In the Equal-Subset-Sum problem, we are given a set @math of @math integers and the problem is to decide if there exist two disjoint nonempty subsets @math , whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in @math time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give @math worst case Monte Carlo algorithm. This answers the open problem from Woeginger's inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in @math time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in @math time and polynomial space.
and showed @math @cite_22 . The first nontrivial upper bound on @math was @math (for sufficiently large @math ) @cite_1 . Subsequently, proved that @math and showed @math . offered 500 dollars for proof or disproof of conjecture that @math for some constant @math .
@cite_1 @cite_22
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We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the @math workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60-62]. In this model there are @math processors @math , each of which is initially given a private input bit @math . The goal is for @math to learn the value of @math , for some specified function @math , using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176-180] gave a noise-resistant protocol that allows @math to learn the entire input with constant probability in @math broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of @math on the depth of a tree that learns the entire input. While the above lower bound is for an @math -bit function, we also show an @math lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least @math for a constant parameter @math (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of @math on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) @math lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts. For any integers @math and @math , we show the existence of a set of @math vectors @math such that any embedding @math satisfying @math must have @math This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of @math of interest, since there is always an isometric embedding into dimension @math (either the identity map, or projection onto @math ). Previously such a lower bound was only known to hold against linear maps @math , and not for such a wide range of parameters @math [LN16]. The best previously known lower bound for general @math was @math [Wel74, Lev83, Alo03], which is suboptimal for any @math . In this paper, we prove tight lower bounds on the smallest degree of a nonzero polynomial in the ideal generated by @math or @math in the polynomial ring @math , @math are coprime, which is called over @math . The immunity of @math is lower bounded by @math , which is achievable when @math is a multiple of @math ; the immunity of @math is exactly @math for every @math and @math . Our result improves the previous bound @math by Green. We observe how immunity over @math is related to @math circuit lower bound. For example, if the immunity of @math over @math is lower bounded by @math , and @math , then @math requires @math circuit of exponential size to compute. We prove the first nontrivial communication complexity lower bound for the problem of estimating the edit distance (aka Levenshtein distance) between two strings. To the best of our knowledge, this is the first computational setting in which the complexity of estimating the edit distance is provably larger than that of Hamming distance. Our lower bound exhibits a trade-off between approximation and communication, asserting, for example, that protocols with @math bits of communication can obtain only approximation @math , where @math is the length of the input strings. This case of @math communication is of particular importance since it captures constant-size sketches as well as embeddings into spaces like @math and squared- @math , two prevailing algorithmic approaches for dealing with edit distance. Indeed, the known nontrivial communication upper bounds are all derived from embeddings into @math . By excluding low-communication protocols for edit distance, we rule out a strictly richer class of algorithms than previous results. Furthermore, our lower bound holds not only for strings over a binary alphabet but also for strings that are permutations (aka the Ulam metric). For this case, our bound nearly matches an upper bound known via embedding the Ulam metric into @math . Our proof uses a new technique that relies on Fourier analysis in a rather elementary way.