Minimum Spanning Trees

1
Minimum Spanning Trees

• Definition
• Two properties of MSTs
• Prim and Kruskals Algorithm
• Proofs of correctness
• Boruvkas algorithm
• Verifying an MST
• Randomized algorithm in linear time

2
Problem Definition

• Input
• Weighted, connected undirected graph G(V,E)
• Weight (length) function w on each edge e in E
• Task
• Compute a spanning tree of G of minimum total
  weight
• Spanning tree
• If there are n nodes in G, a spanning tree
  consists of n-1 edges such that no cycles are
  formed

3
Example
Input
4
Two Properties of MSTs

• Cycle Property For any cycle C in a graph, the
  heaviest edge in C does not appear in the minimum
  spanning tree
• Used to rule edges out
• Cut Property For any proper non-empty subset X
  of the vertices, the lightest edge with exactly
  one endpoint in X belongs to the minimum spanning
  forest
• Used to rule edges in

5
Cycle Property Illustration/Proof

• Proof by contradiction
• Suppose T is an MST with such an edge e.
• Derive a contradiction showing that T is not an
  MST

6
Cut Property Illustration/Proof

• Proof by contradiction
• Suppose T is an MST without such an edge e.
• Derive a contradiction showing that T is not an
  MST

7
Three Classic Greedy Algorithms

• Kruskals approach
• Select the minimum weight edge that does not form
  a cycle
• Prims approach
• Choose an arbitrary start node v
• At any point in time, we have connected component
  N containing v and other nodes V-N
• Choose the minimum weight edge from N to V-N
• Boruvkas approach
• Prim in parallel

8
Example

• Illustrate the execution of Kruskal and Prims
  algorithms on the following input graph. Let D
  be the arbitrary start node for Prims algorithm.

9
Prim Implementation

• Use a priority queue to organize nodes in V-N to
  facilitate finding closest node.
• Extract-Min operation
• Decrease-key operation
• Describe how we could implement Prim using a
  priority queue.

10
Running Time of Prim

• How many extract-min operations will we perform?
• How many decrease-key operations will we perform?
• How much time to build initial priority queue?
• Given binary heap implementation, what is the
  running time?
• Fibonacci heap
• Decrease-key drops to O(1) amortized time

11
Kruskal Implementation

• Kruskals Algorithm
• Adding edge (u,v) to the current set of edges T
  forms a cycle if and only if u and v are in the
  same connected component.

12
Disjoint Set Data Stucture (Ch 21)

• Given a universe U of objects
• Maintain a collection of sets Si such that
• Unioni Si U
• Si intersect Sj is empty
• Find-set(x) Returns set Si that contains x
• Merge(Si, Sj) Returns new set Sk Si union Sj
• Describe how we can implement cycle detection
  with this data structure.

13
Running Time of Kruskal

• How many merges will we perform?
• How many Find-set operations will we perform?
• Each can be implemented in amortized a(V) time
  where a is a very slow growing function.
• What other operations do we need to implement?
• Overall running time?

14
Proofs of Correctness

• Why do we know each edge that is added in
  Kruskals algorithm is part of an MST?
• Why do we know that each edge added in Prims
  algorithm is part of an MST?

15
Boruvkas Algorithm

• Prim in parallel
• Boruvka Step
• We have a graph of vertices
• For each v in V, select the minimum weight edge
  connected to v
• Update
• Contract all selected edges, replacing each
  connected component by a single vertex
• Delete loops, and keep only the lowest weight
  edge in a multi-edge
• Run Boruvka steps until we have a single node

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Boruvka Step Illustration
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Boruvkas Algorithm Analysis

• Correctness
• How can we verify that each edge we add is part
  of an MST?
• Running time
• What is the running time of a Boruvka Step?
• How many Boruvka steps must we implement in the
  worst case?

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Verification Problem

• Input
• Weighted, connected undirected graph G(V,E)
• Weight (length) function w on each edge e in E
• A spanning tree T of G
• Task
• Answer yes/no if T is a minimum spanning tree for
  G
• Two key concepts
• Decision problem problem with yes/no answer
• Verification Is it easier to verify an answer as
  correct as compared to generating an answer?

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Key idea T(u,v)

• Suppose we have a tree T
• For each edge (u,v) not in T, let T(u,v) be the
  heaviest edge on the (u,v) path in T
• If w(u,v) gt w(T(u,v)), then (u,v) should not be
  in T.

• Consider edge (B,F) with weight 7.
• T(B,F) (C,G) which has weight 6.
• (B,F) is appropriately not in T as w(B,F) gt
  w(C,G).

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Verification Algorithm

• For each edge (u,v) not in T, find T(u,v).
• Compare w (u,v) to w(T(u,v))
• If all are ok, then return yes, else return no.
• Running time?
• O(E) time to perform comparisons
• Sophisticated techniques to find all the T(u,v)
  in O(E) time.

21
Randomized Algorithm

• Run Baruvkas Step 2 times
• If there were originally n nodes, how many in
  reduced problem
• Random Sampling
• Make a smaller subgraph by choosing each edge
  with probability ½
• Recursively compute minimum spanning tree (or
  perhaps forest) F on this reduced graph
• Use verification algorithm to eliminate any edges
  (u,v) not in F whose weight is more than weight
  of F(u,v).
• Apply algorithm recursively to the remaining
  graph to compute a spanning tree T
• Knit together tree from step 1 with F to form
  spanning tree

22
Visualization
Graph Go
Step 1 Run 2 Baruvka Steps
Graph G3 G1 E2
Tree T4 T3 union E1
23
Analysis Intuition
Graph Go
Step 1 Run 2 Baruvka Steps
G2 has at most ½ the edges of G0
Step 2 Construct G2 by sampling ½ edges
Graph G3 G1 E2
Step 2 Recursively Construct T2 for graph G2
Step 2 Use T2 to identify edges E2 that cannot
be in MST for G1
G3 edges bounded by ½ nodes of G0
Tree T4 T3 union E1