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Broadcasting with side information
 Proc. of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008
"... A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt be the minimum number of bits that has t ..."
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A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt be the minimum number of bits that has to be transmitted when each block is of length t, and let β be the limit β = limt→ ∞ βt/t. In words, β is the average communication cost per bit in each block (for long blocks). Finding the coding rate β, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding. In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on βt, and prove that for any constant C there is an explicit broadcast setting in which β = 2 but β1> C. One of these examples answers a question of [15]. In addition, we provide examples with the following counterintuitive directsum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the
Topological interference management through index coding
, 2013
"... While much recent progress on interference networks has come about under the assumption of abundant channel state information at the transmitters (CSIT), a complementary perspective is sought in this work through the study of interference networks with no CSIT except a coarse knowledge of the topolo ..."
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Cited by 30 (14 self)
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While much recent progress on interference networks has come about under the assumption of abundant channel state information at the transmitters (CSIT), a complementary perspective is sought in this work through the study of interference networks with no CSIT except a coarse knowledge of the topology of the network that only allows a distinction between weak and significant channels and no further knowledge of the channel coefficients ’ realizations. Modeled as a degreesoffreedom (DoF) study of a partially connected interference network with no CSIT, the problem is found to have a counterpart in the capacity analysis of wired networks with arbitrary linear network coding at intermediate nodes, under the assumption that the sources are aware only of the end to end topology of the network. The wireless (wired) network DoF (capacity) region, expressed in dimensionless units as a multiple of the DoF (capacity) of a single point to point channel (link), is found to be bounded above by the capacity of an index coding problem where the antidotes graph is the complement of the interference graph of the original network and the bottleneck link capacity is normalized to unity. The problems are shown to be equivalent under linear solutions over the same field. An interference alignment
Towards Desynchronization of Multihop Topologies
 IEEE International Conference on SelfAdaptive and SelfOrganizing Systems (SASO 2008
"... In this paper we study desynchronization, a closelyrelated primitive to graph coloring. A valid graph coloring is an assignment of colors to nodes such that no node’s color is the same as a neighbor’s. A desynchronized configuration is an assignment of real values in S1 to nodes such that each node’ ..."
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In this paper we study desynchronization, a closelyrelated primitive to graph coloring. A valid graph coloring is an assignment of colors to nodes such that no node’s color is the same as a neighbor’s. A desynchronized configuration is an assignment of real values in S1 to nodes such that each node’s value is exactly at the midpoint of two of its closest neighbors ’ values. Recent work has shown that a simple, selforganizing algorithm, DESYNC, can solve desynchronization in singlehop networks, with applications to collisionfree wireless broadcast [1] and dutycycling [2]. Here we generalize this work by defining and analyzing desynchronization for multihop networks and experimentally analyzing the DESYNC algorithm’s behavior for multihop networks. We describe desynchronized configurations for several classes of graphs (lines, rings, twocolorable, and hamiltonian cycles) and discuss the relationship with other variants of graph coloring. We extend the DESYNC algorithm and DESYNCbased resource allocation to multihop networks and study the performance and efficiency of resource allocation in simulation. While many applications for graph coloring require synchronization and an agreement on a schedule to be effective, we show that the selforganizing algorithm, DESYNC, does not require either of these to achieve desynchronization and to define a resourceallocation schedule. Although applications to wireless sensor networks pose some unique problems, the results suggest that DESYNC has significant potential as a lightweight method for providing nonoverlapping variablesized slots in adhoc multihop settings. 1
The impact of frequencyagility on dynamic spectrum sharing
 In Proc. of IEEE DySPAN
, 2010
"... Abstract — Designed to adapt spectrum usage onthefly, frequencyagile radios can drastically improve performance of wireless networks. Such flexibility, however, comes with a cost of increased hardware complexity. This motivates us to understand when and why having higher degree of frequencyagili ..."
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Cited by 9 (1 self)
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Abstract — Designed to adapt spectrum usage onthefly, frequencyagile radios can drastically improve performance of wireless networks. Such flexibility, however, comes with a cost of increased hardware complexity. This motivates us to understand when and why having higher degree of frequencyagility helps and how much improvement it can lead to. In this paper, we approach this question by comparing two types of agile radios in the context of dynamic spectrum sharing in any given spectrum chunk. We consider 1agile radios that use a single frequency channel but can adjust the channel’s width and central frequency, and kagile radios that can combine up to k noncontiguously aligned frequency segments into one transmission. We show that, due to inherent demand dynamics and conflict heterogeneity, networks using 1agile radios often face the problem of spectrum fragmentation. But kagile radios can effectively suppress this problem directly at the physical layer. Using theoretical analysis and simulation experiments, we quantify the advantage of kagile radios over 1agile radios in their network spectrum usage. For a fair comparison, we abstract the impact of demand and topology configurations by evaluating the worst case and average case performance. Our results show that in worst cases, the improvement of using fullyagile radios is arbitrarily large, although the improvement of using kagile radios is upper bounded by k. In average cases, the improvement reduces to 10–40 % under typical network configurations. Interestingly, in the context of dynamic spectrum sharing, 2agile radios realize the majority of the improvement brought by fullyagile radios. I.
The fractional chromatic number of trianglefree graphs with ∆ ≤ 3
, 2010
"... Let G be any trianglefree graph with maximum degree ∆ ≤ 3. Staton 5 proved that the independence number of G is at least n. Heckman 14 and Thomas conjectured that Staton’s result can be strengthened into a bound on the fractional chromatic number of G, namely χf(G) ≤ 14 5.. In this paper, we prov ..."
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Let G be any trianglefree graph with maximum degree ∆ ≤ 3. Staton 5 proved that the independence number of G is at least n. Heckman 14 and Thomas conjectured that Staton’s result can be strengthened into a bound on the fractional chromatic number of G, namely χf(G) ≤ 14 5.. In this paper, we prove Recently, Hatami and Zhu proved χf(G) ≤ 3 − 3 64 χf(G) ≤ 3 − 3
Fractional biclique covers and partitions of graphs
"... A biclique is a complete bipartite subgraph of a graph. This paper investigates the fractional biclique cover number, bc ∗ (G), and the fractional biclique partition number, bp ∗ (G), of a graph G. It is observed that bc ∗ (G) andbp ∗ (G) provide lower bounds on the biclique cover and partition numb ..."
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A biclique is a complete bipartite subgraph of a graph. This paper investigates the fractional biclique cover number, bc ∗ (G), and the fractional biclique partition number, bp ∗ (G), of a graph G. It is observed that bc ∗ (G) andbp ∗ (G) provide lower bounds on the biclique cover and partition numbers respectively, and conditions for equality are given. It is also shown that bc ∗ (G) is a better lower bound on the Boolean rank of a binary matrix than the maximum number of isolated ones of the matrix. In addition, it is noted that bc ∗ (G) ≤ bp ∗ (G) ≤ β ∗ (G), the fractional vertex cover number. Finally, the application of bc ∗ (G) andbp ∗ (G) to two different weak products is discussed.
Recent progress in graph pebbling
 Graph Theory Notes N. Y
"... The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1 ..."
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The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1
A Fractional Analogue of Brooks’ Theorem
"... Let ∆(G) be the maximum degree of a graph G. Brooks’ theorem states that the only connected graphs with chromatic number χ(G) = ∆(G)+1 are complete graphs and odd cycles. We prove a fractional analogue of Brooks’ theorem in this paper. Namely, we classify all connected graphs G such that the fracti ..."
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Let ∆(G) be the maximum degree of a graph G. Brooks’ theorem states that the only connected graphs with chromatic number χ(G) = ∆(G)+1 are complete graphs and odd cycles. We prove a fractional analogue of Brooks’ theorem in this paper. Namely, we classify all connected graphs G such that the fractional chromatic number χf(G) is at least ∆(G). These graphs are complete graphs, odd cycles, C 2 8, C5 ⊠K2, and graphs whose clique number ω(G) equals the maximum degree ∆(G). Among the two sporadic graphs, the graph C 2 8 is the square graph of cycle C8 while the other graph C5 ⊠ K2 is the strong product of C5 and K2. In fact, we prove a stronger result; if a connected graph G with ∆(G) ≥ 4 is not one of the graphs listed above, then we have χf(G) ≤ ∆(G) − 2