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CSC 213

Lecture 24:Minimum Spanning Trees

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Final exam is: Thurs. 5/11 from 8:30-10:30AM in Old Main 205

Minimum Spanning Tree (MST)Spanning subgraph

Subgraph including every vertex

Spanning tree Spanning subgraph

that is also a tree

Minimum spanning tree Spanning tree with

minimum total edge weight

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Cycle PropertyCycle Property:

MSTs do not contain cycles

So weight of each edge in MST is lower than weight of edge which completes cycle

Proof: e is edge in G & not in MST C is cycle formed when

adding e to MST For every edge f in C,

weight(f) weight(e) If weight(f) weight(e), then

exists a smaller spanning tree with e instead of f

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Replacing f with e yieldsa better spanning tree

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Partition PropertyPartition Property:

Given partitions of vertices, minimum weight edge is in a MST connecting partitions

Proof: e is edge with smallest

weight connecting partitions, but not in MST

f connects partitions in MST If MST does not contain e,

consider cycle formed by including e with MST

By cycle property,weight(f) ≤ weight(e)

But since e has smallest weight,weight(f) = weight(e)

Create another MST by replacing f with e

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Replacing f with e yieldsanother MST

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Kruskal’s Algorithm

Add edges to MST greedily Priority queue contains edges outside

the cloud Element: edge Key: edge’s weight

Only want edges that will not make a cycle Unfortunately, detecting cycles can be

hard…

Data Structs for Kruskal Algorithm

Algorithm maintains forest of treesEdge accepted when connecting treesUse partition (i.e., collection of disjoint sets): find(u): return set containing u union(u,v): replace sets with u & v

with their union

Partition Representation

Each set is SequenceEach element refers to its set makeSet(u) takes O(1) time find(u) takes O(1) time union(u,v) moves elements of smaller set

to Sequence of larger set and updates references

By having union move smaller set into large set, elements are processed only when set size doubles n calls to union() takes O(log n) time

Kruskal’s Algorithm

Algorithm KruskalMST(Graph G)Q new PriorityQueue()T new Graph()P new Partition()for each vertex v in G

P.makeSet(v)T.insertVertex(v)

for each edge e in GQ.insert(e.getWeight(), e);

while T.edges().getSize() < n-1Edge e = Q.removeMin()Assign u, v to G.endpoints(e)if P.find(u) P.find(v) then

T.insertEdge(e) P.union(P.find(u),P.find(v))

return T

Kruskal Example

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BOS

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PVD

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JFK

BOS

MIA

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LAXDFW

SFO BWI

PVD

8672704

187

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849

144740

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Example

Example

JFK

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MIA

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LAXDFW

SFO BWI

PVD

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Prim-Jarnik’s Algorithm

Similar to Dijkstra’s algorithmPick arbitrary vertex s and grow MST as cloud of verticesLabel each vertex v with d(v) --- smallest weight of edge connecting v to vertex in cloud At each step: Add vertex u with smallest distance to cloud Update labels of vertices adjacent to u

Prim-Jarnik’s Algorithm (cont.)

Priority queue stores vertices outside of cloud Key: distance Element: vertexLocator-based methods insert(k,e) – returns locator replaceKey(l,k) – changes item’s key

Store three labels with each vertex Distance from cloud Edge connecting vertex to cloud (MST edge) Locator in priority queue

Prim-Jarnik’s Algorithm (cont.)

Algorithm PrimJarnikMST(Graph G)Q new PriorityQueue()s G.vertices().elemAtRank(0)for each vertex v in G

if v = sv.setDistance(0)

else v.setDistance()

v.setParent(null)l Q.insert(getDistance(v), v)

setLocator(v,l)while !Q.isEmpty()

u Q.removeMin() for each edge e in G.incidentEdges(u)

z G.opposite(u,e)r e.getWeight()if r < z.getDistance()

z.setDistance(r)z.setParent(e)

Q.replaceKey(z.getLocator(),r)

Example

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Example (contd.)

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Prim-Jarnik’s AnalysisOperations incidentEdges called once per vertex Set/Get each vertex’s labels O(deg(v)) times

Priority queue operations Each vertex inserted once and removed

once Each insertion/removal takes O(log n)

Vertex w’s key modified O(deg(w)) times Each key change takes O(log n) time

Algorithm takes O((n m) log n) time using adjacency list structure

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