Connected Components for Undirected Graphs

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Connected Components for Undirected Graphs

• Irena Pevac

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Connected Components

• Def Connected component of an undirected graph
  is a subgraph such that any two vertices within
  it are connected with a path.
• Relation P is defined as uPv iff u is connected
  with a path to v.
• P is an equivalence relation.
• P is reflexive For every node u uPu
• P is symmetric For every two nodes u,v
  uPvgtvPu
• P is transitive For every three nodes u,v,w
• uPv AND vPw gt uPw

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Equivalence Relation P

• Equivalence relation P partitions the set of
  vertices into equivalence classes.
• The class containing node u is the set of all
  nodes v uPv.
• Example G(V,E) V 1,2,3,4,5,67,8,9
• E1,2, 2,4, 2,6, 4,6, 3,5, 3,9,
  5,9, 7,8
• G has 3 connected components.

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Find Connected Components

• Initially, all nine nodes are in components of
  their own.
• If u and v are in the same component Gi then Gi1
  has the same set of connected components as Gi
• If u and v are in different components, we merge
  the components containing u and v to get
  connected components for Gi1. Whenever we merge
  two trees, the root of a lesser child becomes a
  child of the root with greater height.

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Running Time for Connected Components

• Let graph G(V,E)
• Let V n, and m be the greater of the number
  of edges and the number of nodes.
• The policy of merging lower trees into higher
  ones guarantees that path from any node to its
  root is less than log n.
• Total time is O(m log n).

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Minimal Spanning Tree

• A spanning tree for an undirected graph G is all
  the nodes of G together with a subset of edges of
  G such that
• There is a path between any two nodes using only
  the edges in the spanning tree.
• There are no cycles. (It forms an unrooted
  unordered tree.)
• A minimal spanning tree is a spanning tree with
  minimal sum of edges labels.

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Applications

• Find the cheapest way to connect
• a set of cities by roads
• A set of terminals by wires
• A set of computers by telephone lines
• One undirected connected weighted (labeled)
  graph can have more that one minimal spanning
  tree.

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Kruskals Algorithm

• Kruskals Algorithm for finding minimal spanning
  tree is slight modification of the algorithm to
  find connected components
• At the beginning each node is in its own
  component.
• We consider the edges in the order of their
  labels. (smallest label first)
• If an edge has its ends in different components
  we select that edge for the spanning tree and
  merge components. Otherwise , we do not select
  that edge for the spanning tree.

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Graph G(V,E)
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Edges Ordered by Weight

• Select edge B D
• Select edge C D
• Select edge D F
• Select edge A B
• Do not select edge B C
• Select D E
• Do not select C F
• Do not select E F

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Graph G(V,E)
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Select A B
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Select C D
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Select D F
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Select A B
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Resulting Minimal Spanning Tree
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Greedy Algorithm

• Kruskals algorithm is an example of a greedy
  algorithm.
• In a greedy algorithm we make series of
  decisions, each one doing what seems best at the
  time.
• The local decisions are which edge to add to the
  spanning tree being formed. We pick up edge with
  the least label that does not violate the
  definition of the spanning tree.
• In this algorithm local optimum is also globally
  optimal.

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Running Time for Kruskals Algorithm

• Let graph G(V,E)
• Let V n, and m be the greater of the number
  of edges and the number of nodes. (Typically, the
  number of edges is larger.)
• If graph is represented with adjacency lists, we
  can find edges in O(m) time.
• To sort edges we need at least O(m log m) time.
• To do the algorithm we need O(m logn) time.
• Overall time is O( m(lognlogm) )
• Since mltn2 (there are at most n(n-1) /2 edges)
  it follows log m lt2 log n gt m(lognlogm) lt
  3mlogn
• So, time for Kruskal algorithm is O(m logn).