On Estimating the First Frequency Moment of Data Streams

Estimating the first moment of a data stream defined as F_1 = \sum_{i \in \{1, 2, \ldots, n\}} \abs{f_i} to within $1 \pm \epsilon$-relative error with high probability is a basic and influential problem in data stream processing. A tight space bound of $O(\epsilon^{-2} \log (mM))$ is known from the work of [Kane-Nelson-Woodruff-SODA10]. However, all known algorithms for this problem require per-update stream processing time of $\Omega(\epsilon^{-2})$, with the only exception being the algorithm of [Ganguly-Cormode-RANDOM07] that requires per-update processing time of $O(\log^2(mM)(\log n))$ albeit with sub-optimal space $O(\epsilon^{-3}\log^2(mM))$. In this paper, we present an algorithm for estimating $F_1$ that achieves near-optimality in both space and update processing time. The space requirement is $O(\epsilon^{-2}(\log n + (\log \epsilon^{-1})\log(mM)))$ and the per-update processing time is $O( (\log n)\log (\epsilon^{-1}))$.Comment: 12 page

India and Eastphalia

Eastphalia Emerging?: Asia, International Law, and Global Governance, Symposium. Indiana University Maurer School of Law, Bloomington, Indiana, 200

High Probability Frequency Moment Sketches

We consider the problem of sketching the p-th frequency moment of a vector, p>2, with multiplicative error at most 1 +/- epsilon and with high confidence 1-delta. Despite the long sequence of work on this problem, tight bounds on this quantity are only known for constant delta. While one can obtain an upper bound with error probability delta by repeating a sketching algorithm with constant error probability O(log(1/delta)) times in parallel, and taking the median of the outputs, we show this is a suboptimal algorithm! Namely, we show optimal upper and lower bounds of Theta(n^{1-2/p} log(1/delta) + n^{1-2/p} log^{2/p} (1/delta) log n) on the sketching dimension, for any constant approximation. Our result should be contrasted with results for estimating frequency moments for 1 <= p <= 2, for which we show the optimal algorithm for general delta is obtained by repeating the optimal algorithm for constant error probability O(log(1/delta)) times and taking the median output. We also obtain a matching lower bound for this problem, up to constant factors

The Pathological Alliance

Few of America’s bilateral ties have been as fraught as those with Pakistan. Since 1958, when the two countries forged a military pact, the relationship has been based on dubious premises, false expectations, periodic quarrels, and episodic tensions. Apart from a small community of scholars and policy makers, few are aware that it was Pakistan, not the United States, which had avidly courted the other. Deft Pakistani diplomacy, almost immediately after the creation of the state, persuaded the naïve administration of Dwight D. Eisenhower that Pakistan could swiftly emerge as a staunch anticommunist ally. British officials, still smarting from the end of their empire in India and keen on retaining influence in the region, disingenuously encouraged this belief