Nikhil Karamchandani

Hierarchical Coded Caching

caching of popular content during off-peak hours is a strategy to reduce network loads during peak hours. Recent work has shown significant benefits of designing such caching strategies not only to locally deliver the part of the content, but also to provide coded multicasting opportunities even among users with different demands. Exploiting both of these gains was shown to be approximately optimal for caching systems with a single layer of caches. Motivated by practical scenarios, we consider, in this paper, a hierarchical content delivery network with two layers of caches. We propose a new caching scheme that combines two basic approaches. The first approach provides coded multicasting opportunities within each layer; the second approach provides coded multicasting opportunities across multiple layers. By striking the right balance between these two approaches, we show that the proposed scheme achieves the optimal communication rates to within a constant multiplicative and additive gap. We further show that there is no tension between the rates in each of the two layers up to the aforementioned gap. Thus, both the layers can simultaneously operate at approximately the minimum rate.

Effect of number of users in multi-level coded caching

It has been recently established that joint design of content delivery and storage (coded caching) can significantly improve performance over conventional caching. This has also been extended to the case when content has non-uniform popularity through several models. In this paper we focus on a multi-level popularity model, where content is divided into levels based on popularity. We consider two extreme cases of user distribution across caches for the multi-level popularity model: a single user per cache (single-user setup) versus a large number of users per cache (multi-user setup). When the capacity approximation is universal (independent of number of popularity levels as well as number of users, files and caches), we demonstrate a dichotomy in the order-optimal strategies for these two extreme cases. In the multi-user case, sharing memory among the levels is order-optimal, whereas for the single-user case clustering popularity levels and allocating all the memory to them is the order-optimal scheme. In proving these results, we develop new information-theoretic lower bounds for the problem.

Fundamental limits of secretive coded caching

Recent work by Maddah-Ali and Niesen introduced coded caching which demonstrated the benefits of joint design of storage and transmission policies in content delivery networks. They studied a setup where a server communicates with a set of users, each equipped with a local cache, over a shared error-free link and proposed an order-optimal caching and delivery scheme. In this paper, we introduce the problem of secretive coded caching where we impose the additional constraint that a user should not be able to learn anything, from either the content stored in its cache or the server transmissions, about a file it did not request. We propose a feasible scheme for this setting and demonstrate its order-optimality with respect to information-theoretic lower bounds.

Coded Caching for Multi-level Popularity and Access

To address the exponentially rising demand for wireless content, the use of caching is emerging as a potential solution. It has been recently established that joint design of content delivery and storage (coded caching) can significantly improve performance over conventional caching. Coded caching is well suited to emerging heterogeneous wireless architectures which consist of a dense deployment of local-coverage wireless access points (APs) with high data rates, along with sparsely-distributed, large-coverage macro-cell base stations (BS). This enables design of coded caching-and-delivery schemes that equip APs with storage, and place content in them in a way that creates coded-multicast opportunities for combining with macro-cell broadcast to satisfy users even with different demands. Such coded-caching schemes have been shown to be order-optimal with respect to the BS transmission rate, for a system with single-level content, i.e., one where all content is uniformly popular. In this paper, we consider a system with non-uniform popularity content which is divided into multiple levels, based on varying degrees of popularity. The main contribution of this paper is the derivation of an order-optimal scheme which judiciously shares cache memory among files with different popularities. To show order-optimality we derive new information-theoretic lower bounds, which use a sliding-window entropy inequality, effectively creating a non-cut-set bound. We also extend the ideas to when users can access multiple caches along with the broadcast. Finally, we consider two extreme cases of user distribution across caches for the multi-level popularity model: a single user per cache (single-user setup) versus a large number of users per cache (multi-user setup), and demonstrate a dichotomy in the order-optimal strategies for these two extreme cases.

On the clustering properties of exponential random networks

We consider the clustering properties of one-dimensional sensor networks where the nodes are randomly deployed. Unlike most other work on randomly deployed networks, ours assumes that the node locations are drawn from a non uniform distribution. Specifically, we consider an exponential distribution. We first obtain the probability that there exists a path between two labeled nodes in a randomly deployed network and obtain the limiting behavior of this probability. The probability mass function (pmf) for the number of components in the network is then obtained. We show that the number of components in the network converges in distribution. We also derive the probabilities for different locations of the components. We then obtain the probability for the existence of a k-sized component and components of size /spl ges/k. Asymptotics in the number of nodes in the network are computed for these probabilities. An interesting result is that, as the number of nodes, n, in the network tends to infinity, a giant component, in which a specific fraction, /spl alpha/, of the nodes form a component, almost surely does not exist for any 0</spl alpha/<1. However, the probability converges to a non-zero value for /spl alpha/=1. Another result is that for 0</spl alpha/<1, we can find an n/sub 0/ such that for n>n/sub 0/, the network almost surely does not have a giant component.

Content replication in large distributed caches

In this paper, we consider the algorithmic task of content replication and request routing in a distributed caching system consisting of a central server and a large number of caches, each with limited storage and service capabilities. We study a time-slotted system where in each time-slot, a large batch of requests has to be matched to a large number of caches, where each request can be served by any cache which stores the requested content. All requests which cannot be served by the caches are served by fetching the requested content from the central server. The goal is to minimize the transmission rate from the central server. We use a novel mapping between our content replication problem and the Knapsack problem to prove a lower bound on the transmission rate for any content replication policy. Using insights obtained from the mapping, we propose a content replication policy — Knapsack Storage — which achieves this lower bound. While it intuitively makes sense to replicate the more popular contents on a larger number of caches, surprisingly, in certain cases, the Knapsack Storage policy chooses not to replicate the most popular contents on the caches at all.

Rate and delay for coded caching with carrier aggregation

Motivated by the ability of modern terminals to receive simultaneously from multiple networks (e.g., WLAN and Cellular), we extend the single shared link network with caching at the user nodes to the case of r parallel partially shared links, where users in different classes receive from the server simultaneously and in parallel through different set of links. For this setting, we give an order-optimal rate and (maximal) delay region characterization for the case of r = 2 links with two classes of users, one receiving only from link 1 and the other from both links 1 and 2. We also extend these results to r = 3 with three classes of users, receiving from link 1, from links 1 and 2, and from links 1 and 3, respectively.

Coded caching with partial adaptive matching

We study the coded caching problem when we are allowed to match users to caches based on their requested files. We focus on the case where caches are divided into clusters and each user can be assigned to a unique cache from a specific cluster. We show that neither the coded delivery strategy (approximately optimal when the user-cache assignment is pre-fixed) nor the uncoded replication strategy (approximately optimal when all caches belong to a single cluster) is sufficient for all memory regimes. We propose a hybrid solution that combines ideas from both schemes and that performs at least as well as either strategy in most memory regimes. Finally, we show that this hybrid strategy is approximately optimal in most memory regimes.

Temporally Agnostic Rumor-Source Detection

We revisit the problem of inferring the source of a rumor on a network, given a snapshot of the extent of its spread. We differ from prior work in two aspects: We consider settings where additional relative information about the infection times of a fraction of node pairs is also available to the estimator and instead of only considering the most likely spreading pattern, we take a complementary approach where our estimator for general networks ranks each node based on counting the number of possible spreading patterns with a given node as root that are compatible with the observations. We first consider the case where additional information is available about infection incidents in the form of a set of directed pairs of neighboring nodes with the implication that the first is known to have infected the second. Under this hypothesis, we derive an estimator for the most likely rumor source based on the Markov chain tree theorem and propose a Markov Chain Monte Carlo scheme to find it. Empirical studies of various performance measures are provided, along with comparisons with the popular scheme of Shah and Zaman adapted to this framework. A further variant of the problem considered is when such pairwise temporal precedence is known for a fraction of pairs of nodes which, however, are not necessarily neighbors. For this case, we again propose a Markov Chain Monte Carlo based scheme for rumor source detection, which combines Aldouss algorithm for uniform sampling of arborescences with rejection sampling.

Randomized Kaczmarz for rank aggregation from pairwise comparisons

We revisit the problem of inferring the overall ranking among entities in the framework of Bradley-Terry-Luce (BTL) model, based on available empirical data on pairwise preferences. By a simple transformation, we can cast the problem as that of solving a noisy linear system, for which a ready algorithm is available in the form of the randomized Kaczmarz method. This scheme is provably convergent and has excellent empirical performance. Convergence, convergence rate, and error analysis of the proposed algorithm are presented and several numerical experiments are conducted whose results validate our theoretical findings.