data structures based on expander graphs. We show that any one probe scheme with the properties of the previous data structure from [OP02] is indeed space optimal. We then construct four different dictionary data structures for various models of parallel external memory. All o ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
We consider dictionary data structures based on expander graphs. We show that any one probe scheme with the properties of the previous data structure from [OP02] is indeed space optimal. We then construct four different dictionary data structures for various models of parallel external memory. All of them allows lookups using a single parallel probe. In the following n denotes the number of keys in the dictionary, and u the universe of possible keys. ∆opt denotes the space in bits required to store the n keys and their satellite data without any type of compression and d = O(log(u/n)). • A static dictionary data structure with error correcting codes using O(∆opt) bits of space, and one requiring O(ndlog d + ∆opt) bits of space without using error correcting codes. • A dynamic dictionary data structure for the parallel disk head model using O(ndlog n + ∆opt) bits of space, where updates take O(1) I/O’s amortized. • A dynamic dictionary data structure for the parallel disk model, with
Fast Algorithms for Interactive Coding
"... Consider two parties who wish to communicate in order to execute some interactive protocol π. However, the communication channel between them is noisy: An adversary sees everything that is transmitted over the channel and can change a constant fraction of the bits as he pleases, thus interrupting th ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Consider two parties who wish to communicate in order to execute some interactive protocol π. However, the communication channel between them is noisy: An adversary sees everything that is transmitted over the channel and can change a constant fraction of the bits as he pleases, thus interrupting the execution of π (which was designed for an errorless channel). If π only contains a single long message, then a good error correcting code would overcome the noise with only a constant overhead in communication. However, this solution is not applicable to interactive protocols consisting of many short messages. Schulman (FOCS 92, STOC 93) presented the notion of interactive coding: A simulator that, given any protocol π, is able to simulate it (i.e. produce its intended transcript) even with constant rate adversarial channel errors, and with only constant (multiplicative) communication overhead. Until recently, however, the running time of all known simulators was exponential (or subexponential) in the communication complexity of π (denoted N in this work). Brakerski and Kalai (FOCS 12) recently presented a simulator that runs in time poly(N). Their simulator is randomized (each party flips private coins) and has failure probability roughly 2 −N. In this work, we improve the computational complexity of interactive coding. While at least N computational steps are required (even just to output the transcript of π), the BK simulator runs in time ˜ Ω(N 2). We present two efficient algorithms for interactive coding: The first with computational complexity O(N log N) and exponentially small (in N) failure probability; and the second with computational complexity O(N), but failure probability 1/poly(N). (Computational complexity is measured in the RAM model.)
Correlation Clustering in . . .
, 2005
"... We consider the following general correlationclustering problem [1]: given a graph withreal nonegative edge weights and a h+i/hi edge labeling, partition the vertices into clusters to minimize the total weight of cut h+i edges and uncut hi edges. Thus, h+i edges withlarge weights (representing s ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We consider the following general correlationclustering problem [1]: given a graph withreal nonegative edge weights and a h+i/hi edge labeling, partition the vertices into clusters to minimize the total weight of cut h+i edges and uncut hi edges. Thus, h+i edges withlarge weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while hi edges with large weights encourage the endpointsto belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both ofthese parameters are effectively chosen to be the best possible by the problem definition. Correlation clustering was introduced by Bansal, Blum, and Chawla [1], motivated by bothdocument clustering and agnostic learning. They proved NPhardness and gave constantfactor approximation algorithms for the special case in which the graph is complete (full information)and every edge has the same weight. We give an O(log n)approximation algorithm for thegeneral case based on a linearprogramming rounding and the "regiongrowing " technique. We also prove that this linear program has a gap of \Omega (log n), and therefore our approximation istight under this approach. We also give an O(r3)approximation algorithm for Kr,rminorfreegraphs. On the other hand, we show that the problem is equivalent to minimum multicut, and
7.1 Lecture Outline
, 2005
"... In this lecture we will see three explicit constructions of expanders. By an “explicit construction”, we mean a construction with the following three properties: 1. We can build the entire Nvertex graph in poly(N) time. 2. From a vertex v, we can find the ith neighbor in poly(log N, log D) time wh ..."
Abstract
 Add to MetaCart
In this lecture we will see three explicit constructions of expanders. By an “explicit construction”, we mean a construction with the following three properties: 1. We can build the entire Nvertex graph in poly(N) time. 2. From a vertex v, we can find the ith neighbor in poly(log N, log D) time where D is the degree of the graph. 3. Given vertices u and v, we can determine if they are adjacent in poly(log N) time. The first two constructions will be presented without proof, but we will see the proof in the case of the zigzag construction. 1. The first construction is due to Margulis and GaberGalil. 2. The second construction is due to Lubotsky, Phillips, and Sarnak, and achieves optimal spectral expansion λ ≈ 2 / √ d. 3. The third construction is due to Reingold, Vadhan, and Wigderson. These socalled zigzag expanders are built via repeated applications of two basic operations that jointly increase the number of nodes but keep the degree and expansion λ small. These operations are graph squaring and the zigzag product. The proof that these graphs are expanders will use the tensor product of two vectors. 7.2 The First Two Constructions Construction 7.1 (Margulis [Mar]) Fix a positive integer M and let [M] = {1, 2,..., M}. Define the bipartite graph G = (V, E) as follows. Let V = [M] 2 ∪[M] 2, where vertices in the first partite set are denoted (x, y)1 and vertices in the second partite set are denoted (x, y)2.
www.theoryofcomputing.org Quantum Expanders: Motivation and Constructions
, 2008
"... Abstract: We define quantum expanders in a natural way and give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Phillips ..."
Abstract
 Add to MetaCart
Abstract: We define quantum expanders in a natural way and give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Phillips, and Sarnak (1988). The second construction is combinatorial, and is based on a quantum variant of the ZigZag product introduced by Reingold, Vadhan, and Wigderson (2000). Both constructions are of constant degree, and the second one is explicit. Using another construction of quantum expanders by Ambainis and Smith (2004), we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK–complete (where QSZK is the class of languages having a zeroknowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof follows the classical proof structure that the entropy difference problem is SZK– complete, but crucially depends on the use of quantum expanders.
Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply
"... We consider profit maximization pricing problems, where we are given a set of m customers and a set of n items. Each customer c is associated with a subset Sc ⊆ [n] of items of interest, together with a budget Bc, and we assume that there is an unlimited supply of each item. Once the prices are fixe ..."
Abstract
 Add to MetaCart
We consider profit maximization pricing problems, where we are given a set of m customers and a set of n items. Each customer c is associated with a subset Sc ⊆ [n] of items of interest, together with a budget Bc, and we assume that there is an unlimited supply of each item. Once the prices are fixed for all items, each customer c buys a subset of items in Sc, according to its buying rule. The goal is to set the item prices so as to maximize the total profit. We study the unitdemand minbuying pricing (UDPMIN) and the singleminded pricing (SMP) problems. In the former problem, each customer c buys the cheapest item i ∈ Sc, if its price is no higher than the budget Bc, and buys nothing otherwise. In the latter problem, each customer c buys the whole set Sc if its total price is at most Bc, and buys nothing otherwise. Both problems are known to admit O(min {log(m + n), n})approximation algorithms. We prove that they are log 1−ɛ (m+n) hard to approximate for any constant ɛ, unless NP ⊆ DTIME(n logδ n where δ is a constant depending on ɛ. Restricting our attention to approximation factors depending only on n, we show that these problems are 2log1−δ nhard to approximate for any δ> 0 unless NP ⊆ ZPTIME(nlogδ ′ n ′), where δ is some constant depending on δ. We also prove that
Key words and phrases: expander, zigzag, kpage graphs, pushdown graphs
, 2010
"... Abstract: The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. consta ..."
Abstract
 Add to MetaCart
Abstract: The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. constantdegree dimension expanders in finite fields, resolving a question of Barak, Impagliazzo, Shpilka, and Wigderson (2004); 2. O(1)page and O(1)pushdown expanders, resolving a question of Galil, Kannan, and Szemerédi (1986) and leading to tight lower bounds on simulation time for certain Turing Machines. Recently, Bourgain (2009) gave a rather involved construction of such constantdegree monotone expanders. The first application (1) above follows from a reduction due to Dvir and Shpilka (2007). We sketch Bourgain’s construction and describe the reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that
On the Role of Expander Graphs in Key Predistribution Schemes for Wireless Sensor Networks
"... Abstract. Providing security for a wireless sensor network composed of small sensor nodes with limited battery power and memory can be a nontrivial task. A variety of key predistribution schemes have been proposed which allocate symmetric keys to the sensor nodes before deployment. In this paper we ..."
Abstract
 Add to MetaCart
Abstract. Providing security for a wireless sensor network composed of small sensor nodes with limited battery power and memory can be a nontrivial task. A variety of key predistribution schemes have been proposed which allocate symmetric keys to the sensor nodes before deployment. In this paper we examine the role of expander graphs in key predistribution schemes for wireless sensor networks. Roughly speaking, a graph has good expansion if every ‘small ’ subset of vertices has a ‘large ’ neighbourhood, and intuitively, expansion is a desirable property for graphs of networks. It has been claimed that good expansion in the product graph is necessary for ‘optimal ’ networks. We demonstrate flaws in this claim, argue instead that good expansion is desirable in the intersection graph, and discuss how this can be achieved. We then consider key predistribution schemes based on expander graph constructions and compare them to other schemes in the literature. Finally, we propose the use of expansion and other graphtheoretical techniques as metrics for assessing key predistribution schemes and their resulting wireless sensor networks.