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Unformatted text preview: 6.851: Advanced Data Structures Spring 2010 Lecture 18 April 14, 2010 Prof. Erik Demaine 1 Overview In the last lecture we introduced Euler tour trees , dynamic data structures that can perform link-cut tree operations in O (lg n ) time. We then showed how to implement an ecient dynamic connectivity algorithm using a spanning forest of Euler tour trees, as demonstrated in . This yielded an amortized time bound of O (lg 2 n ) for update operations (such as edge insertion and deletion), and O (lg n/ lg lg n ) for querying the connectivity of two vertices. In this lecture, we switch to examining the lower bound of dynamic connectivity algorithms. Until recently, the best lower bound for dynamic connectivity operations was (lg n/ lg lg n ), as described by Fredman and Henzinger in  and independently by Miltersen in . However, we will show below that it is possible to prove (lg n ), using the method given by Patra scu and Demaine in . In fact, we prove this lower bound even if the connected components of our graph are always some paths. We use a weak assumption here that the word size is O (log n ). This is equivalent to assuming that the space of our algorithm (our problem) is polynomial in the size of the input which is a reasonable assumption. Note that handling super-polynomial space probably needs a super-polynomial time algorithm. 2 Cell Probe Complexity Model The (lg n ) bound relies on a model of computation called the cell probe complexity model , originally described in the context of proving dynamic lower bounds by Fredman and Saks in . The cell probe model views a data structure as a sequence of cells , or words, each containing a w-bit field. The model calculates the complexity of an algorithm by counting the number of reads and writes to the cells; any additional computation is free. This makes the model comparable to a RAM model with constant time access. Because computation is free, the model is not useful for determining upper bounds, only lower bounds. We empirically assume that the size of each cell, w , is at least lg n . This is because we would like the cells to store pointers to each of our n vertices, and information theory tells us that we need lg n bits to address n items. For the the following proof, we will further assume that w = (lg n ). In this sense, the cell probe model is a transdichotomous model , as it provides a bridge between the problem size, n , and the cell or machine size, w . 3 Dynamic Connectivity Lower Bound for Paths The lower bound we are trying to determine is the best achievable worst-cast time for a sequence of updates and queries to a path. We will prove the following: 1 Theorem 1. Under the cell probe model, the lower bound worst-case cost is (lg n ) per operation....
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- Spring '10
- Data Structures