# Jaikumar Radhakrishnan

## Contact Details

NameJaikumar Radhakrishnan |
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## Pubs By Year |
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## Pub CategoriesQuantum Physics (9) Computer Science - Computational Complexity (6) Computer Science - Data Structures and Algorithms (6) Computer Science - Discrete Mathematics (3) Computer Science - Distributed; Parallel; and Cluster Computing (3) Computer Science - Information Theory (3) Mathematics - Information Theory (3) Physics - Statistical Mechanics (1) Computer Science - Computer Science and Game Theory (1) Mathematics - Combinatorics (1) |

## Publications Authored By Jaikumar Radhakrishnan

In this work we study the problem of communication over the quantum wiretap channel. For this channel there are three parties Alice (sender), Bob (legitimate receiver) and Eve (eavesdropper). We obtain upper and lower bounds on the amount of information Alice can communicate to Bob such that Eve gets to know as little information as possible about the transmitted messages. Read More

We consider the non-adaptive bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form "Is x in S?" by non-adaptively probing the bit vector at t places. Let s_N(m,n,t) be the minimum number of bits of storage needed for such a scheme. In this work, we show existence of non-adaptive and adaptive schemes for a range of t that improves an upper bound of Buhrman, Miltersen, Radhakrishnan and Srinivasan (2002) on s_N(m,n,t). Read More

The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson \cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and quantum query complexities, exact and approximate polynomial degrees, etc. In particular, the widest possible (quadratic) separations between deterministic and zero-error randomized query complexity, as well as between bounded-error and zero-error randomized query complexity, have been obtained by considering {\em variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function. However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the precise zero-error complexity of the original pointer function was not known. Read More

We consider space-efficient algorithms for two-coloring $n$-uniform hypergraphs $H=(V,E)$ in the streaming model, when the hyperedges arrive one at a time. It is known that any such hypergraph with at most $0.7 \sqrt{\frac{n}{\ln n}} 2^n$ hyperedges has a two-coloring [Radhakrishnan & Srinivasan, RSA, 2000], which can be found deterministically in polynomial time, if allowed full access to the input. Read More

Let $f : \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ be a 2-party function. For every product distribution $\mu$ on $\{0,1\}^n \times \{0,1\}^n$, we show that $$\mathsf{CC}^\mu_{0.49}(f) = O\left(\left(\log \mathsf{prt}_{1/8}(f) \cdot \log \log \mathsf{prt}_{1/8}(f)\right)^2\right),$$ where $\mathsf{CC}^\mu_\varepsilon(f)$ is the distributional communication complexity of $f$ with error at most $\varepsilon$ under the distribution $\mu$ and $\mathsf{prt}_{1/8}(f)$ is the {\em partition bound} of $f$, as defined by Jain and Klauck [{\em Proc. Read More

We initiate the study of a quantity that we call coordination complexity. In a distributed optimization problem, the information defining a problem instance is distributed among $n$ parties, who need to each choose an action, which jointly will form a solution to the optimization problem. The coordination complexity represents the minimal amount of information that a centralized coordinator, who has full knowledge of the problem instance, needs to broadcast in order to coordinate the $n$ parties to play a nearly optimal solution. Read More

We consider the bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form "Is x in S?" by adaptively probing the bit vector at t places. Let s(m,n,t) be the minimum number of bits of storage needed for such a scheme. Several recent works investigate s(m,n,t) for various ranges of the parameter; we obtain improvements over some of the bounds shown by Buhrman, Miltersen, Radhakrishnan, and Srinivasan (2002) and Alon and Feige (2009). Read More

We present a technique of proving lower bounds for noisy computations. This is achieved by a theorem connecting computations on a kind of randomized decision trees and sampling based algorithms. This approach is surprisingly powerful, and applicable to several models of computation previously studied. Read More

We show a tight lower bound of $\Omega(N \log\log N)$ on the number of transmissions required to compute the parity of $N$ input bits with constant error in a noisy communication network of $N$ randomly placed sensors, each having one input bit and communicating with others using local transmissions with power near the connectivity threshold. This result settles the lower bound question left open by Ying, Srikant and Dullerud (WiOpt 06), who showed how the sum of all the $N$ bits can be computed using $O(N \log\log N)$ transmissions. The same lower bound has been shown to hold for a host of other functions including majority by Dutta and Radhakrishnan (FOCS 2008). Read More

We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of smooth min and max Renyi divergences. Read More

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 K\H{o}v\'{a}ri, S\'{o}s and Tur\'{a}n on the minimum number of edges in a bipartite graph that has no large independent set, and the theorem of Hansel (also Katona and Szemer\'{e}di, 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. As an application of our results, we show how they unify the underlying combinatorial principles developed in the proof of tight lower bounds for depth-two superconcentrators. Read More

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 {\em 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 $T_X$ 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. Read More

We study the complexity of the following problems in the streaming model. Membership testing for \DLIN We show that every language in \DLIN\ can be recognised by a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error, and by a deterministic p-pass $O(n/p)$ space algorithm. We show that these algorithms are optimal. Read More

We provide proofs of the following theorems by considering the entropy of random walks: Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple graph with n vertices, girth g, minimum degree at least 2 and average degree d: Odd girth: If g=2r+1,then n \geq 1 + d*(\Sum_{i=0}^{r-1}(d-1)^i) Even girth: If g=2r,then n \geq 2*(\Sum_{i=0}^{r-1} (d-1)^i) Theorem 2.(Hoory) Let G = (V_L,V_R,E) be a bipartite graph of girth g = 2r, with n_L = |V_L| and n_R = |V_R|, minimum degree at least 2 and the left and right average degrees d_L and d_R. Read More

We show optimal Direct Sum result for the one-way entanglement-assisted quantum communication complexity for any relation f subset of X x Y x Z. We show: Q^{1,pub}(f^m) = Omega(m Q^{1,pub}(f)), where Q^{1,pub}(f), represents the one-way entanglement-assisted quantum communication complexity of f with error at most 1/3 and f^m represents m-copies of f. Similarly for the one-way public-coin classical communication complexity we show: R^{1,pub}(f^m) = Omega(m R^{1,pub}(f)), where R^{1,pub}(f), represents the one-way public-coin classical communication complexity of f with error at most 1/3. Read More

We prove the following theorem about relative entropy of quantum states. "Substate theorem: Let rho and sigma be quantum states in the same Hilbert space with relative entropy S(rho|sigma) = Tr rho (log rho - log sigma) = c. Then for all epsilon > 0, there is a state rho' such that the trace distance ||rho' - rho||_t = Tr sqrt{(rho' - rho)^2} <= epsilon, and rho'/2^{O(c/epsilon^2)} <= sigma. Read More

In this article we study relationship between three measures of distinguishability of quantum states called as divergence, relative entropy and the substate property. Read More

The hidden subgroup problem (HSP) provides a unified framework to study problems of group-theoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an efficient solution in the abelian case, not much is known for general non-abelian groups. Recently, some authors raised the question as to whether post-processing the Fourier spectrum by measuring in a random orthonormal basis helps for solving the HSP. Read More

In the quantum database search problem we are required to search for an item in a database. In this paper, we consider a generalization of this problem, where we are provided d identical copes of a database each with N items which we can query in parallel. Then, given k items, we are required to determine the locations where these items are stored. Read More

In this paper, we consider the partial database search problem where given a database on N items, we are required to determine the first k bits of an address x such that f(x)=1. We derive an algorithm and a lower bound for this problem in the quantum circuits model. Let q(k,N) be the minimum number of queries needed to find the first k bits of the required address x. Read More

We prove lower bounds for the direct sum problem for two-party bounded error randomised multiple-round communication protocols. Our proofs use the notion of information cost of a protocol, as defined by Chakrabarti, Shi, Wirth and Yao and refined further by Bar-Yossef, Jayram, Kumar and Sivakumar. Our main technical result is a `compression' theorem saying that, for any probability distribution $\mu$ over the inputs, a $k$-round private coin bounded error protocol for a function $f$ with information cost $c$ can be converted into a $k$-round deterministic protocol for $f$ with bounded distributional error and communication cost $O(kc)$. Read More

We consider the class of functions whose value depends only on the intersection of the input X_1,X_2, ... Read More

We consider the problem of computing the second elementary symmetric polynomial S^2_n(X) using depth-three arithmetic circuits of the form "sum of products of linear forms". We consider this problem over several fields and determine EXACTLY the number of multiplication gates required. The lower bounds are proved for inhomogeneous circuits where the linear forms are allowed to have constants; the upper bounds are proved in the homogeneous model. Read More

We study the quantum complexity of the static set membership problem: given a subset S (|S| \leq n) of a universe of size m (m \gg n), store it as a table of bits so that queries of the form `Is x \in S?' can be answered. The goal is to use a small table and yet answer queries using few bitprobes. This problem was considered recently by Buhrman, Miltersen, Radhakrishnan and Venkatesh, where lower and upper bounds were shown for this problem in the classical deterministic and randomized models. Read More

We analyze quantitatively several strategies for better utilization of the {\em cache} or the {\em {fast access}} memory in computers. We define a performance factor $\alpha$ that denotes the fraction of the cache area utilized when the main memory is accessed at random. We calculate $\alpha$ exactly for different competing strategies, including the hash-rehash and the skewed-associative strategies which were earlier analyzed via simulations. Read More