# Slides

```CSC 213 – Large Scale Programming
LECTURE 35:
SPANNING TREES
Minimum Spanning Tree (MST)
 Spanning subgraph
ORD
 Subgraph w/ all vertices
10
1
PIT
DEN
9
7
3
STL
4
8
DFW
6
DCA
5
2
ATL
Minimum Spanning Tree (MST)
 Spanning subgraph
ORD
 Subgraph w/ all vertices
10
1
PIT
DEN
7
9
4
DCA
STL
5
DFW
2
ATL
Minimum Spanning Tree (MST)
 Spanning subgraph
ORD
 Subgraph w/ all vertices
 Spanning tree
1
DEN
 Combines
spanning subgraph + tree
4
10
PIT
6
STL
7
DCA
2
DFW
ATL
Minimum Spanning Tree (MST)
 Spanning subgraph
ORD
 Subgraph w/ all vertices
 Spanning tree
1
DEN
 Combines
6
7
3
spanning subgraph + tree
 MST
PIT
4
STL
DCA
2
 Spanning tree which
minimizes sum of edge
weights
DFW
ATL
No Cycles in MST
 Edge in MST cheaper than one making cycle




Assume there exists an edge e not in MST
Cycle, C, occurs after adding e to min. spanning tree
Assume that f in C heavier than e
Shrink MST using e and dropping f from C
8
f
8
f
4
2
9
6
3
8
7
4
2
e
7
9
6
3
8
7
e
7
Partition Property
 Given partitioning of vertices in a graph
 Required that each vertex in exactly 1 partition
 Use smallest edge to connect each of the
 To complete following MST, can use either f or e
7
2
9
5
8
8
7
4
3
Kruskal’s Algorithm
 Similar to Prim-Jarnik, including finding MST
 Also like Prim-Jarnik, adds edges greedily
 But Kruskal processes Edges using PriorityQueue
 Check if Edge makes cycle before adding to MST
 No cycles in an MST, so add Edge if no cycle occurs
 Detecting cycles hard, however
Kruskal’s Algorithm
 Similar to Prim-Jarnik, including finding MST
 Also like Prim-Jarnik, adds edges greedily
 But Kruskal processes Edges using PriorityQueue
 Check if Edge makes cycle before adding to MST
 No cycles in an MST, so add Edge if no cycle occurs
 Detecting cycles hard, however
Kruskal’s Algorithm
 Similar to Prim-Jarnik, including finding MST
 Also like Prim-Jarnik, adds edges greedily
 But Kruskal processes Edges using PriorityQueue
 Check if Edge makes cycle before adding to MST
 No cycles in an MST, so add Edge if no cycle occurs
 Detecting cycles hard, however
Structure for Kruskal’s Algorithm
 Kruskal’s needs to maintain collection of trees
 Accept Edge connecting 2 trees &amp; reject it otherwise
 Data structure named Partition relied upon
 Store disjoint Set instances within a Partition
 For Kruskal’s, each Set holds tree from graph
 Methods supporting Sets defined by Partition
 find(u): find and returns Set containing u
 union(p,r): replace p &amp; r with their union
 makeSet(u): creates Set for u &amp; adds to Partition
Kruskal’s Algorithm
Algorithm KruskalMST(Graph G)
Q  new PriorityQueue()
T  new Graph()
P  new Partition()
for (Vertex v : G.vertices())
P.makeSet(v)
T.insertVertex(v)
for (Edge e : G.edges())
Q.insert(e.getWeight(), e);
while (T.numEdges() &lt; T.numVertices()-1)
Edge e = Q.removeMin().value()
Assign u, v to G.endpoints(e)
if P.find(u)  P.find(v)
T.insertEdge(e)
P.union(P.find(u),P.find(v))
return T
Kruskal Example
2704
867
849
ORD
740
1846
621
SFO
JFK
PVD
144
184
802
1464
187
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
ORD
740
1846
621
SFO
JFK
PVD
144
184
802
1464
187
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
ORD
740
1846
621
SFO
JFK
PVD
144
184
802
1464
187
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
ORD
740
1846
621
SFO
JFK
PVD
144
184
802
1464
187
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
ORD
740
1846
621
SFO
JFK
PVD
144
184
802
1464
187
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
ORD
740
1846
621
SFO
JFK
PVD
144
184
802
1464
187
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
ORD
740
1846
621
SFO
JFK
PVD
144
184
802
1464
187
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
187
ORD
1846
621
SFO
PVD
144
184
802
1464
JFK
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
187
ORD
1846
621
SFO
PVD
144
184
802
1464
JFK
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
849
187
ORD
1846
621
SFO
PVD
144
184
802
1464
JFK
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
187
ORD
1846
621
SFO
PVD
144
184
802
1464
JFK
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
867
187
ORD
1846
621
SFO
PVD
144
184
802
1464
JFK
BOS
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
BOS
187
ORD
1846
621
SFO
144
184
802
1464
JFK
PVD
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
BOS
187
ORD
1846
621
SFO
144
184
802
1464
JFK
PVD
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
BOS
187
ORD
1846
621
SFO
144
184
802
1464
JFK
PVD
1391
1258
BWI
1090
337
LAX
1235
946
DFW
1121
MIA
2342
Kruskal Example
2704
BOS
187
ORD
1846
621
184
802
SFO
1464
JFK
1391
BWI
337
LAX
1235
946
DFW
1121
MIA
2342
PVD
144
1258
Kruskal Example
2704
BOS
187
ORD
1846
621
184
802
SFO
1464
JFK
1391
BWI
337
LAX
1235
DFW
946
MIA
2342
PVD
144
1258
Kruskal Example
2704
BOS
187
ORD
1846
621
184
802
SFO
1464
JFK
1391
BWI
337
LAX
1235
DFW
946
MIA
2342
PVD
144
1258
Kruskal Example
2704
BOS
187
ORD
1846
621
184
802
SFO
1464
JFK
1391
BWI
337
LAX
1235
DFW
946
MIA
2342
PVD
144
1258
Kruskal Example
2704
BOS
187
ORD
1846
621
184
802
SFO
1464
JFK
1391
BWI
337
LAX
1235
DFW
946
MIA
2342
PVD
144
Kruskal Example
2704
BOS
187
ORD
1846
621
802
SFO
1464
JFK
184
BWI
337
LAX
1235
DFW
946
MIA
2342
PVD
144
Kruskal Example
BOS
187
ORD
621
802
SFO
JFK
184
BWI
337
LAX
1235
DFW
946
MIA
PVD
144
For Next Lecture
 Weekly assignment available on Angel
 Due at special time: before next Monday’s quiz
 Programming assignment #3 designs due today
 Graph Quiz will be on Monday
 Started before test, so could include implementations
 Bring notes, templates, &amp; anything else you want
 Weekly assignment question review highly encouraged
```