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 Equivalence Relations
 The Dynamic Equivalence Problem
 Simple algorithms for solving:
 Find\Union
 Smart Union Algorithms
 Path Compression
 Example
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 Relation which satisfies the three properties:
 Reflexive Reflexive ‐ a R a, for all a ‫ א‬S
 Symmetric ‐ a R b if and only if b R a
 Transitive – a R b and b R c implies that a R c
 Examples:
 Electrical Connectivity
 Same country: Two cities are related if they belong to the same country
t
 Given an equivalence relation ~, decide if a~b for any a and b
 Solving via a 2‐d array of booleans
 What if relation is implicit?
 Equivalence class of an element a ‫ א‬S is a subset of S which contains all the elements that are related to a
 Check if a and b belong to the same equivalence class
 Disjoint sets
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 find: returns the equivalence class for a given element
 union: adds relations
 Merges two equivalence classes
 Dynamic : the sets change during the course of the algorithm
 Observations:
 Values don’t matter, location matters
 find(a) == find(b) matters
 find in constant worst‐time?
 union in constant worst
union in constant worst‐time?
time?
 Consider:
 Each set as a tree
 Initially – all elements are disjoint
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 union : make parent link of root of one tree to the root f
of the other tree
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 union‐by‐size
 Always make smaller tree subtree of the larger
 No extra space : instead of ‐1 set as ‐(size of tree) for root
 union‐by‐height: Make a shallow tree a subtree of a p
deep tree
 Update height only if two equally deep trees are joined
 Instead of ‐1 , ‐(height) for root
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 Independent of union algorithm
 Make every node on the path from x to root have its parent changed to root.
 Note that: union‐by‐height => union‐by‐rank
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Start with walls everywhere, except for the entrance and exit
2. Continually, choose a wall randomly, and knock it down if the cells that the wall separates are not already connected to each other.
3. Repeat this process until the starting and ending cells are connected 4. We have a maze.
1.
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 Start with walls everywhere, except for the entrance and exit.
2. Continually, choose a wall randomly, and knock it p
down if the cells that the wall separates are not already connected to each other.
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3.
Repeat this process until the starting and ending cells are connected
4. We have a maze
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