Lecture 7 Greedy algorithms: minimum spanning tree

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Lecture 7 Greedy algorithmsminimum spanning
tree

• COMP 523 Advanced Algorithmic Techniques
• Lecturer Dariusz Kowalski

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Overview

• Previous lectures
• Greedy algorithms
• Interval scheduling
• This lecture
• Minimum spanning tree - many greedy approaches

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Greedy algorithms paradigm

• Algorithm is greedy if
• it builds up a solution in small steps
• it chooses a decision at each step myopically to
  optimize some underlying criterion
• Analyzing optimal greedy algorithms by showing
  that
• in every step it is not worse than any other
  algorithm, or
• every algorithm can be gradually transformed to
  the greedy one without hurting its quality

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Minimum spanning tree

• Input weighted graph G (V,E)
• every edge in E has its positive weight
• Output finding the spanning tree such that the
  sum of weights is not bigger than the sum of
  weights of any other spanning tree
• Spanning tree subgraph with
• no cycle, and
• connected (every two nodes in V are connected by
  a path)

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Properties of minimum spanning trees MST

• Spanning trees
• n nodes
• n - 1 edges
• at least 2 leaves (leaf - a node with only one
  neighbor)
• MST cycle property
• After adding an edge we obtain exactly one cycle
  and all the edges from MST in this cycle have no
  bigger weight than the weight of the added edge

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cycle
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Crucial observation about MST

• Consider sets of nodes A and V - A
• Let F be the set of edges between A and V - A
• Let a be the smallest weight of an edge from F
• Theorem
• Every MST must contain at least one edge of
  weight a
• from set F

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Proof of the observation

• Let e be the edge in F with the smallest weight -
  for simplicity
• assume that there is unique such edge. Suppose to
  the contrary that
• e is not in some MST. Choose one such MST.
• Add e to MST - obtain the cycle, where e is
  (among) smallest weights.
• Since two ends of e are in different sets A and V
  - A,
• there is another edge f in the cycle and in F.
  Remove f from the tree
• (with added edge e) - obtain a spanning tree with
  the smaller weight
• (since f has bigger weight than e). This is a
  contradiction with MST.

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Greedy algorithm finding MST

• Kruskals algorithm
• Sort all edges according to the weights
• Choose n - 1 edges one after another as follows
• If a new added edge does not create a cycle with
  previously selected then we keep it in (partial)
  MST, otherwise we remove it
• Remark we always have a partial forest

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Greedy algorithm finding MST

• Prims algorithm
• Select a node as a root arbitrarily
• Choose n - 1 edges one after another as follows
• Look on all edges incident to the currently build
  (partial) tree and which do not create a cycle in
  it, and select one which has the smallest weight
• Remark we always have a connected partial tree

root
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Why the algorithms work?

• Follows from the crucial observation
• Kruskals algorithm
• Suppose we add edge v,w. This edge has the
  smallest weight among edges between the set of
  nodes already connected with v (by a path in
  selected subgraph) and other nodes.
• Prims algorithm
• Always chooses an edge with the smallest weight
  among edges between the set of already connected
  nodes and free nodes.

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Time complexity

• There are implementations using
• Union-find data structure (Kruskals algorithm)
• Priority queue (Prims algorithm)
• achieving time complexity
• O(m log n)
• where n is the number of nodes and m is the
• number of edges

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Conclusions

• Greedy algorithms for finding minimum
• spanning tree in a graph, both in time
• O(m log n)
• Kruskals algorithm
• Prims algorithm
• Remains to design the efficient data structures!

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Exercises

• Solved Exercise 3 from textbook - Chapter 4
  Greedy Algorithms