Skip to content

DynamicConnectivity.md

Latest commit

 

History

History
137 lines (92 loc) · 3.57 KB

DynamicConnectivity.md

File metadata and controls

137 lines (92 loc) · 3.57 KB
Raw

Given a set of N objects

- Union command: connect two objects

- Find/connected query: is there a path connecting two objects?

Suppose we give:

union(4,3) union(3,8) union(6,5) union(9,4) union(2,1)

this forms such a structure:

0 1 ---------- 2 3---------4 | | 5-----6 7 8 9

now we give a query connected(0,7) - returns false connected(8,9) - returns true (because 8 is connected to 3, which is connected to 4, which is connected to 9; therefore 8 is connected to 9)

now let's say we say: union(5,0) union(7,2) union(6,1) union(1,0)

we get a new structure:

0 --- 1 ---------- 2 3---------4 | | | | | 5-----6 7 8 9

connected(0,7) - returns true

Applications of this data structure involves manipulating objects of all types:

  • Pixels in a digital photo
  • Computers in a network
  • Friends in a social network
  • Transistors in a computer chip
  • Elements in a mathematical set
  • Variable names in Fortran program(Abstract things)
  • Metallic sites in a composite system(Physical things)

When programming, convenient to name objects 0 to N-1:

  • use integers as array index
  • Suppress details not relevant to union-find

We need few abstract properties these connections need to satisfy

We assume "is connected to" is an equivalence relation:

  • Reflexive: p is connected to p
  • Symmetric: if p is connected to q, then q is connected to p
  • Transitive: if p is connected to q, and q is connected to r, then p is connected to r

When we have an equivalence relation a set of objects and connections divide into subsets called connected components.

A connected component is a maximal set of objects that's mutually connected.

take a structure:

0 1 2---3 /| | /| / | | / | 4---5 6 7

{0} {1,4,5} {2,3,6,7}

Implementation:

Find query: Check if two objects are in the same component Union command: Replace components containing two objects with their union

Suppose we have

0 1 2---3 0 1 2 3 /| | /| ==========> / /| /| / | | / | union(2,5) / / | / | 4---5 6 7 4----5 6 7

{0} {1,4,5} {2,3,6,7} {0} {1,2,3,4,5,6,7}

Union-find datatype(API)

Goal: Design efficient data structure for Union-find

  • Number of objects N can be huge
  • Number of operations M can be huge
  • Find queries and union commands may be intermixed.

public class UF UF(int N) initialize union-find data structure with N objects(0 to N-1)

void union(int p, int q)    add connection between p and q

boolean connected(int p, int q) are p and q in the same component?

int find(int p) component identifier for p(0 to N-1)

int count() number of components

(1,2,7,8,9)(3,4)(0,5,6)

Dynamic-connectivity client

Read in number of objects N from standard input. ・Repeat: – read in pair of integers from standard input – if they are not yet connected, connect them and print out pair

Solution:

public static void main(String[] args) { int N = StdIn.readInt(); UF uf = new UF(N); while (!StdIn.isEmpty()) { int p = StdIn.readInt(); int q = StdIn.readInt(); if (!uf.connected(p, q)) { uf.union(p, q); StdOut.println(p + " " + q); } } }