Arash Saber Tehrani

Efficient Algorithms for Renewable Energy Allocation to Delay Tolerant Consumers

We investigate the problem of allocating energy from renewable sources to flexible consumers in electricity markets. We assume there is a renewable energy supplier that provides energy according to a time-varying (and possibly unpredictable) supply process. The plant must serve consumers within a specified delay window, and incurs a cost of drawing energy from other (possibly non-renewable) sources if its own supply is not sufficient to meet the deadlines. We formulate two stochastic optimization problems: The first seeks to minimize the time average cost of using the other sources (and hence strives for the most efficient utilization of the renewable source). The second allows the renewable source to dynamically set a price for its service, and seeks to maximize the resulting time average profit. These problems are solved via the Lyapunov optimization technique. Our resulting algorithms do not require knowledge of the statistics of the time-varying supply and demand processes and are robust to arbitrary sample path variations.

Instantly decodable network codes for real-time applications

We consider the scenario of broadcasting for realtime applications and loss recovery via instantly decodable network coding. Past work focused on minimizing the completion delay, which is not the right objective for real-time applications that have strict deadlines. In this work, we are interested in finding a code that is instantly decodable by the maximum number of users. First, we prove that this problem is NP-Hard in the general case. Then we consider the practical probabilistic scenario, where users have i.i.d. loss probability, and the number of packets is linear or polynomial in the number of users. In this case, we provide a polynomial-time (in the number of users) algorithm that finds the optimal coded packet. Simulation results show that the proposed coding scheme significantly outperforms an optimal repetition code and a COPE-like greedy scheme.

Bipartite index coding

We analyze a generalized index coding problem that allows multiple users to request the same packet. For this problem we introduce a novel coding scheme called partition multicast. Our scheme can be seen as a natural generalization of clique cover for directed index coding problems. Further, partition multicast corresponds to an achievable scheme for the generalized bipartite index coding problem that we introduce in this paper. Our scheme partitions the nodes into groups and solves a multicasting problem within each group. We show that Partition Multicast is optimal for a few families of graphs and generalizes previous achievable schemes, namely directed cycle covers. We also show that finding the best partition is computationally intractable to compute in general.

Dynamic index coding for wireless broadcast networks

We consider a wireless broadcast station that transmits packets to multiple users. The packet requests for each user may overlap, and some users may already have certain packets. This presents a problem of broadcasting in the presence of side information, and is a generalization of the well known (and unsolved) index coding problem of information theory. Rather than achieving the full capacity region, we develop a code-constrained capacity region, which restricts attention to a pre-specified set of coding actions. We develop a dynamic max-weight algorithm that allows for random packet arrivals and supports any traffic inside the code-constrained capacity region. Further, we provide a simple set of codes based on cycles in the underlying demand graph. We show these codes are optimal for a class of broadcast relay problems.

Dynamic Index Coding for Wireless Broadcast Networks

We consider a wireless broadcast station that transmits packets to multiple users. The packet requests for each user may overlap, and some users may already have certain packets. This presents a problem of broadcasting in the presence of side information, and is a generalization of the well-known (and unsolved) index coding problem of information theory. We represent the problem by a bipartite demand graph. Uncoded transmission is optimal if and only if this graph is acyclic. Next, we define a code-constrained capacity region that restricts attention to any prespecified set of coding actions. A dynamic max-weight algorithm that acts over variable length frames is developed. The algorithm allows for random packet arrivals and supports any traffic inside the code-constrained capacity region. A simple set of codes that exploit cycles in the demand graph are shown to be optimal for a class of broadcast relay problems.

SigSag: Iterative Detection Through Soft Message-Passing

The multiple-access framework of ZigZag decoding (Gollakota and Katabi 2008) is a useful technique for combating interference via multiple repeated transmissions, and is known to be compatible with distributed random access protocols. However, in the presence of noise this type of decoding can magnify errors, particularly when packet sizes are large. We show that ZigZag decoding can be seen as an instance of belief propagation in the high signal-to-noise ratio (SNR) limit. Building on this observation, we present a simple soft-decoding version, called SigSag, that improves performance. We show that for two users, collisions result in a cycle-free factor graph that can be optimally decoded via belief propagation. For collisions between more than two users, we show that if a simple bit-permutation is used then the graph is locally tree-like with high probability, and hence belief propagation is near-optimal. Further, we introduce the joint channel-collision decoding which decodes the collided packets while the packets are coded by an LDPC code. Through simulations we show that our scheme performs better than coordinated collision-free time division multiple access (TDMA) and the ZigZag decoder. Furthermore, we investigate the performance of the joint channel-collision decoder in different scenarios and show that it performs better than TDMA and ZigZag decoder accompanied by sum-product decoding.

Optimal deterministic compressed sensing matrices

We present the first deterministic measurement matrix construction with an order-optimal number of rows for sparse signal reconstruction. This improves the measurements required in prior constructions and addresses a known open problem in the theory of sparse signal recovery. Our construction uses adjacency matrices of bipartite graphs that have large girth. The main result is that girth (the length of the shortest cycle in the graph) can be used as a certificate that a measurement matrix can recover almost all sparse signals. Specifically, our matrices guarantee recovery “for-each” sparse signal under basis pursuit. Our techniques are coding theoretic and rely on a recent connection of compressed sensing to LP relaxations for channel decoding.

Explicit matrices for sparse approximation

We show that girth can be used to certify that sparse compressed sensing matrices have good sparse approximation guarantees. This allows us to present the first deterministic measurement matrix constructions that have an optimal number of measurements for ℓ1/ℓ1 approximation. Our techniques are coding theoretic and rely on a recent connection of compressed sensing to LP relaxations for channel decoding.

Finding three transmissions is hard

We investigate a fundamental wireless broadcast problem called Index coding. We focus on binary linear scalar index coding problems when it is a-priori known that the optimal solution requires three transmissions. For this case, we characterize the relation of the clique cover number of the side information graph G and the optimal index code. This allows us to show that even when there is a-priori knowledge that three transmissions can solve a given index coding problem, finding these transmissions is NP-hard. Another implication of our results is that for three solvable undirected problems, the benefit of interference alignment solutions is at most one compared to a simple clique cover.

SigSag: Iterative detection through soft message-passing

The multiple-access framework of ZigZag decoding [1] is a useful technique for combating interference via multiple repeated transmissions, and is known to be compatible with distributed random access protocols. However, in the presence of noise this type of decoding can magnify errors, particularly when packet sizes are large. We present a simple soft-decoding version, called SigSag, that improves performance. We show that for two users, collisions result in a cycle-free factor graph that can be optimally decoded via belief propagation. For collisions between more than two users, we show that if a simple bit-permutation is used then the graph is locally tree-like with high probability, and hence belief propagation is near optimal. Through simulations we show that our scheme performs better than coordinated collision-free time division multiple access (TDMA) and the ZigZag decoder.