Element Distinctness, Frequency Moments, and Sliding Windows

We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than previous lower bounds for comparison-based algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and space efficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows, where the task is to take an input of length 2n-1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a time-space tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to compute the number of distinct elements over sliding windows. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k, k neq 1. This shows that those frequency moments and the decision problem F_0 mod 2 are strictly harder than element distinctness. We complement this lower bound with a T in O(n^2/S) comparison-based deterministic RAM algorithm for exactly computing F_k over sliding windows, nearly matching both our lower bound for the sliding-window version and the comparison-based lower bounds for the single-window version. We further exhibit a quantum algorithm for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437

Finding the Median (Obliviously) with Bounded Space

We prove that any oblivious algorithm using space $S$ to find the median of a list of $n$ integers from $\{1,...,2n\}$ requires time $\Omega(n \log\log_S n)$. This bound also applies to the problem of determining whether the median is odd or even. It is nearly optimal since Chan, following Munro and Raman, has shown that there is a (randomized) selection algorithm using only $s$ registers, each of which can store an input value or $O(\log n)$-bit counter, that makes only $O(\log\log_s n)$ passes over the input. The bound also implies a size lower bound for read-once branching programs computing the low order bit of the median and implies the analog of $P \ne NP \cap coNP$ for length $o(n \log\log n)$ oblivious branching programs

Proceedings Community Leader's Litter Control Workshop

PDF pages: 5

Intellectual Property Policy Online: A Young Person’s Guide

This is an edited version of a presentation to the Intellectual Property Online panel at the Harvard Conference on the Internet and Society, May 28-31, 1996. The panel was a reminder of both the importance of intellectual property and the dangers of legal insularity. Of approximately 400 panel attendees, 90% were not lawyers. Accordingly, the remarks that follow are an attempt to lay out the basics of intellectual property policy in a straighforward and non-technical manner. In other words, this is what non-lawyers should know (and what a number of government lawyers seem to have forgotten) about intellectual property policy on the Internet. The legal analysis which underlies this discussion is set out in the Appendix

Reducing Racial and Ethnic Disparities in Jails: Recommendations for Local Practice

People of color are overrepresented in our criminal justice system. One in three African American men born today will be incarcerated in his lifetime. In some cities, African Americans are ten times more likely to be arrested when stopped by police. With the national debate national focused on race, crime, and punishment, criminal justice experts are examining how to reduce racial disparities in our prisons and jails, which often serve as initial entry points for those who become entangled in the criminal justice system.This report, which relies on input from 25 criminal justice leaders, pinpoints the drivers of racial disparities in our jails, lays out common sense reforms to reduce this disparity, including increasing public defense representation for misdemeanor offenses, encouraging prosecutors to prioritize serious and violent offenses, limiting the use of pretrial detention, and requiring training to reduce racial bias for all those involved in running our justice system

Development of Variant of Software Architecture Implementation for Low-power General Purpose Microcontrollers by Finite State Machines

As a result of the research, two directions for development of software architecture for low-power general purpose microcontrollers (LPGPM) are identified. The first, classical approach is the development using standard State patterns. The second is the development of programs, algorithms and structures based on mathematical analysis.The first direction is chosen in the work. The variant of the implementation of a typical pattern for development of software architecture (SA) in the form of a finite state machine (FSM) is proposed to discussion. This pattern allows to divide the development of the architectural part of the program for LPGPM and programming the LPGPM hardware. This approach makes it possible to divide the work of the software architect and the work of LPGPM hardware specialists. Advantage of the solution in comparison with the real time operating system (RTOS) is the saving of LPGPM hardware resources. In addition, it improves the readability of code and good testing prospects. The resulting architecture makes it possible to easily accompany the software and switch to other types of microcontroller. The disadvantage is an increase in the required amount of RAM with an increase in the number of states. It is this disadvantage that requires the application not only of experimental and engineering-intuitive methods, but also to continue research in the second direction