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In theoretical computer science, the term isolation lemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist. This is achieved by constructing random constraints such that, with non-negligible probability, exactly one solution satisfies these additional constraints if the solution space is not empty. Isolation lemmas have important applications in computer science, such as the Valiant–Vazirani theorem and Toda's theorem in computational complexity theory. The first isolation lemma was introduced by , albeit not under that name. Their isolation lemma chooses a random number of random hyperplanes, and has the property that, with non-negligible probability, the intersection of any fixed non-empty solution space with the chosen hyperplanes contains exactly one element. This suffices to show the Valiant–Vazirani theorem: there exists a randomized polynomial-time reduction from the satisfiability problem for Boolean formulas to the problem of detecting whether a Boolean formula has a unique solution. introduced an isolation lemma of a slightly different kind: Here every coordinate of the solution space gets assigned a random weight in a certain range of integers, and the property is that, with non-negligible probability, there is exactly one element in the solution space that has minimum weight. This can be used to obtain a randomized parallel algorithm for the maximum matching problem. Stronger isolation lemmas have been introduced in the literature to fit different needs in various settings. For example, the isolation lemma of has similar guarantees as that of Mulmuley et al., but it uses fewer random bits. In the context of the exponential time hypothesis, prove an isolation lemma for k-CNF formulas. Noam Ta-Shma〔Noam Ta-Shma (2015); (''A simple proof of the Isolation Lemma'' ), in ''eccc''〕 gives an isolation lemma with slightly stronger parameters, and gives non-trivial results even when the size of the weight domain is smaller than the number of variables. ==The isolation lemma of Mulmuley, Vazirani, and Vazirani== :Lemma. Let and be positive integers, and let be an arbitrary family of subsets of the universe . Suppose each element in the universe receives an integer weight , each of which is chosen independently and uniformly at random from . The weight of a set ''S'' in is defined as :: :Then, with probability at least , there is a ''unique'' set in that has the minimum weight among all sets of . It is remarkable that the lemma assumes nothing about the nature of the family : for instance may include ''all'' nonempty subsets. Since the weight of each set in is between and on average there will be sets of each possible weight. Still, with high probability, there is a unique set that has minimum weight. Suppose we have fixed the weights of all elements except an element ''x''. Then ''x'' has a ''threshold'' weight ''α'', such that if the weight ''w''(''x'') of ''x'' is greater than ''α'', then it is not contained in any minimum-weight subset, and if , then it is contained in some sets of minimum weight. Further, observe that if , then ''every'' minimum-weight subset must contain ''x'' (since, when we decrease ''w(x)'' from ''α'', sets that do not contain ''x'' do not decrease in weight, while those that contain ''x'' do). Thus, ambiguity about whether a minimum-weight subset contains ''x'' or not can happen only when the weight of ''x'' is exactly equal to its threshold; in this case we will call ''x'' "singular". Now, as the threshold of ''x'' was defined only in terms of the weights of the ''other'' elements, it is independent of ''w(x)'', and therefore, as ''w''(''x'') is chosen uniformly from , : |