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Minimum Spanning Trees

Minimum Spanning Trees. Definition Two properties of MST’s Prim and Kruskal’s Algorithm Proofs of correctness Boruvka’s algorithm Verifying an MST Randomized algorithm in linear time. Problem Definition. Input Weighted, connected undirected graph G=(V,E)

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Minimum Spanning Trees

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  1. Minimum Spanning Trees • Definition • Two properties of MST’s • Prim and Kruskal’s Algorithm • Proofs of correctness • Boruvka’s 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. 1 4 8 A 1 B 4 C D 8 A B C D 7 Output: 6 2 9 3 7 6 2 5 9 3 E F G 5 E F G Example Input:

  4. Two Properties of MST’s • 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. 1 4 8 A B C D 7 6 2 9 3 5 E F G 1 4 8 A B C D 7 6 2 9 3 5 E F G 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. 1 4 8 A B C D 7 6 2 9 3 5 E F G 1 4 8 A B C D 7 6 2 9 3 5 E F G 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 • Kruskal’s approach • Select the minimum weight edge that does not form a cycle • Prim’s 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 • Boruvka’s approach • Prim “in parallel”

  8. 1 4 8 A B C D 7 6 2 9 3 5 E F G Example • Illustrate the execution of Kruskal and Prim’s algorithms on the following input graph. Let D be the arbitrary start node for Prim’s algorithm.

  9. 1 4 8 A B C D 7 6 2 9 3 5 E F G 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. 1 4 8 A B C D 7 6 2 9 3 5 E F G Kruskal Implementation • Kruskal’s 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. 1 4 8 A B C D 7 6 2 9 3 5 E F G 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 Kruskal’s algorithm is part of an MST? • Why do we know that each edge added in Prim’s algorithm is part of an MST?

  15. Boruvka’s 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

  16. 1 9 8 A B C D 7 6 2 4 3 5 E F G 1 9 8 A B C D 7 7 A’ B’ 6 2 4 3 5 E F G Boruvka Step Illustration

  17. Boruvka’s 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?

  18. 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?

  19. 1 4 8 A B C D 7 6 2 9 3 5 E F G 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) > 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) > w(C,G).

  20. 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 Baruvka’s 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. Contracted Graph G1 List of edges E1 Step 2: Construct G2 by sampling ½ edges Step 2: Recursively Construct T2 for graph G2 Step 2: Use T2 to identify edges E2 that cannot be in MST for G1 Step 3: Recursively Construct T3 for graph G3 Step 3: Return T3 melded with edges E1 as final tree T4 Visualization Graph Go Step 1: Run 2 Baruvka Steps: Graph G3 = G1 – E2 Tree T4 = T3 union E1

  23. Contracted Graph G1 List of edges E1 Step 3: Recursively Construct T3 for graph G3 Step 3: Return T3 melded with edges E1 as final tree T4 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

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