EECS 311: Chapter 8 Notes Chris Riesbeck EECS Northwestern

2  Unless otherwise noted, all tables, graphs and code from Mark Allen Weiss' Data Structures and Algorithm Analysis in C++, 3 rd ed, copyright © 2006 by Pearson Education, Inc.

3 Maze Maker Problem

4 Maze Maker Algorithm  Represent maze with 2D array of cells  Randomly knock down walls until all cells connected by some path.

5 Maze Maker Algorithm  To avoid short paths, don't knock down wall if 2 cells already connected. E.g., leave wall between 8 and 13  Problem: how to quickly tell if two cells already connected

6 Relations  Given a set of objects a, b, c, …  A relation R is a binary predicate between objects electricallyConnected(component 1, component 2 ) semanticallyRelated(document 1, document 2 )  i.e., shares many terms ancestorOf(person 1, person 2 )  In mathematics, a relation is a set of pairs  In computation, representing a relation as all pairs is often infeasible

7 Equivalence Relations  R is an equivalence relation if R is Reflexive: R(a, a) is always true Symmetric: R(a, b)  R(b, a) Transitive: R(a, b) and R(b, c)  R(a, c)  Which of these are equivalence relations? electricallyConnected semanticallyRelated ancestorOf Yes Not transitive Not symmetric

8 Equivalence Class  The equivalence class for an object a and relationship R is the set { x | R(a, x) }  Every object belongs to one and only one equivalence class  Equivalence classes are disjoint  E.g., the sets of all connected components

9 Union/Find  find(x): return the equivalence class for x for relation R  find(x) = find(y) if and only if R(x, y)  union(set 1, set 2 ): destructively merge two equivalence classes  What represents an equivalence class can be anything as long as the above works.  Often, one object is used to represent each class

10 Implementing Equivalence Classes  Since we know a, b, c, … in advance  Make O(1) mapping from a, b, c, … to 0, 1, 2, …, N-1  Make array A[N] Let A[0] represent { a } Let A[1] represent { b } …  Initially, every object in its own class

11 Unioning Sets union(4, 5) union(6, 7)

12 Unioning Sets union(4, 6)

13 Representing Equivalence Classes  A[i] = j if element i points to j  A[i] = -1 for a "root" element

14 union(int x, int y) // x,y are roots a[y] = x find(int x) if a[x] < 0 return x else return find(a[x]) Implementing union/find What are the complexities of union() and find()? What is an iterative version of find()?

15 A Union Problem union(3, 4) Assuming find() more frequent than union(), how avoid this?

16 Union by Size (or Height) union(3, 4) Join smaller set to the larger

17 Representing Size/Height size N = -N height N = -N

18 Path Compression  Shorten connections during find(x)  Reset every node between x and root to root find(x) if a[x] < 0 return x; else return a[x] = find( a[x] )

19 Path Compression find(14) worst case using union by size