Spanning Trees â€” Minimum-weight spanning trees

Spanning Trees — §7.1 116

Minimum-weight spanning trees

Motivation: Create a connected network as cheaply as possible.

� Think: Setting up electrical grid or road network.

� Some connections are cheaper than others.

� Only need to minimally connect the vertices.







Definition: A weighted graph consists of a graph G = (V ,E )and weight function w : E → R defined on the edges of G .The weight of a subgraph H of G is the sum of the edges in H.

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Definition: A weighted graph consists of a graph G = (V ,E )and weight function w : E → R defined on the edges of G .The weight of a subgraph H of G is the sum of the edges in H.

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Definition: For a graph G , a spanning tree T is a subgraph of Gwhich is a tree and contains every vertex of G .

Goal: For a weighted graph G , find a minimum-weight spanning tree.


Kruskal’s algorithm

Kruskal’s Algorithm finds a minimum-weight spanning tree in aweighted graph.

1 Initialization: Order the edges from lowest to highest weight:

w(e1) ≤ w(e2) ≤ w(e3) ≤ · · · ≤ w(ek).





w(e1) ≤ w(e2) ≤ w(e3) ≤ · · · ≤ w(ek).

2 Step 1: Define T = {e1} and grow the tree as follows:





w(e1) ≤ w(e2) ≤ w(e3) ≤ · · · ≤ w(ek).


3 Step i : Determine if adding ei to T would create a cycle.

� If not, add ei to the set T .� If so, do nothing.





w(e1) ≤ w(e2) ≤ w(e3) ≤ · · · ≤ w(ek).


3 Step i : Determine if adding ei to T would create a cycle.

� If not, add ei to the set T .� If so, do nothing.

If you have a spanning tree, STOP. You have a m.w.s.t.Otherwise, continue onto step i + 1.



Example. Run Kruskal’s algorithm on the following graph:

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Notes on Kruskal’s algorithm

� Proof of correctness similar to homework. Must additionallyverify that the spanning tree is indeed minimum-weight.




� Kruskal’s algorithm is an example of a greedy algorithm.(It chooses the cheapest edge at each point.)




� Kruskal’s algorithm is an example of a greedy algorithm.(It chooses the cheapest edge at each point.)

� Greedy algorithms don’t always work.

The Traveling Salesman Problem 120

The traveling salesman problem

Motivation: Visit all nodes and return home as cheaply as possible.




� Least cost trip flying between five major cities.

� Optimal routes for delivering mail, collecting garbage.

� Finding a trip to all buildings on campus, return.




� Least cost trip flying between five major cities.

� Optimal routes for delivering mail, collecting garbage.

� Finding a trip to all buildings on campus, return.

Goal: Find a minimum-weight Hamiltonian cycle in a weighted graph.




� Least cost trip flying between five major cities.� Optimal routes for delivering mail, collecting garbage.� Finding a trip to all buildings on campus, return.


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We can not use a greedy algorithm to find this TSP tour!



� It is hard to find an optimum solution.




� Goal: Create an easy-to-find pretty good solution.





Theorem. When the edge weights satisfy the triangle inequality,the tree shortcut algorithm finds a tour that costs at most twicethe optimum tour.

Recall. The triangle inequality says that if x , y , and z arevertices, then wt(xy) + wt(yz) ≤ wt(xz).






Recall. The triangle inequality says that if x , y , and z arevertices, then wt(xy) + wt(yz) ≤ wt(xz).

Example: Euclidean distances

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Finding a good TSP-tour

The Tree Shortcut Algorithm to find a good TSP-tour

1 Find a minimum-weight spanning tree (Use Kruskal’s Algorithm)2 Walk in a circuit around the edges of the tree.3 Take shortcuts to find a tour.

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Proof of theorem



Proof of theorem


Proof. Define:

� TSPA: TSP tour from shortcutting spanning tree

� CIRCA: Circuit constructed by doubling spanning tree

� MST : Minimum-weight spanning tree

� TSP∗: Minimum-weight TSP tour


Proof of theorem


Proof. Define:

� TSPA: TSP tour from shortcutting spanning tree

� CIRCA: Circuit constructed by doubling spanning tree

� MST : Minimum-weight spanning tree

� TSP∗: Minimum-weight TSP tour

Then,

wt(TSPA) ≤ wt(CIRCA) = 2wt(MST ) ≤ 2wt(TSP∗).

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Spanning Trees â€” Minimum-weight spanning trees