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The notion of fault local mending was suggested as a paradigm for designing fault tolerant algorithms that scale to large networks. For such algorithms the complexity of recovering is proportional to the number of faults. We refine this notion by introducing the concept of tight fault locality to deal with problems whose complexity (in the absence of faults) is sublinear in the size of the network. For a function whose complexity on an n-node network is f(n), a tightly fault local algorithm recovers a legal global state in O(f(x)) time when the (unknown) number of faults is x. We illustrate this concept by presenting a general transformation for MIS algorithms to make them fault local. In particular, our transformation yields an O(logx) randomized mending algorithm and a 2/sup /spl radic//spl beta/logx/ deterministic mending algorithm for MIS. Similar results are obtained for other local functions such as a /spl Delta/+1 coloring. We also present the first tight fault local mending algorithm for global functions, using our results for MIS. This improves (by a logarithmic factor) the complexity of a previous fault-local mending algorithm for global functions.