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Sublinear Algorihms for Big Data

Sublinear Algorihms for Big Data. Exam Problems. Grigory Yaroslavtsev http://grigory.us. Problem 1: Modified Chernoff Bound. Problem: Derive Chernoff Bound #2 from Chernoff Bound #1. Explain your answer in detail.

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Sublinear Algorihms for Big Data

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  1. SublinearAlgorihms for Big Data Exam Problems GrigoryYaroslavtsev http://grigory.us

  2. Problem 1: Modified Chernoff Bound • Problem: Derive Chernoff Bound #2 from Chernoff Bound #1. Explain your answer in detail. • (Chernoff Bound #1) Let be independent and identically distributedr.vs with range [0, 1] and expectation . Then if and • (Chernoff Bound #2) Let be independent and identically distributedr.vs with range and expectation . Then if and

  3. Problem 2: Modified Chebyshev • Problem: Derive Chebyshev Inequality #2 from Chebyshev Inequality #1. Explain your answer in detail. • (Chebyshev Inequality #1) For every : • (Chebyshev Inequality #2) For every

  4. Problem 3: Sparse Recovery Error • Definition (frequency vector): = frequency of in the stream = # of occurrences of value • Sparse Recovery: Find such that is minimized among s with at most non-zero entries. • Definition: • Problem: Show that where are indices of largest . Explain your answer formally and in detail.

  5. Problem 4: Approximate Median Value • Stream: elements from universe • (all distinct) and let • Median , where + 1 ( odd). • Algorithm: Return the median of a sample of size taken from (with replacement). • Problem: Does this algorithm give a approximate value of the median with probability , i.e. such that if ? Explain your answer.

  6. Problem 5: Lower bound on • Definition (frequency vector): = frequency of in the stream = # of occurrences of value • Define (frequency moment): • Problem: Show that for all integer it holds that: • Hint: worst-case when . Use convexity of

  7. Problem 6: Approximate MST Weight • Let be a weighted graph with weights of all edges being integers between 1 and W. • Definition: Let be the # of connected components if we remove all edges with weight . • Problem: For some constant show the following bounds on the weight of the minimum spanning tree

  8. Evaluation • Deadline: September 1st , 2014, 23:59 GMT • Submissions in English: grigory@grigory.us • Submission Title (once): Exam + Space + “Your Name” • Question Title: Question + Space + “Your Name” • Submission format • PDF from LaTeX (best) • PDF • You can work in groups (up to 3 people) • Each group member lists all others on the submission • Point system • Course Credit = 2 points • Problems 1-3 = 0.5 point each • Problem 4 = 1 point • Problems 5-6 = 2 point each

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