Algorithms

1.5 Union-Find

public class UF {

void union(int p, int q)

boolean connected(int p, int q)

int find(int p)

int count()

}

Dynamic connectivity 

1.5.1 Quick-Find <initialize N, union N, connect 1>  too slow, for huge bata
Quick-Find defect: union too expensive.(N array accesses).

takes N^2 array accesses to process sequence of N union commands

trees are flat, but too expensive to keep them flat.

public class QuickFindUF {

private int[] id;

public QuickFindUF(int N){

id = new int[N];

for (int i=0;i < N; i++) 

id[i] = i;

}

}

1.5.2 Quick-Union [lazy approach] <initialize N, union N, connect N>

Quick-Union defect: Trees can get tall

  Find too expensive(could be N array accesses)

1.5.3 Improvement:

1. weighted quick-union:<initialize N, union logN, connect logN>

same as quick-union data structure, but maintain extra array sz[i] to count number of objects in the tree rooted at i 

we make the root of the smaller tree points to the root of the larger tree

Run time: find: depth of p and q. 

union: constant time, given roots.

2. path compression:quick union with path compression

private int root(int i){

while(i != id[i]){

id[i] =id[id[i]];    <—only one extra line 

i = id[i];

}

}

algorithm        worst-case time  (M union-find operations on a set of N objects)

quick-find           MN

quick-union        MN

weighted QU     N + MlogN

QU + path compression   N + MlogN

weighted QU +path compression N + MlgN

1.5.4 Application.

Social Network, (people know other people)

Percolation. (top to bottom) blocked or not blocked, N * N grid

Monte Carlo simulation

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