Union By Rank Ackermann

1
Union By RankAckermanns FunctionGraph
Algorithms

• Rajee S Ramanikanthan
• Kavya Reddy Musani

2

• Union by Rank
• In Union, have parent of shallower tree point to
  other tree.
• Maintain rank(x) as an upper bound on the depth
  of the tree rooted at x.
• Consider the following example

x
y
b
d
s
c
m
l
h
a
3

• Rank of x is 3, and rank of y is 2
• Union (x,y) results in with the rank of the
  resultant tree greater rank

x
y
b
d
s
c
m
l
h
a
4

• If the two trees are of same rank then the rank
  of the resultant tree increases by one

x
y
d
s
c
l
h

• The resultant rank of the union is rank of x 1.

x
d
y
l
s
c
h
5
Algorithm for Path Compression

• 1st walk Find the name of the set . Take a walk
  until we reach the root.
• 2nd walk Retrace the path and join all the
  elements along the path to the root using another
  pointer.
• This enables future finds to take shorter paths.

6
Path Compression

• Find the root(x) by traversing parent pointers.
• Set each node traversed to the resulting root(x)

a
c
b
d
e
f
x
7
Path Compression

• For union by rank,

Best Case
1
Worst Case
Log n
8
Analysis

• Union by rank only
• Q (m lgn) where m of operations
• n MakeSet operations in m
• Path compression only
• Q(n f lgn) fltn
• f FindSet operations
• n MakeSet operations
• There are always lt n-1 Unions
• Union by Rank and Path Compression
• For n-1 union and m finds the running time is
• (n m  a (n))
• a (n) is inverse of Ackermann's function

9
Amortised analysis for Path Compression and Union
by Rank

• Time for n-1 unions and n finds
• O(n logn)

logn is a slow growing function
n log n log n
22 2 1
222 4 2
2222 222 3
22222 2222 4
10
Ladder Function

• This is a very fast growing function,even faster
  than exponential
• Ladder (n) 222...2

n times
n 2n Ladder(n)
1 2 4
2 4 16
3 8 256
4 16 65536
5 32 4294967296
11
Ackermann's Function

• Ai (j) is Ackermanns function, it mimics a
  ladder function for higher values of i
• Ai(j) A i-1 A i-1 A i-1............ A i-1(j)

j1 times

• Ai (j) j1 if i0
• 2j1 if i1
• (j1) 2j1 -1
• B(j)Aj(j)
• a(n)Inverse of B

12
Comparisons of functions
F(n)n
Log n
Logn
4
?(n)
13
Graph Algorithms
14
KONIGSBERG BRIDGES

A
B
C
The town of Konigsberg( now kalliningrad) lay on
the banks and on two islands of the Predal
river. The city was connected by 7 bridges. The
puzzle whether it was possible to start walking
from anywhere in town and return to the starting
point by crossing all bridges exactly once.
15
GRAPHS

• A Graph G(V,E) is a finite nonempty set V of
  objects called vertices together with a set E of
  unordered pairs of distinct vertices of G called
  edges.
• A directed graph(digraph) G(V,E) is a finite
  nonempty set V of vertices together with a set E
  of ordered pairs of vertices of G called arcs. A
  directed graph is not a symmetric matrix.
• Weighted Graph A graph having a weight, or
  number, associated with each edge.A weighted
  graph is usually implemented using adjacency
  matrix.

16
CONNECTIVITY PROBLEMS

• BICONNECTIVITY PROBLEMS In a biconnected graph
  if a path fails , then we have another path. A
  biconnected graph is a graph from which at least
  two nodes have to be deleted to break it up into
  disconnected pieces.
• REACHABILITY/COVERING PROBLEMS Given a graph,
  then we need to find the whether we can cover
  the required path.
• REPRESENTATION PROBLEM Representing graphs in
  computer.

17
GRAPH TERMINOLOGIES
M
A
D
N
E
O
B
C
P
Q
DIRECTED GRAPH
UNDIRECTED GRAPH
18
PATHS AND CYCLES

• A path from vertex v1 to vk is a sequence of
  vertices v1,v2, , vk that are connected by edges
  (v1,v2), (v2,v3), , (vk-1,vk).
• A path is simple if each vertex in it appears
  only once.
• Vertex u is said to be reachable from v if there
  is a path from v to u.
• A circuit is a path whose first and last vertices
  are the same.
• A simple circuit is a cycle if except for the
  first (and last) vertex, no other vertex appears
  more than once.
• A Hamiltonian cycle of a graph G is a cycle that
  contains all the vertices of G

19
GRAPH REPRESENTATIONS ADJACENCY MATRIX

• The adjacency matrix for a finite graph G on n
  vertices
• is an nn matrix where the nondiagonal entry
  a(i,j) is the
• number of edges joining vertex i and vertex
  j, and the
• diagonal entry a(i,i) is either twice the
  number of loops at
• vertex i or just the number of loops. There
  exists a
• unique adjacency matrix for each graph. If
  the graph is
• undirected, the adjacency matrix is
  symmetric. For
• dense graphs, that is graph with more edges,
  an
• adjacency matrix is often preferred.

20
ADJACENCY LIST

• An adjacency list is the representation of all
  edges or
• arcs in a graph as a list. If a graph is
  undirected, every
• entry is a set of two nodes containing the
  two ends of the
• corresponding edge if it is directed, every
  entry is a
• tuple of two nodes, one denoting the source
  node and
• the other denoting the destination node of
  the
• corresponding arc. Adjacency lists are
  unordered. For a
• graph with a sparse adjacency matrix an
  adjacency list
• representation of the graph occupies less
  space,
• because it does not use any space to
  represent edges
• which are not present.

21
GRAPH REPRESENTATIONS
22
DEPTH FIRST SEARCH

• DFS is an uninformed search that progresses by
  expanding the first child node of the search tree
  that appears and thus going deeper and deeper
  until a goal state is found, or until it hits a
  node that has no children.
• Then the search backtracks and starts off on the
  next node.
• In a non-recursive implementation, all freshly
  expanded nodes are added to a stack for
  expansion.
• Time complexity is equal to the number of
  vertices plus the number of edges in the graphs
  they traverse.
• When searching large graphs that can not be fully
  contained in memory, DFS suffers from
  non-termination when the length of a path in the
  search tree is infinite. This can be solved by
  maintaining an increasing limit on the depth of
  the tree, which is called iterative deepening
  depth first search.

23
(No Transcript)
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
BREADTH FIRST SEARCH

• BFS is an uninformed search method that aims to
  expand and examine all nodes of a tree
  systematically in search of a search of a
  solution. In other words, it exhaustively
  searches the entire tree without considering the
  goal until it finds it. It does not use a
  heuristic.
• All child nodes obtained by expanding a node are
  added to a FIFO queue. In typical
  implementations, nodes that have not yet been
  examined for their neighbors are placed in some
  container called open and then once examined
  are placed in the container closed.
• Time complexity is equal to the number of
  vertices plus the number of edges in the graphs
  they traverse.
• BFS has space complexity linear in size of the
  tree/graph searched as it needs to store all
  expanded nodes in memory.

28
(No Transcript)
29
(No Transcript)
30
C
B
31
(No Transcript)
32
REFERENCES

• Dr.Kumars notes
• Dr.Cooks notes
• Fall 2004 notes
• www.mathworld.com
• en.wikipedia.org