Chinmoy Dutta

Split and Join: Strong Partitions and Universal Steiner Trees for Graphs

We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph G and a root node r, we seek a single spanning tree T of minimum stretch, where the stretch of T is defined to be the maximum ratio, over all terminal sets X, of the cost of the minimal sub-tree TX of T that connects X to r to the cost of an optimal Steiner tree connecting X to r in G. Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications. graphs with 2O(√log n)-stretch. We also give a polynomial time We provide a polynomial time UST construction for general polylog(n)-stretch construction for minor-free graphs. One basic building block of our algorithms is a hierarchy of graph partitions, each of which guarantees small strong diameter for each cluster and bounded neighbourhood intersections for each node. We show close connections between the problems of constructing USTs and building such graph partitions. Our construction of partition hierarchies for general graphs is based on an iterative cluster merging procedure, while the one for minor-free graphs is based on a separator theorem for such graphs and the solution to a cluster aggregation problem that may be of independent interest even for general graphs. To our knowledge, this is the first subpolynomial-stretch (o(nε) for any ε >; 0) UST construction for general graphs, and the first polylogarithmic-stretch UST construction for minor-free graphs.

Lower Bounds for Noisy Wireless Networks using Sampling Algorithms

We show a tight lower bound of Omega(N\log\log N) on the number of transmissions required to compute several functions (including the parity function and the majority function) in a network of N randomly placed sensors, communicating using local transmissions, and operating with power near the connectivity threshold. This result considerably simplifies and strengthens an earlier result of Dutta, Kanoria Manjunath and Radhakrishnan (SODA 08) that such networks cannot compute the parity function reliably with significantly fewer than N\log \log N transmissions, thereby showing that the protocol with O(N\log \log N) transmissions due to Ying, Srikant and Dullerud (WiOpt 06) is optimal. We also observe that all the lower bounds shown by Evans and Pippenger (SIAM J. on Computing, 1999) on the average noisy decision tree complexity for several functions can be derived using our technique simply and in a unified way.

Tradeoffs in depth-two superconcentrators

An N-superconcentrator is a directed graph with N input vertices and N output vertices and some intermediate vertices, such that for k=1, 2, ..., N, between any set of k input vertices and any set of k output vertices, there are k vertex disjoint paths. In a depth-twoN-superconcentrator each edge either connects an input vertex to an intermediate vertex or an intermediate vertex to an output vertex. We consider tradeoffs between the number of edges incident on the input vertices and the number of edges incident on the output vertices in a depth-two N-superconcentrator. For an N-superconcentrator G, let a(G) be the average degree of the input vertices and b(G) be the average degree of the output vertices. Assume that b(G) ≥ a(G). We show that there is a constant k1 > 0 such that

More on a Problem of Zarankiewicz

a(G)log (\frac{2b(G)}{a(G)}) log b(G) \geq k_1 \cdot log^2 N

On the complexity of information spreading in dynamic networks

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A tight lower bound for parity in noisy communication networks

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical results in graph theory: the theorem of Kővari, Sos and Turan on the minimum number of edges in a bipartite graph that has no large independent set, and the theorem of Hansel (also Katona and Szemeredi, Krichevskii) on the sum of the sizes of bipartite graphs that can be used to construct a graph (non-necessarily bipartite) that has no large independent set. Our results unify the underlying combinatorial principles developed in the proof of tight lower bounds for depth-two superconcentrators.