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Sublinear Algorihms for Big Data

Solve sublinear algorithms for big data exam problems including Chernoff Bound, Chebyshev Inequality, Sparse Recovery Error, Median Value, Lower Bound, and MST Weight. Provide detailed explanations and formal answers. Submit by September 1st, 2014.

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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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