Introduction Outline The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation...
-
date post
22-Dec-2015 -
Category
Documents
-
view
216 -
download
0
Transcript of Introduction Outline The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation...
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
The Problem Domain
General network design problem:
Given a set S of n points in Rd, how to construct a good network that connects these points ?
Property 1:Any network connecting a set of n points must have at least n-1 edges.
Sparse network – the number of edges is linear to the number of points. (i.e. O(n) edges)
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Network Design
Size
Network quality measures:
Stretch factor
Weight
Degree Diameter Connectivity - Fault Tolerance
Network quality measures (cont.)
Load Factor
Number of Steiner points
e
(To be covered)
Bottle Neck
Motivation by examples
Scandinavian rail network (prior 2000)
Copenhagen
Malmo
Motivation by examples
Stretch FactorImagine set of highways connecting cities.We need to upgrade the highway system in the cost-effective manner.
Should we upgrade ALL existing highways ?Upgrade only carefully selected subset of existing highways !
Significance of Network Analysis
What is the size, weight, stretch factor, diameter, degree or connectivity of the network ?
Queries examples:
What is the farthest pair of cites in the network ?For which pair of the cites the stretch factor is the largest ?Which edge (or K edges) should be added to the network to achieve the greatest decrease in the stretch factor and/or load factor ?
What is the total length of the edges to be added to the network to achieve a desired fault-tolerance and stretch factor without destroying the planarity ?
Network Design Tradeoffs
Bounded degree vs. the small diameter
Small stretch factor vs. the small size/weight
Thus, network quality measures can be thought of as multicriteria optimization problems
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Spanning Trees
Spanning Tree: given set S of n points,
ST(S) is an acyclic connected graph on these points.
Spanning tree is good in sense that it has minimum number of edges
Sylvester(1857),Cayley(1889) formula:A set of n points has exactly nn-2 spanning trees
Spanning Trees Let T be a spanning tree of the set S.
Weight ωt(T) is defined to be the sum of the length of its edges
length of an edge {p,q} is the Euclidian distance |pq| between p and q
Minimum Spanning Tree - MST(S) of set S is a spanning tree of minimum weight.
MST Properties
A MST(S) is a shortest network connecting the points of S
In MST(S), each point has degree at most 6.
Let S be a set of n points in the plane, then:
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Steiner Trees
We allow additional points (Steiner points) to be added to the graph, in order to reduce the total length
We define: Steiner Minimum Tree - SMT(S) be a graph
with minimum total length(weight)
Interconnect set S of points by a network (graph) of shortest length, where the length is the sum of the lengths of all edges. MST…?
Steiner Trees Ex.1
C
A B
XMST(S) =SMT(S) =
Steiner Trees
Ex.2
C
A B
X1
MST(S) =SMT(S) =
D
X2
Steiner Trees Lemma (1.1.4)
Let S be a finite set of points in Rd, then:
ωt(MST(S)) ≤ 2 ∘ ωt(SMT(S)) Proof :
Let : SMT(S)=T
we will construct a spanning tree T* of S having at most twice the weight of T
Steiner Trees (cont.)
Proof (construction):
C
A B
X
SMT(S)=TW=
Euler’s tour: W’
ωt(W’ ) = ωt(W ) = 2∘ ωt(T )
Steiner Trees (cont.)
As result we have:T* that visits each point of S exactly onceωt(T*) ≤ 2∘ωt(T)
remove all Steiner points
For each point p of S, remove all visits to p except the first one
We will construct T* of points of S from W’ (Euler’s tour) applying the short-cuts technique:
Steiner Trees (cont.)
Recall:
C
A B
X
Euler tour: W’
?
Steiner Trees (cont.) Is factor 2 is the best possible…?
23
Gilbert & Pollack (1968) conjectured that: is the best possible
That is, for any finite set S of points of R2 :
ωt(MST(S)) ≤ ∘ ωt(SMT(S))2
3
After 20 years, Du&Hwang settled the conjecture
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Triangulation Define:
Convex Hull of set of points S in a real vector space V is the minimal Convex Set containing S
Y
XX
Y
In Euclidian Space, an object is convex if :
for every pair of points within the object, every point on the straight line segment that joins them is also within the object
A Convex set A non-Convex set
Triangulation Convex Hull
S =
Triangulation
Let S be a set of points in the planeA triangulation is a partition of the convex hull
into triangles , such that vertices of these triangles are exactly points of S
General:Subdivision of geometric object into simplices
In particular: In the plane it is a subdivision into triangles,
hence the name.
Triangulation Ex.3
S =
Delaunay Triangulation
Boris Delaunay (1934)
Delaunay triangulation for a set S of points in the plane is a triangulation DT(S) that: no point of S, is inside the circumcircle of any
triangle in DT(S)
Property:DT(S) contains MST(S)
Delaunay Triangulation
Example
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Spanners Spanner Definition.
Let S be a set of n points in Rd and let t ≥ 1 be a real number. A t-spanner for S is an undirected graph G with vertex set S, such that for any two points p and q of S, there is a path in G between p and q, whose length is less than or equal to t|pq|. Any path satisfying this condition is called a t-spanner path between p and q.
Spanners
1-spanner 1.5-spanner 3-spanner
Spanners
Stretch Factor Definition.
Let S be a set of n points in Rd and let G be a Euclidean graph with vertex set S. The stretch factor of G is the smallest real number t such that G is a t-spanner of S.
Spanners
Spanner Property.Let S be a set of n points in Rd and let t ≥ 1 be a real number. Any t-spanner of S has:
1. Number of edges - at least n – 1
2. Weight - at least ωt(MST(S))
Spanners
Basic Spanner Problem
Let S be a set of n points in Rd and let t ≥ 1 be a real number.
Does there exist a t-spanner for S having at most ctdn edges, where ctd is a real number that depends only on t and d?
If so, how much time does it take to compute such a t-spanner?
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Spanners Application Approximating Minimum Spanner Trees
Let S be a set of n points in the plane.
Since any MST(S) of S is contained in Delanay triangulation DT(S) of S, we can compute MST(S) in the following way:
1. Compute DT(S) in O(n log n) time.
2. Compute MST of DT(S), which is in fact an Euclidean MST of the set S.
Since DT(S) contains only linear number of edges, entire algorithm runs O(n log n)
Theorem
Let S be a set of n points in Rd, let t ≥ 1 be a real number and let G be an arbitrary t-spanner for S. A minimum spanning tree of G is t-approximate minimum spanning tree of S, that is the weight of any MST of G is at most
t∘ωt(MST(S))
Proof
Let T be a minimum spanning tree of S, and number its edges arbitrarily as e1, e2,…en-1. Consider edge ei. Since G is a t-spanner for S, there exists a t-spanner path Pi in G between the endpoints of ei. Thus the length wt(Pi) of Pi is at most t times the length of ei.
It follows that:
1
1
1
1
))((*)(*)(n
i
n
iii SMSTwttewttPwt
Proof (cont.) Let G’ be the subgraph of G, whose edge set is
the union of the edge sets of the paths Pi, 1 ≤ i ≤ n-1.
Then G’ is a connected graph with vertex set S and its weight is at most t*wt(MST(S)).
Since the weight of the minimum spanning tree of G is less than of equal to the weight of G’, the proof is complete.
Introduction Outline
The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm
Greedy Spanner AlgorithmAlgorithm PathGreedy (S, t)
Input: Set S of n points in Rd , t > 1
Output: G=(S, E) – a t-spanner for S
Sort the ( ) pairs of distinct points in nondecreasing order of their distances and store them in list L
E {} G (S, E) for each pair (u,v) є L (considering pairs in sorted order)
let dG(u,v) be the length of the shortest path in G between u and v
if dG(u,v) > t |u,v| add (u,v) to E
Output G
2
n
Greedy Spanner Algorithm (example 1)
A
CD
B Consider:
S = {A, B, C, D}, t = 2
AD = BC = 3
AB = DC = 4
AC = BD = 5
Sorting Pairs of S:
L = {AD, BC, AB, DC, AC, BD}
Deciding (D, C): (dG(D,C) = 10 > 2 * 4)? True! DC is added to G
Deciding (A, C): (dG(A,C) = 7 > 2 * 5)? False! AC is not added to G
Deciding (B, D): (dG(B,D) = 7 > 2 * 5)? False! BD is not added to G
G
Greedy Spanner Algorithm (example 2)
A
CD
B Consider:
S = {A, B, C, D}, t = 1.3
AD = BC = 3
AB = DC = 4
AC = BD = 5
Sorting Pairs of S:
L = {AD, BC, AB, DC, AC, BD}
Deciding (D, C): (dG(D,C) = 10 > 1.3 * 4)? True! DC is added to G
Deciding (A, C): (dG(A,C) = 7 > 1.3 * 5)? True! AC is added to G
Deciding (B, D): (dG(B,D) = 7 > 1.3 * 5)? True! BD is added to G
G