Dynamic Equivalence and Two Applications in Java

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are in the same equivalence class; if they are not, we apply u n i o n ' This operation merges the two equivalence classes containing a and b into a new equivalence class In terms of sets the result is a new set Sk = Si u S j , which we create by simultaneously destroying the originals and preserving the disjointedness of all the sets The data structure to do this is often called the disjoint set uniodfind data structure The unionfind algorithm is executed by processing unionlfind requests within the disjoint set data structure The algorithm is d,vrzarnic because, during the course of algorithm execution, the sets can change via the u n i o n operation The algorithm must also operate as an online algorithm so that, when a f i n d is performed, an answer must be given before the next query can be viewed Another possibility is an offline algorithm in which the entire sequence of u n i o n and f i n d requests are made visible The answer it provides for each f i n d must still be consistent with all the unions performed before the f i n d However, the algorithm can give all its answers after it has dealt with all the questions This distinction is similar to the difference between taking a written exam (which is generally offline because you only have to give the answers before time expires) and taking an oral exam (which is online because you must answer the current question before proceeding to the next question) Note that we do not perform any operations to compare the relative values of elements but merely require knowledge of their location For this reason we can assume that all elements have been numbered sequentially, starting from 0, and that the numbering can be determined easily by some hashing scheme Before describing how to implement the u n i o n and f i n d operations, we provide three applications of the data structure
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In an online algorithmqan answer must be provided for ,uervbefore the next query can be viewed
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The set elements are sequentially, starting
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2421 Application: Generating Mazes
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An example of the use of the uniontfind data structure is to generate mazes, such as the one shown in Figure 241 The starting point is the top-left comer, and the ending point is the bottom-right corner We can view the maze as a 50 x 88 rectangle of cells in which the top-left cell is connected to the bottomright cell, and cells are separated from their neighboring cells via walls A simple algorithm to generate the maze is to start with walls everywhere (except for the entrance and exit) We then continually choose a wall randomly, and knock it down if the cells that the wall separates are not already connected to each other If we repeat this process until the starting and ending cells are connected, we have a maze Continuing to knock down
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I Unfortunately union is a (little-used) keyword in C++ We use union throughout this section but when we write code we use unionsets as the function name
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Figure 242 Initial state: All walls are up, and all cells are in their own sets
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walls until every cell is reachable from every other cell is actually better because doing so generates more false leads in the maze We illustrate the algorithm with a 5 x 5 maze, and Figure 242 shows the initial configuration We use the unionlfind data structure to represent sets of cells that are connected to each other Initially, walls are everywhere, and each cell is in its own equivalence class Figure 243 shows a later stage of the algorithm, after a few walls have been knocked down Suppose, at this stage that we randomly target the wall that connects cells 8 and 13 Because 8 and 13 are already connected (they are in the same set), we would not remove the wall because to do so would simply
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