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Binary Decision Diagrams
"... Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and timespace tradeoffs. Due to the lack of powerful ..."
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Decision diagrams are a natural representation of finite functions. The obvious complexity measures are length and size which correspond to time and space of computations. Decision diagrams are the right model for considering space lower bounds and timespace tradeoffs. Due to the lack of powerful lower bound techniques, various types of restricted decision diagrams are investigated. They lead to new lower bound techniques and some of them allow efficient algorithms for a list of operations on boolean functions. Indeed, restricted decision diagrams like ordered binary decision diagrams (OBDDs) are the most common data structure for boolean functions with many applications in verification, model checking, CAD tools, and graph problems. From a complexity theoretical point of view also randomized and nondeterministic decision diagrams are of interest. 1
The communication complexity of addition
, 2011
"... Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai ..."
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Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give a matching (publiccoin) Ω(k lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Ω(k lg lg k) for protocols that are oneway, like ours. We give a protocol to determine if ∑ xi> s with error 1 % and communication O(k lg k) lg n. For k ≥ 3 this improves on Nisan’s O(k lg 2 n) bound. A similar improvement holds for computing degree(k − 1) polynomialthreshold functions in the numberonforehead model. We give a (publiccoin, 2player, tight) Ω(lg n) lower bound to determine if x1> x2. This improves on the Ω ( √ lg n) bound by Smirnov (1988).
Lower bounds for predecessor searching in the cell probe model ∗
, 2003
"... We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form “What is the predecessor of x in S? ” can be answered efficiently. We study this problem in the cell probe model introduced by Yao ..."
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We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form “What is the predecessor of x in S? ” can be answered efficiently. We study this problem in the cell probe model introduced by Yao [Yao81]. Recently, Beame and Fich [BF99] obtained optimal bounds on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only n O(1) cells of word size (log m) O(1) bits. We give a new lower bound proof for this problem that matches the bounds of Beame and Fich. Our lower bound proof has the following advantages: it works for randomised query schemes too, while Beame and Fich’s proof works for deterministic query schemes only. In addition, it is simpler than Beame and Fich’s proof. In fact, our lower bound for predecessor searching extends to the ‘quantum addressonly ’ query schemes that we define in this paper. In these query schemes, quantum parallelism is allowed only over the ‘address lines ’ of the queries. These query schemes subsume classical randomised query schemes, and include many quantum query algorithms like Grover’s algorithm [Gro96]. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson [MNSW98]. Using tools from information theory, we prove a strong round elimination lemma for communication complexity that enables us to obtain a tight lower bound for the predecessor problem. Our strong round elimination lemma also extends to quantum communication complexity. We also use our round elimination lemma to obtain a rounds versus communication tradeoff for the ‘greaterthan’ problem, improving on the tradeoff in [MNSW98]. We believe that our round elimination lemma is of independent interest and should have other applications. 1