Algorithm Design and Analysis - University of Delawareudel.edu/~amotong/teaching/algorithm...

Algorithm Design and Analysis

Summer 2018 Amo G. Tong 1

Lecture 11Greedy Strategy• Minimum Spanning Tree


• Consider a weighted undirected graph 𝐺 = (𝑉, 𝐸).

• A spanning tree 𝑇 of 𝐺 is a connected subgraph with no cycle.• A subgraph is defined by a subset of the edges.

• A graph is connected if there is a path between each pair of nodes.

Minimum Spanning Tree




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• Build channels such that the stations are able to connected with each other.



2D map




2D map

• The following definitions are equivalent.

• A spanning tree is

• a connected subgraph with no cycle. (original)






• a connected subgraph consisting of two trees connected by a single edge. (recursive definition)







• A connected graph with exactly 𝑛 − 1 edges.








• A subgraph where there is a unique path between each pair of two vertices.








• A subgraph where there is a unique path between each pair of two vertices.

• Extra credits: prove the above four definitions are equivalent to each other.



• Consider a weighted undirected graph 𝐺 = (𝑉, 𝐸).

• A spanning tree 𝑇 is connected subgraph with no cycle.

• A cost of a tree is the sum of the weights of the edges.

• Problem: find a spanning tree of the minimum cost.






2D map

• Naïve algorithms.• Enumerate all the subgraphs

• Check if it is a spanning tree.

• Find the one with the minimum cost.

• How many subgraphs are there?

• Enumerate all the subgraphs with n-1 edges.• Check if it is a spanning tree.

• Find the one with the minimum cost.

• How many subgraphs with n-1 edges are there?

• …



• Two greedy algorithms.• Kruskal’s algorithm

• Prim’s algorithm

• Conditions for a spanning tree to be minimum.• Cut-optimality condition

• Path-optimality condition



• Kruskal’s algorithm• Order the edges by the weight non-decreasingly.




• Let (𝑒1, … , 𝑒𝑚) be the sorted sequence.





• 𝑇 = ∅

• Consider each edge e in (𝑒1, … , 𝑒𝑚) in order. Add 𝑒 to 𝑇 if 𝑇 ∪{𝑒} does not have a cycle.





• 𝑇 = ∅




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• 𝑇 = ∅




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• Exchange Property.• (𝑒1, … , 𝑒𝑚)

• There exists a minimum spanning tree containing 𝑒1.



• Self-reducibility• For an edge 𝑒 and a minimum spanning tree 𝑇 containing 𝑒, 𝑇 ∖ 𝑒 is

a minimum spanning tree of graph 𝐺 ’ where 𝐺 ’ is obtained from 𝐺by shrinking edge 𝑒 into a point.



e

'TT

• Exchange Property.• (𝑒1, … , 𝑒𝑚)

• There exists a minimum spanning tree containing 𝑒1.

• Self-reducibility• For an edge 𝑒 and a minimum spanning tree 𝑇 containing 𝑒, 𝑇 ∖ 𝑒 is

a minimum spanning tree of graph 𝐺 ’ where 𝐺 ’ is obtained from 𝐺by shrinking edge e into a point.

• Kruskal’s algorithm is optimal.



• Implement Kruskal’s algorithm • Order the edges by the weight non-decreasingly.


• 𝑇 = ∅




• Disjoint sets Q: maintain a set of elements by a partition.• 𝑄 = {𝑆1, … , 𝑆𝑘} with 𝑆𝑖 ∩ 𝑆𝑗 = ∅. Each set 𝑆𝑖 is associated with a

pointer (id or index).

• Operations:

• Make-Set(𝑥): create a new set containing only 𝑥

• 𝑄 = {𝑆1, … , 𝑆𝑘} −> 𝑄 = {𝑆1, … , 𝑆𝑘 , {𝑥}}

• Union(𝑥, 𝑦): for 𝑥 and 𝑦 where there exist 𝑆𝑖 and 𝑆𝑗 containing 𝑥 and 𝑦, respectively, union 𝑆𝑖 and 𝑆𝑗.

• Find-Set(𝑥): return the pointer to the set containing 𝑥.





• 𝑇 = ∅

• Consider each edge 𝑒 in (𝑒1, … , 𝑒𝑚) in order. Add 𝑒 to 𝑇 if 𝑇 ∪{𝑒} does not have a cycle.



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• 𝑇 = ∅




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• 𝑇 = ∅




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𝑒 = (𝑢, 𝑣) results a cycle iff 𝑢 and 𝑣 are in the same set.



• 𝑇 = ∅


• Kruskal’s algorithm• Order the edges by the weight non-decreasingly. (𝑒1, … , 𝑒𝑚)

• For each node 𝑣, Make-Set(𝑣).

• Consider each edge 𝑒 = (𝑢, 𝑣) in (𝑒1, … , 𝑒𝑚) in order. If Find-Set(𝑢) ≠ Find-Set(𝑣), Union(𝑢, 𝑣) and add 𝑒 to 𝑇.






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• For each node 𝑣, Make-Set(𝑣). 𝑇 = ∅.



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• Consider each edge 𝑒 = (𝑢, 𝑣) in (𝑒1, … , 𝑒𝑚) in order. If Find-Set(𝑢) ≠Find-Set(𝑣), Union(𝑢, 𝑣) and add 𝑒 to 𝑇.



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{1}, {2}, {3}, {4}, {5}, {6}






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{1}, {2}, {3}, {4}, {5}, {6}{1, 2}, {3}, {4}, {5}, {6}






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{1}, {2}, {3}, {4}, {5}, {6}{1, 2}, {3}, {4}, {5}, {6}{1, 2}, {3, 5}, {4}, {5, 6}






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{1}, {2}, {3}, {4}, {5}, {6}{1, 2}, {3}, {4}, {5}, {6}{1, 2}, {3, 5}, {4}, {5, 6}{1, 2, 3, 5}, {4}, {6}






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{1}, {2}, {3}, {4}, {5}, {6}{1, 2}, {3}, {4}, {5}, {6}{1, 2}, {3, 5}, {4}, {5, 6}{1, 2, 3, 5}, {4}, {6}{1, 2, 3, 4, 5}, {6}






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{1}, {2}, {3}, {4}, {5}, {6}{1, 2}, {3}, {4}, {5}, {6}{1, 2}, {3, 5}, {4}, {5, 6}{1, 2, 3, 5}, {4}, {6}{1, 2, 3, 4, 5}, {6}{1, 2, 3, 4, 5, 6}

• Prim’s algorithm• Maintain two sets 𝑆 and 𝑇. Initially, S={1} and 𝑇 = ∅.

• While 𝑆 ≠ 𝑉

• Fine the edge 𝑒 = (𝑢, 𝑣) with minimum weight such that 𝑢 ∈𝑆 and 𝑣 ∈ 𝑉\S

• Add 𝑣 to 𝑆 and 𝑒 to 𝑇.

• End







• End



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Algorithm Design and Analysis - University of Delawareudel.edu/~amotong/teaching/algorithm...

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