A binary sequence A = A(0)A(1). .. is called infinitely often (i.o.) Turing-autoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don't-know symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truth-table-autoreducible. We obtain the somewhat counterintuitive result that every Martin-Löf random sequence, in fact even every rec-random or p-random sequence, is i.o. truth-table-autoreducible. Furthermore, we investigate the question of how dense the set of guessed bits can be when i.o. autoreducing a random sequence. We show that rec-random sequences are never i.o. truth-table-autoreducible such that the set of guessed bits has positive constant density in the limit and that a similar assertion holds for Martin-Löf random sequences and i.o. Turing autoreducibility. On the other hand, we show that for any rational-valued computable function r that goes nonascendingly to zero, any rec-random sequence is i.o. truth-table-autoreducible such that on any prefix of length m at least a fraction of r(m) of the m bits in the prefix are guessed. We include a self-contained account of the hat problem, a puzzle that has received some attention outside of theoretical computer science. The hat problem asks for guessing bits of a finite sequence, thus illustrating the notion of i.o. autoreducibility in a finite setting. The solution to the hat problem is then used as a module in the proofs of the positive results on i.o. autoreducibility.
In their seminal work on Go With the Winners (GWW) algorithms, D. Aldous and U. Vazirani [3] proved a sufficient condition for the number of particles needed for reaching the bottom of a tree with high probability via a GWW random walk. However, to use this result in practice would require knowledge of the entire search tree which is infeasible for most problems. In this paper we improve slightly on this situation by deriving a recurrence relation that provides an upper-bound for a tree's imbalance in terms of the imbalance between tree levels that are close to one another, provided that these latter imbalances can be measured with sufficient accuracy.We then turn our attention to the problem of finding both frequent and infrequent patterns in a database. One of the most widely used algorithms for finding frequent patterns in memory-resident databases is a randomized algorithm first proposed by Gunopulos et al. [12]. We show that such an algorithm is precisely one for which the GWW paradigm was designed to improve on. Experimental results using the Splice-junction Gene Sequences Database [4] are also provided and lend empirical evidence of the benefits of using GWW.
To the Editor Health care group purchasing organizations (GPOs) play an integral role in the health care supply chain by helping hospitals, surgery centers, clinics, long-term care facilities, and other health care organizations source the best products and services at the best value. Unfortunately, I think the Viewpoint by Mr Bruhn and colleagues 1 arrived at flawed conclusions about GPOs.GPOs are cost-savings partners that help hospitals source lifesaving medical products for the patients they serve. GPOs save the health care system up to $55 billion annually. 2 Independent industry-and nonindustry-funded analyses of GPOs have consistently found that GPOs deliver billions of dollars in cost savings every year to the health care delivery system. 3,4 Despite the authors' claims regarding vendor fees, GPOs disclose all administrative fees earned on their purchases to members in writing. There are no undisclosed fees that result in better listings in GPO catalogs, and all GPO contracts are voluntary and the product of open competition and evaluation by member clinical advisory committees. GPO cost savings, administrative structure, and business practices have all been thoroughly reviewed by the US Government Accountability Office (GAO), the US Department of Justice, the Federal Trade Commission (FTC), the US Supreme Court, the US Court of Appeals for the Eighth Circuit, academia, and most US hospitals.The authors suggested a link between GPOs and drug shortages, but stated that "there is limited evidence to support the direct link between GPOs and drug shortages." 1 GPOs work with health care organizations, manufacturers, and distributors to help prevent and mitigate drug shortages. The US Food and Drug Administration (FDA) has repeatedly identified quality control problems, manufacturing issues, and barriers to getting new suppliers on line as the primary causes of drug shortages-not GPOs.Former FTC chairman Jon Leibowitz and coauthors examined GPOs and found that they operate in a competitive environment and reduce health care costs for patients, hospitals, payers, Medicare and Medicaid, and taxpayers. 5 They also found that repealing the GPO safe harbor exemption would cause both short-and long-term disruptions to the supply chain that could jeopardize the ability of physicians and hospitals to effectively treat patients, and would provide no benefit.Joining a GPO is completely voluntary and hospitals that become GPO members frequently purchase off contract, yet they continue to choose GPOs to help source lifesaving medical products for the patients they serve.
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