CMSC 858T: Randomized Algorithms
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چکیده
One of the basic principles behind conditioning is that “conditioning on a high-probability event keeps things reasonably unchanged; however, if we condition on a low-probability event, then all bets could be off”. More precisely, suppose we wish to estimate Pr[A | B]. Note, in general, that this could be less than, equal to, or greater than Pr[A]. We now prove that if Pr[B] is “high”, then Pr[A | B] is “approximately close” to the unconditional probability Pr[A]; on the other hand, we give simple examples to show that if Pr[B] is “low”, then Pr[A | B] can be very different from Pr[A]. First, suppose Pr[B] is “high”; i.e., Pr[B] = 1 − for some small . Let us see why Pr[A | B] ∼ Pr[A] in such a case. As for upper bounds, we have
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CMSC 858T: Randomized Algorithms
A basic problem that arises often in the design and analysis of randomized algorithms is to get a good estimate (upper bound, lower bound, or both) of Pr( ∨ i∈[m]Ei), for some given events Ei. Equivalently, a good lower bound, upper bound or both, is required for Pr( ∧ i∈[m]Ei). As we have seen, one approach would be to use the union bound (“counting sieve”), which is unfortunately quite weak i...
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1Copyright, David M. Mount, 2012 Dept. of Computer Science, University of Maryland, College Park, MD, 20742. These lecture notes were prepared by David Mount for the course CMSC 451, Design and Analysis of Computer Algorithms, at the University of Maryland. Permission to use, copy, modify, and distribute these notes for educational purposes and without fee is hereby granted, provided that this ...
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