MSTs 1

Minimum Spanning Trees

CSCI 2720

Spring 2005

MSTs 2

Extended Dijkstra // extending Dijkstra’s algorithm to compute the edges of the shortest paths

Algorithm Dijkstra(G, s) // same implementation choices (heap or not) as for Dijkstra// initialize // same performance as for Dijkstrafor all v G.vertices() // same code except for the two red lines

if (v s) setDistance(v, 0) else setDistance(v, )

v is not in cloud setParent(v, NULL)

while there are nodes not in cloud

u node not in cloud with min. distance add u to cloud // update neighbors of u

for all e G.adjacentEdges(u) z G.getOpposite(e, u) d_new getDistance(u) weight(e)if d_new getDistance(z)

updateDistance(z, d_new) updateParent(z, u)

MSTs 3

Example (same graph as before)

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A.parent = NULLB.parent = EC.parent = AD.parent = CE.parent = CF.parent = D

MSTs 4

All-Pairs Shortest PathsFind the distance between every pair of vertices in a weighted directed graph G.

Create a matrix, i’th row = distances for vertex vi

(and another matrix if we want to store parent values)

We can make n calls to Dijkstra’s algorithm (if no negative edges), one for each vertex as source. Each call fills in one row of the matrix.

This takes O(nm log n) time with heap implementation (of sparse graph with adjacency lists), O(n3) otherwise.

MSTs 5

DAGs and Topological Ordering

A directed acyclic graph (DAG) is a digraph that has no directed cycles

Halfway between trees and digraphs Used instead of expression trees, to

“fold together” common subexpressions

Often used to represent dependencies:

tasks/jobs -- (x,y) means x must be done before y

course requirements – (x,y) means x must be taken before y

preferences – (x,y) means that x is preferable to y

In each case, acyclicity is important

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DAG G

MSTs 6

Topological SortingNumber vertices, so that (u,v) in E implies u < v

Often needed in DAG applications tasks/jobs – the order in which to do jobs course requirements – the order in which to take courses preferences – the overall “rating

Theorem A digraph admits a topological ordering if and only if it is a DAG (why?)

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MSTs 7

Whenever there is a choice between multiple vertices with no outgoing edges, the output is not unique.

Naïve implementation is O(n2). Why?

Algorithm for Topological Sorting

Algorithm TopologicalSort(G) H G // Temporary copy of G n G.numVertices() while H is not empty do

v find vertex with no outgoing edgesLabel v nn n - 1Remove v from H

MSTs 8

Topological sort can be implemented in O(n + m). (Is that always faster?)

This is done as a variant of depth-first search

We label every node at the end of visiting it (before returning).

The node labels are in descending order (from highest down).

Result is the same as with naïve version before.

Topological Sorting using DFS

MSTs 9

Topological Sorting using DFS

Algorithm topologicalDFS(G, v)Input graph G and a start vertex v of G Output labeling of the vertices of G

in the connected component of v setLabel(v, VISITED)for all e G.incidentEdges(v)

if getLabel(e) UNEXPLORED

w opposite(v,e)if getLabel(w) UNVISITED

setLabel(e, DISCOVERY)topologicalDFS(G, w)

else{e is a forward or cross edge}

Set the topological order of v to n n n – 1 // end of forall loop

Algorithm topologicalDFS(G)Input dag GOutput topological ordering of G

n G.numVertices()for all u G.vertices()

setLabel(u, UNVISITED)for all e G.edges()

setLabel(e, UNEXPLORED)for all v G.vertices()

if getLabel(v) UNEXPLORED

topologicalDFS(G, v)

Except for blue lines, this is

the same DFS code as before.

MSTs 10

Topological Sorting Example

MSTs 11

Topological Sorting Example

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MSTs 12

Topological Sorting Example

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MSTs 13

Topological Sorting Example

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MSTs 14

Topological Sorting Example

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MSTs 15

Topological Sorting Example

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MSTs 16

Topological Sorting Example

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MSTs 17

Topological Sorting Example

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MSTs 18

Topological Sorting Example

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MSTs 19

Topological Sorting Example

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MSTs 20

Minimum Spanning TreeSpanning subgraph

Subgraph of a graph G containing all the vertices of G

Spanning tree Spanning subgraph that is

itself a (free) treeMinimum spanning tree (MST)

Spanning tree of a weighted graph with minimum total edge weight

Applications Communications networks:

Connect all servers, at least cost

Transportation networks:Connect all cities, at least cost

ORD

PIT

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DEN

DFW

DCA

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MSTs 21

U V

Partition PropertyPartition Property:

Consider a partition of the vertices of G into subsets U and V

Let e be an edge of minimum weight across the partition

There is a minimum spanning tree of G containing edge e

Proof (by contradiction): Suppose no MSTs of G contain e. Let T be an MST of G; T does not

contain e Consider the cycle C formed by e

with T and let f be an edge of C across the partition

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

By replacing f with e, we obtaina better spanning tree!

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Replacing f with e yieldsanother MST

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MSTs 22

MST algorithms

2 algorithms to compute MST of a given weighted graph

Prim-Jarnik (similar to Dijkstra’s) Kruskal (uses a new ADT, “union-find”)

Both are greedy greedy: An algorithm that always takes the best

immediate, or local, solution while finding an answer. Greedy algorithms find the overall, or globally, optimal solution for some optimization problems, but may find less-than-optimal solutions for some instances of other problems

Prim-Jarnik greedily chooses nodes for MST

Kruskal’s algorithm greedily chooses edges for MST.

MSTs 23

Prim-Jarnik’s AlgorithmSimilar to Dijkstra’s algorithm; the only difference is what distance means (now it’s just the edge weight).

We pick an arbitrary vertex s and we grow the MST as a cloud of vertices, starting from s

We store with each vertex v a label d(v) = the smallest weight of an edge connecting v to a vertex in the cloud

At each step: We add to the cloud the vertex

u outside the cloud with the smallest distance label

We update the labels of the vertices adjacent to u

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MSTs 24

Example

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MSTs 25

Example (contd.)

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MSTs 26

Pseudocode Algorithm Prim-Jarnik(G) // same implementation choices (heap or not) as for Dijkstra

// initialize // same performance as for Dijkstra s a vertex of G // same code except for the two red lines for all v G.vertices()

if (v s) setDistance(v, 0) else setDistance(v, )

v is not in cloud setParent(v, NULL)

while there are nodes not in cloud

u node not in cloud with min. distance add u to cloud // update neighbors of u

for all e G.adjacentEdges(u) z G.getOpposite(e, u) d_new weight(e)if d_new getDistance(z)

updateDistance(z, d_new) updateParent(z, u)