How many paths from one corner to the diagonally opposite corner in an n x n matrix do not cross the diagonal?

The Britney Spears Problem ,Tracking who's HOT and who's not , presents an algorithmic challenge ?

 By Brian Hayes

• Back in 1999, the operators of the Lycos Internet portal began publishing a weekly list of the 50 most popular queries submitted to their Web search engine. Britney Spears—initially tagged a "teen songstress," later a "pop tart"—was No. 2 on that first weekly tabulation. She has never fallen off the list since then—440 consecutive appearances when I last checked. Other perennials include ­Pamela Anderson and Paris Hilton. What explains the enduring popularity of these celebrities, so famous for being famous? That's a fascinating question, and the answer would doubtless tell us something deep about modern culture. But it's not the question I'm going to take up here. What I'm trying to understand is how we can know Britney's ranking from week to week. How are all those queries counted and categorized? What algorithm tallies them up to see which terms are the most frequent?

                                  

• One challenging aspect of this task is simply coping with the volume of data. Lycos reports processing 12 million queries a day, and other search engines, such as Google, handle orders of magnitude more. But that's only part of the problem. After all, if you have the computational infrastructure to answer all those questions about Britney and Pamela and Paris, then it doesn't seem like much of an added burden to update a counter each time some fan submits a request. What makes the counting difficult is that you can't just pay attention to a few popular subjects, because you can't know in advance which ones are going to rank near the top. To be certain of catching every new trend as it unfolds, you have to monitor all the incoming queries—and their variety is unbounded.
• In the past few years the tracking of hot topics has itself become a hot topic in computer science. Algorithms for such tasks have a distinctive feature: They operate on a continuous and unending stream of data, rather than waiting for a complete batch of information to be assembled. Like a worker tending a conveyor belt, the algorithm has to process each element of the stream in sequence, as soon as it arrives. Ideally, all computations on one element are finished before the next item comes along.
• Much of the new interest in stream algorithms is inspired by the Internet, where streams of many kinds flow copiously. It's not just a matter of search-engine popularity contests. A similar algorithm can help a network manager monitor traffic patterns, revealing which sites are generating most of the volume. The routers and switches that actually direct the traffic also rely on stream algorithms, passing along each packet of data before turning to the next. A little farther afield, services that filter spam from e-mail can use stream algorithms to detect messages sent in thousands or millions of identical copies.
• Apart from the Internet, stream algorithms are also being applied to flows of financial data, such as stock-market transactions and credit-card purchases. If some government agency wanted to monitor large numbers of telephone calls, they too might have an interest in stream algorithms. Finally, the designers of software for some large-scale scientific experiments adopt a stream-oriented style of data processing. Detectors at particle-physics laboratories produce so much data that no machine can store it all, even temporarily, and so preliminary filtering is done by programs that analyze signals on the fly....................................................to be contd. :)

Interested People Can Read Full Article Here
http://amsciadmin.eresources.com/libraries/documents/2008631225466814-2008-07Hayes.pdf

Resilient Binary Search-search an element in O(log n) time in the perturbed array?

You are about to search for an element in a sorted array A[1..n] using binary search when the array suddenly gets perturbed. By perturbed, we mean a number in ith position in the array can now be either in i, i-1 or i+1th position; but all the numbers are still in A[1..n]. Can you still search an element in O(log n) time in the perturbed array?

Valid Strings-erase the smallest possible number of characters such that the remaining string after the erasures is valid

A string over the characters (, [, ] and ) is said to be valid if one can reduce the string to the null string by repeatedly erasing two consecutive characters of the form () or []. For example, the string [([])[]] is valid but neither ([)] nor ([ is valid.
Give a polynomial time algorithm to solve the following problem: Given a string, erase the smallest possible number of characters such that the remaining string after the erasures is valid. For example, given the string [(], we can erase ( to get [].

Nab the thief-prove that Alice can win in at most 3n moves.

Alice and Bob play the following game: Two cops(c) and a thief(t) are positioned at some vertices of an nxn grid. Alice controls the two cops and Bob controls the thief. The players take turns to play, starting with Alice. In one turn, Alice can move each cop to a neighboring vertex, i.e. left, right, up or down or keep the cop in the same place. In his turn, Bob can move the thief similarly.
If the thief and a cop end up in the same vertex, Alice wins.
For any arbitrary positioning of the two cops and the thief, prove that Alice can win in at most 3n moves.

Estimating Distance-Find maximum distance between any pair of points in a plane

A set of n points are on a plane. Let d denote the maximum distance between any pair of points. Design a linear time algorithm to guess the value of d. Your guess is good if it lies between 0.7d and d. also proove correctness of algorithm.

Beaded Necklace-Find a sequence of such operations with minimum total cost for a given initial distribution of beads.

You are given a circular necklace containing n beads. Each bead maybe black or white. You have to rearrange the beads so that beads of the same color occur consecutively.

The only operation allowed is to cut the necklace at any point, remove some number of beads from both ends and put them back in any order, and join the cut ends. The cost of this operation is the number of beads removed. Any number of such operations can be used. You have to find a sequence of such operations with minimum total cost for a given initial distribution of beads.

For example, if the initial string is wbwbwb, this can be done by a single operation of cost 4, or two operations of cost 2 as shown.

Describe a polynomial-time algorithm for this problem; with short proof of correctness. You get points only if you match (or better) the judge solution's time complexity.