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The fact that xt+1 is added to S with probability m/(t + 1) concludes the proof Instead of ipping a coin for each element (that requires to generate n random values), the reservoir sampling algorithm randomly generates the number of elements to be skipped before the next element is added to S Special care is taken to generate these skip numbers, so as to guarantee the same properties that we discussed in Theorem 1 for the na ve coin-tossing approach The implementation based on skip numbers has the advantage that the number of random values to be generated is the same as the number of updates of the sample S We refer to the work by Vitter [63] for the details and the analysis of this implementation.

here is no way to sugarcoat it: Current economic conditions have the earmarks not only of a severe US recession but dare I say it the potential of a global depression That is about as dire as it can get However, for me, as tragic and ominous as that prospect may be, it does not represent the worst consequence of today s global economic conditions I fear the Law of Selective Gravity a cousin of one of Murphy s Laws which postulates: An object will fall so far as to do the most damage As the world knows, a couple of weeks ago, Treasury Secretary Henry Paulson asked Congress to approve a $700 billion rescue of the banking industry Without this sudden, massive infusion of federal cash, we were told, economic disaster loomed.

We remark that reservoir sampling works well for insert and updates of the incoming data, but runs into dif culties if the data contain deletions In many applications, however, the timeliness of data is important, since outdated items expire and should be no longer used when answering queries Other sampling techniques have been proposed that address this issue: see, for example, [8,35,52] and the references therein Another limitation of reservoir sampling derives from the fact that the stream may contain duplicates, and any value occurring frequently in the sample is a wasteful use of the available space: concise sampling overcomes this limitation representing elements in the sample by pairs (value, count) As described by Gibbons and Matias [33], this natural idea can be used to compress the samples and allows it to solve, for example, the top-k problem, where the k most frequent items need to be identi ed.

In the rest of this section, we provide a concrete example of how sampling can be effectively applied to certain nontrivial streaming problems However, as we will see in Section 84, there also exist classes of problems for which sampling-based approaches are not effective, unless using a prohibitive (almost linear) amount of memory 8312 An Application of Sampling: Frequent Items Following an approach proposed by Manku and Motwani [51], we will now show how to use sampling to address the problem of identifying frequent items in a data stream, that is, items whose frequency exceeds a user-speci ed threshold Intuitively, it should be possible to estimate frequent items by a good sample The algorithm that we discuss, dubbed.

Prompt approval, on the other hand, would assure the solvency of the nancial sector, thaw frozen credit ows, and give investors a badly needed dose of con dence Faced with the prospect of rising unemployment, a plunging stock market, and the inability of corporate America to borrow, Congress approved a revised package on Friday, October 3, 2008 This gave investors the weekend to contemplate the economic value of the federal action Were it able to inspire some con dence and halt for a moment or two the bloodletting, one might be a bit more charitable in assessing the panic-driven action by the captains of American capitalism On Monday, the stock market plunged into an abyss and the turmoil spread to Europe, Asia, and South America The plan which Secretary Paulson and Fed chairman Bernanke told us we must approve to prevent a market crash did nothing of the sort.

Figure 2-2 shows how web work ideally lets us work from wherever we want even, in the future, on an airplane and stay loosely connected not just to teammates but to many people with whom we can work and socialize. Here you see how the web bursts your work life out of its familiar confines.