Spanning Trees
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1 Spanning Trees Longin Jan Latecki Temple University based on slides by David Matuszek, UPenn, Rose Hoberman, CMU, Bing Liu, U. of Illinois, Boting Yang, U. of Regina
description
Spanning Trees. Longin Jan Latecki Temple University based on slides by David Matuszek, UPenn, Rose Hoberman, CMU, Bing Liu, U. of Illinois, Boting Yang, U. of Regina. A connected, undirected graph. Four of the spanning trees of the graph. Spanning trees. - PowerPoint PPT Presentation
Transcript of Spanning Trees
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Spanning Trees
Longin Jan Latecki
Temple University
based on slides by
David Matuszek, UPenn,
Rose Hoberman, CMU,
Bing Liu, U. of Illinois,
Boting Yang, U. of Regina
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Spanning trees
Suppose you have a connected undirected graph Connected: every node is reachable from every other node Undirected: edges do not have an associated direction
...then a spanning tree of the graph is a connected subgraph in which there are no cycles
A connected,undirected graph
Four of the spanning trees of the graph
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Theorem
A simple graph is connected iff it has a spanning tree.
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Finding a spanning tree To find a spanning tree of a graph,
pick an initial node and call it part of the spanning tree do a search from the initial node:
each time you find a node that is not in the spanning tree, add to the spanning tree both the new node and the edge you followed to get to it
An undirected graph One possible result of a BFSstarting from top
One possible result of a DFSstarting from top
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Graph Traversal Algorithm
To traverse a tree, we use tree traversal algorithms like pre-order, in-order, and post-order to visit all the nodes in a tree
Similarly, graph traversal algorithm tries to visit all the nodes it can reach.
If a graph is disconnected, a graph traversal that begins at a node v will visit only a subset of nodes, that is, the connected component containing v.
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Two basic traversal algorithms
Two basic graph traversal algorithms: Depth-first-search (DFS)
After visit node v, DFS strategy proceeds along a path from v as deeply into the graph as possible before backing up
Breadth-first-search (BFS) After visit node v, BFS strategy visits every node adjacent to v before
visiting any other nodes
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Depth-first search (DFS)
DFS strategy looks similar to pre-order. From a given node v, it first visits itself. Then, recursively visit its unvisited neighbors one by one.
DFS can be defined recursively as follows.
procedure DSF(G: connected graph with vertices v1, v2, …, vn)T:=tree consisting only of the vertex v1visit(v1)mark v1 as visited;procedure visit(v: vertex of G)for each node w adjacent to v not yet in Tbegin
add vertex w and edge {v, w} to Tvisit(w);mark w as visited;
end
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DFS example
Start from v3
v1
v4
v3
v5
v2
G
v31
v22
v1
3v4
4
v5
5
xxxx x
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Breadth-first search (BFS) BFS strategy looks similar to level-order. From a given node v, it
first visits itself. Then, it visits every node adjacent to v before visiting any other nodes.
1. Visit v 2. Visit all v’s neighbors 3. Visit all v’s neighbors’ neighbors …
Similar to level-order, BFS is based on a queue.
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Algorithm for BFS
procedure BFS(G: connected graph with vertices v1, v2, …, vn)T:= tree consisting only of vertex v1L:= empty listput v1 in the list L of unprocessed vertices;while L is not empty
beginremove the first vertex v from Lfor each neighbor w of v
if w is not in L and not in T thenbegin
add w to the end of the list Ladd w and edge {v, w} to T
endend
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BFS example Start from v5
v5
1
v32
v4
3
v2
4
v1
5
v1
v4
v3
v5
v2
G
x
Visit Queue (front to back)
v5 v5
empty
v3 v3
v4 v3, v4
v4
v2 v4, v2
v2
empty
v1 v1
empty
x
xx x
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Backtracking Applications
Use backtracking to find a subset of{31, 27, 15, 11, 7, 5}with sum equal to 39.
