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ps8 - Conditional Expetations Let G be a simple graph with...

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6.851 Advanced Data Structures (Spring’10) Prof. Erik Demaine Dr. Andr´ e Schulz TA: Aleksandar Zlateski Problem 8 Due: Thursday, Apr. 8 Be sure to read the instructions on the assignments section of the class web page. Cuckoo Hashing. We pick two hash-functions f, g : [ u ] [ m ] uniformly at random. Let S be the set of keys we want to store by cuckoo hashing. We define the cuckoo graph as done in the lecture: its nodes are the cells of the table, and we have an edge ( f ( x ) , g ( x )), for all x S . Assume further, that the size of the table is m = 6 | S | . Show that with probability at least 1 / 2 the cuckoo graph contains no cycle. Hint: Use the analysis by counting (similar to the “2-cycle case” in the cuckoo hashing analysis).
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Unformatted text preview: Conditional Expetations. Let G be a simple graph with vertex set V and edge set E . A cut of a set of vertices V ⊆ V is the number of edges that have one endpoint in V and the other in V \ V . The NP-complete MaxCut problem asks for the largest cut. A simple randomized approximation problem works as follows: Throw for every vertex a coin. If we got “tails” we add it to V otherwise not. In the end an edge is with probability 1 / 2 in the cut, so the expected value of the cut for V is | E | / 2. Since every cut is at most | E | we have a 2-approximation. Use the concept of conditional expectations to de-randomize this algorithm. 1...
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