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Minimum Spanning Trees(some material adapted from slides by Peter Lee)
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Problem: Laying Telephone Wire
Central office
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Wiring: Nave Approach
Central office
Expensive!
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Wiring: Better Approach
Central office
Minimize the total length of wire connecting the customers
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Minimum Spanning Tree (MST)(see Weiss, Section 24.2.2)
it is a tree (i.e., it is acyclic)
it covers all the vertices V
contains |V| - 1 edges
the total cost associated with treeedges is the minimum among allpossible spanning trees
not necessarily unique
A minimum spanning tree is a subgraph of anundirected weighted graph G, such that
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Applications of MST
Any time you want to visit all vertices in agraph at minimum cost (e.g., wire routingon printed circuit boards, sewer pipelayout, road planning)
Internet content distribution$$$, also a hot research topic
Idea: publisher produces web pages, contentdistribution network replicates web pages to
many locations so consumers can access athigher speed
MST may not be good enough!
content distribution on minimum cost tree may take
a long time!
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How Can We Generate a MST?
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Prims Algorithm
Let V ={1,2,..,n} and U be the set ofvertices that makes the MST and T be theMST
Initially : U = {1} and T =
while (U V)
let (u,v) be the lowest cost edge such that
u U and v V-UT = T {(u,v)}
U = U {v}
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Prims Algorithm implementation
Initializationa. Pick a vertexr to be the root
b. SetD(r) = 0, parent(r) = null
c. For all vertices v V, v r, setD(v) =
d. Insert all vertices into priority queueP,using distances as the keys
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Vertex Parent
e -
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Prims Algorithm
WhileP
is not empty:
1. Select the next vertex u to add to the tree
u = P.deleteMin()
2. Update the weight of each vertex w adjacent tou which is not in the tree (i.e., w P)
Ifweight(u,w)< D(w),
a.parent(w) = u
b.D(w) = weight(u,w)c. Update the priority queue to reflect
new distance forw
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Prims algorithm
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Vertex Parent
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The MST initially consists of the vertex e, and we update
the distances and parent for its adjacent vertices
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Prims algorithm
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Prims algorithm
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Vertex Parent
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Prims algorithm
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Vertex Parent
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Prims algorithm
Vertex Parent
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The final minimum spanning tree
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Prims Algorithm Invariant
At each step, we add the edge (u,v)s.t. the weight of(u,v) is minimumamong all edges where u is in thetree and v is not in the tree
Each step maintains a minimumspanning tree of the vertices that
have been included thus far
When all vertices have beenincluded, we have a MST for thegraph!
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Running time of Prims algorithm
Initialization of priority queue (array): O(|V|)
Update loop: |V| calls Choosing vertex with minimum cost edge: O(|V|)
Updating distance values of unconnectedvertices: each edge is considered only onceduring entire execution, for a total of O(|E|)
updates
Overall cost: O(|E| + |V| 2)
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Another Approach Kruskals
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Create a forest of trees from the vertices Repeatedly merge trees by adding safe edges
until only one tree remains
A safe edge is an edge of minimum weight which
does not create a cycle
forest: {a}, {b}, {c}, {d}, {e}
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Kruskals algorithm
Initializationa. Create a set for each vertex v V
b. Initialize the set of safe edgesA
comprising the MST to the empty set
c. Sort edges by increasing weight
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{a}, {b}, {c}, {d}, {e}
A =
E = {(a,d), (c,d), (d,e), (a,c),
(b,e), (c,e), (b,d), (a,b)}
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Kruskals algorithm
For each edge (u,v) E in increasing orderwhile more than one set remains:
Ifu and v, belong to different sets
a.A = A{(u,v)}
b. merge the sets containing u and v
ReturnA
Use Union-Find algorithm to efficientlydetermine ifuand v belong to differentsets
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Kruskals algorithm
E = {(a,d), (c,d), (d,e), (a,c),
(b,e), (c,e), (b,d), (a,b)}
Forest
{a}, {b}, {c}, {d}, {e}
{a,d}, {b}, {c}, {e}
{a,d,c}, {b}, {e}{a,d,c,e}, {b}
{a,d,c,e,b}
A
{(a,d)}
{(a,d), (c,d)}{(a,d), (c,d), (d,e)}
{(a,d), (c,d), (d,e), (b,e)}
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After each iteration, every tree in theforest is a MST of the vertices it connects
Algorithm terminates when all vertices areconnected into one tree
Kruskals Algorithm Invariant
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Greedy Approach
Like Dijkstras algorithm, both Prims andKruskals algorithms are greedyalgorithms
The greedy approach works for the MSTproblem; however, it does not work formany other problems!
