minimum spanning trees - graz university of technology · 2020-01-31 · birgit vogtenhuber minimum...

Birgit Vogtenhuber Minimum Spanning Trees1

Minimum Spanning Trees

Design & Analysis of Algorithms WS 2019/20

Birgit Vogtenhuber Minimum Spanning Trees2 v

Trees in Graphs

• Given an unweighted connected graph G = (V,E), wecan compute a tree T with all shortest paths from aroot s to the other vertices using breadth first search.

• This does not work if G is a weighted graph!

• Given an unweighted connected graph G = (V,E) withn vertices, every subtree with n vertices has the sametotal edge length n− 1.

• This is not true if G is a weighted graph!

This topic: Trees in weighted graphs with minimum totaledge length (edge weight / edge cost).

Birgit Vogtenhuber Minimum Spanning Trees3 iv

Problem Definition

Basic task:Create connections between n locations with minimal cost.

Definition:Given a set S of points, a minimum spanning tree of S isa tree that connects all points and minimizes the total edgelength among all trees on S.

spanning tree

Example:

Birgit Vogtenhuber Minimum Spanning Trees3 vi

Problem Definition



minimum spanning tree

Example:

Observation:every (Euclidean)minimum spanning treeis a plane tree

Birgit Vogtenhuber Minimum Spanning Trees3 ix

Problem Definition



Example applications:bicycle path network, electrical circuits, telephone network.

Possible problem:Direct connection from A to B not always possible or notproportional to the distance (example: mountain road).

⇒ Consider weighted graphs instead.

Birgit Vogtenhuber Minimum Spanning Trees3 x

Problem Definition


Definition:Given a weighted graph G = (V,E,w), a minimumspanning tree of G is a tree T = (V,E′) with E′ ⊆ Eand with minimal total edge length:

w(T ) =∑

e∈E′ w(e)

is minimized over all trees with vertex set V .

Birgit Vogtenhuber Minimum Spanning Trees3 xi

Problem Definition



1

38

7

4

5

6

92

Example:

weighted graph G

Birgit Vogtenhuber Minimum Spanning Trees3 xii

Problem Definition



minimumspanning tree of G

Example:

1

38

7

4

5

6

92

Birgit Vogtenhuber Minimum Spanning Trees3 xiii

Problem Definition



w(T ) =∑

e∈E′ w(e)

is minimized over all trees with vertex set V .

Create connections between n locations with minimal cost:

Consider the complete weighted graph whose vertices arethe locations.

Birgit Vogtenhuber Minimum Spanning Trees4 ii

How Many Different Trees?

Answers:

• Complete graph Kn: nn−2 different spanning trees!!

• Plane graphs on n vertices: O(5.2852n) spanning trees

• Bounds on the number of different plane spanning treeson n points: Ω(6.75n), O(229.33n)

• Point sets known with Ω(12.52n) plane spanning trees

⇒ We can’t try them all !!

Questions:

• How many different spanning trees for a graph G?

• How many different plane spanning trees for n points?


Iterative Algorithm Idea

Idea:Build a minimum spanning tree (MST) for a graphG = (V,E,w) by iteratively inserting edges:

E′ = 0while |E′| < n− 1 do

select an edge e ∈ E \ E′ which is ‘good’ for E′

E′ = E′ ∪ eodwrite E′

Edge e ∈ E \E′ is ‘good’ for E′ if E′ ∪ e is a subset of anMST of G (there can be more than one MST of G).

Birgit Vogtenhuber Minimum Spanning Trees6 v

Cuts in Graphs

A cut of a graph G = (V,E) is a partition of V into V1, V2.

V1 V2

external edges

An edge e is called external for the cut (V1, V2) if it hasone endpoint in V1 and one in V2; otherwise e is calledinternal.


Characterization of Good Edges

Theorem: Let E′ be a subset of edges of an MST ofG = (V,E,w). Let (V1, V2) be a cut of G for which alledges of X are internal. Then the external edge of the cutwith minimum weight is a good edge for E′.

e

V1

V2

set E′: green

external edges: black

good edge for E′: e

Example:w = Eucl. distance

Birgit Vogtenhuber Minimum Spanning Trees7 xii


Proof:Assume there is an MST T with E′ and without e.

⇒ e closes a cycle C in T .

The cycle C contains at least one edge f of the tree T thatis external for the cut.

By definition of e, theweight w(f) ≥ w(e).

⇒ T\f ∪ e is anMST of G.

If w(f) > w(e) thenT is not an MST of G.

e

V1

V2

C

f

Birgit Vogtenhuber Minimum Spanning Trees7 xiv


Theorem: Let E′ be a subset of edges of an MST ofG = (V,E,w). Let (V1, V2) be a cut of G for which alledges of X are internal. Then the external edge of the cutwith minimum weight is a good edge for E′.

e

V1

V2

C

f

Next:Two different greedyalgorithms that use thistheorem to efficientlycompute an MST

Difference:Choice of the cuts


Prim’s Algorithm

• Start with an arbitrary vertex s of G and iteratively‘grow’ an MST T from s.

• Iterative step:Choose the ‘cheapest’ edge with exactly one node in T .

• Cut: V1 = vertices of T , V2 = vertices not yet in T .

• For each vertex v 6∈ T we maintain: Priority p(v): weight of the shortest edge from v to a

vertex in T (initially: ∞). Nearest n(v): vertex in T realizing p(v): w(v, n(v)) is

min. among neighbors of v in T (initially no vertex).

• A queue Q contains all vertices not yet in T , organizedby priorities (e.g., in a heap; initially all vertices).


Prim’s Algorithm

PRIM-MST (G,s)p(s) = 0p(v) =∞ ∀ v ∈ V \sn(s) = nilfor all v ∈ V do INSERT(v,Q) odwhile Q 6= 0 do

u = MIN(Q)write u, n(u)for all neighbors v of u do

if v ∈ Q and p(v) > w(u, v) thenp(v) = w(u, v), update Qn(v) = u

fiod

od

MST isfinishedhere

n(u)

X

u


Prim’s Algorithm

Run-time-analysis:

n ... number of vertices of Gm ... number of edges of Gd(v) ... Degree of vertex v in G

• Initialization, construction of the heap Q: Θ(n)• n times removing the minimum from Q: O(n log n)• Update priorities for all neighbors of v: O(d(v) · log n)⇒ Alltogether:

O(n + n log n +∑

v∈V d(v) log n) = O(m log n)

Memory requirements:O(m + n) = O(m)


Prim’s Algorithm

Remarks:

• MST always begins at the start vertex s and growsfrom there as a connected tree.

• Shrinking p(v) causes v to move up the heap.⇒ O(log n) time.

• Test for v ∈ Q in O(1) time when bit vector is used tostore which vertices are already in T .

• Runtime can be changed to O(n2). This is useful fordense graphs (see notes on Dijkstra’s algorithm in thenext chapter).

Birgit Vogtenhuber Minimum Spanning Trees12 i

Kruskal’s Algorithm

• Start with empty edge set E′.

• Sort edge set E of G = (V,E,w) in increasing orderof their weights (edges will be inserted in this order):e1, e2, ...em with w(e1) < w(e2) < ... < w(em).

• Iterative step:

E′ forms a forest F (= set of disjoint subtrees,acyclic) in the MST to be constructed.

Edge e that is added to E′ is the shortest edge inE \ E′ that does not close a cycle in E′.



Use a UNION-FIND data structure on V for thecomponents (subtrees) M1,M2, ...Mt of F :

• Label the vertex set of G as v1, v2, ...vn (arbitrary)

• Initially there are n disjoint sets M1,M2, ...Mn

(each with one vertex)

• FIND (v): returns index i if vertex v is in Mi

• UNION (i, j): join sets Mi and Mj : Mi = Mi ∪Mj

• End of the algorithm: one component M1 with allvertices of G.

Birgit Vogtenhuber Minimum Spanning Trees12 iii


A UNION-FIND data structure can be realized such that:

• Creating a 1-element set needs Θ(1) time.

• f FIND and u UNION operations need O(f + u log u)time in total.

• The total memory requirement of the data structure islinear in the number of initial 1-element sets.

Birgit Vogtenhuber Minimum Spanning Trees13 i


KRUSKAL-MST(G)sort edges by weight: e1, e2, . . . emfor i = 1 to n do Mi = vi odfor k = 1 to m do

(u, v) = eki =FIND(u)j =FIND(v)if i 6= j then

write ekUNION(i, j)

fiod



Runtime analysis:

• Sorting of the edges: O(m logm)

• Initialize UNION-FIND data structure for vertices: Θ(n)

• In total 2m FIND operations and n− 1 UNIONoperations: O(m + n log n)

⇒ Altogether O(m logm) time.

⇒ Sorting of the edges dominates the runtime.

Memory requirements:

Θ(n + m) = Θ(m) in total.


Concluding Remarks

Remarks:

• Both algorithms also work if edge weights can benegative.

• If the calculation of the MST for a point set in a planewith Euclidean distance function is to be carried out(geometric version), this is possible in Ω(n log n) time.The main observation is that the MST is a subgraph ofthe Delaunay triangulation of the point set, which canbe computed in Ω(n log n) time.

• For both algorithms, animated versions are available(see course webpage).

minimum spanning trees - graz university of technology · 2020-01-31 · birgit vogtenhuber minimum...

Documents