F96943167 ???

1
Special Topics on Graph Algorithms Finding the
Diameter in Real-World GraphsExperimentally
Turning a Lower Bound into an Upper Bound

• F96943167 ???
• F97943070 ???
• R98943086 ???
• R98943090 ???

R98943088 ??? R98921072 ? ? R99921040
??? R99942061 ???
2
Outline
Introduction
R98943086 ???

• R98943090 ???
• R98943088 ???

Previous Work
R99921040 ??? R99942061 ??? R98943086 ???
Finding the Diameter in Real-World Graphs
F96943167 ??? F97943070 ???
Other Related Topics
Conclusion and Future Work
R98921072 ? ?
3
Diameter

• The length of the "longest shortest path" between
  any two vertices in a graph or a tree
• Given a connected graph G (V,E) with nV
  vertices and mE edges
• the diameter D is Max d(u,v) for u,v in V, where
  d(u,v) denotes the distance between node u and v

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2
3
3
1
3
1
2
2
5
5
3
3
5
2
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A Tree, D 13
A Graph, D 9
3
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Diameter of a Tree

• The diameter of a tree can be computed by
  applying double-sweep algorithm
• 1. Choose a random vertex r, run a BFS at r, and
  find a vertex
• a farthest from r
• 2. Run a BFS at a and find a vertex b farthest
  from a
• 3. Return D d(a,b)

0
8
r
3
3
2
2
3
11
3
3
2
10
1
1
3
5
2
2
3
11
3
3
5
5
5
13
b
6
8
a
8
0
a
4
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Diameter of a Graph

• Double-sweep algorithm might not correctly
  compute the diameter of a graph
• It provides a lower bound instead

r
b
a
a
D 9
5
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Outline
Introduction
R98943086 ???

• R98943090 ???
• R98943088 ???

Previous Work

• R99921040 ???
• R99942061 ???
• R98943086 ???

Finding the Diameter in Real-World Graphs
F96943167 ??? F97943070 ???
Other Related Topics
Conclusion and Future Work
R98921072 ? ?
7
Naïve Algorithm

• Perform n breadth-first searches (BFS) from each
  vertex to obtain distance matrix of the graph
• T(n(nm)) time and T(m) space
• By using matrix multiplication, the distance
  matrix can be computed in O(M(n)logn) time and
  T(n2) space Seidel, ACM STC92
• M(n) the complexity for matrix multiplication
  involving small integers only (O(n2.376))
• ? Is too slow for massive graphs and has a
  prohibitive space cost

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All Pairs Shortest Path

• Compute the distances between all pairs of
  vertices without resorting to matrix products
• Feder, ACM STC91 T(n3 / logn) time and O(n2)
  space
• Chan, ACM-SIAM06 O(n2(loglogn)2 / logn) time
  and O(n2) space
• ? Still too slow and space consuming for massive
  graphs

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All Pairs Almost Shortest Path (1/2)

• Compute almost shortest paths between all pairs
  of vertices Dor, ECCC97
• Additive error 2 ?
• Treat high-degree vertices and low-degree
  vertices separately

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All Pairs Almost Shortest Path (2/2)

• Additive error 2 apasp2
• O(min(n3/2m1/2, n7/3)logn) time and T(n2) space
• ? Still too expensive

11
Self-checking Heuristics

• Too expensive to obtain the exact value or
  accurate estimations of the diameter for massive
  graphs
• ? Empirically establish some lower and upper
  bounds by executing a suitable small number of
  BFS
• L ? D ? U
• Obtain the actual value of D for G when L U
• ? Self-checking heuristics

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Self-checking Heuristics

• No guarantee of success for every feasible input,
  BUT
• 1) It requires few BFSes in practice, and thus
  its complexity is linear Magnien, JEA09
• 2) An empirical upper bound is possible
• 3) Large graphs can be analyzed
• since BFS has a good external-memory
  implementation Mayer, AESA02 and works on
  graphs stored in compressed format Vigna,
  IWWWC04

13
A Comparing Work

• Fast Computation of Empirically Tight Bounds for
  the Diameter of Massive Graphs Magnien, JEA09
• Various bounds to confine the solution range
• Trivial bounds
• Double sweep lower bound
• Tree upper bound
• Iterative algorithm to obtain the actual diameter

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Trivial Bounds

• The eccentricity of any vertex v gives trivial
  bounds of the diameter ecc(v) D 2ecc(v)
• Trivial bounds can be computed in T(m) space and
  time, where m is the number of edges in the graph
• D 2ecc(v)
• If D gt 2ecc(v), then max(ecc(v)) gt 2ecc(v)
• We can choose a center point in the diameter that
  contradicts the derived inequality
• Therefore, D 2ecc(v)

15
Double Sweep Lower Bound

• On chordal graphs, AT-free graphs, and tree
  graphs, if a vertex v is chosen such that d(u, v)
  ecc(u) for a vertex u, then D ecc(u) (i.e. v
  is among the vertices which are at maximal
  distance from u) Corneil01, Handler73
• The diameter may therefore be computed by a BFS
  from any node u and then a BFS from a node at
  maximal distance from u, thus in T(m) space and
  time, where m is the number of edges.
• Generally, the value obtained in this way may
  different from the diameter, but still better
  than trivial lower bounds

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Double Sweep Lower Bound An Example
D 2
actual diameter
D 4
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Tree Upper Bound

• The diameter of any spanning connected subgraph
  of G is larger than or equal to the diameter of G
• Tree diameter can be obtain in T(m) time and
  space Handler73, where m is the number of
  edges in G
• Spanning trees of G, are good candidates for
  obtaining an upper bound
• A tree upper bound is the diameter of a BFS tree
  from a vertex
• It is always better than the corresponding
  trivial upper bound

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Tree Upper Bound An Example
D 5
actual diameter
D 4
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Tighten the Bounds

• Iteratively choosing different initial vertices
  for tighter bounds (for tree upper bounds)
• Random tree upper bound (rtub)
• Iterate the tree upper bound from random vertices
• Highest degree tree upper bound (hdtub)
• Consider vertices in decreasing order of degrees
  when iterating the algorithm

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The Iterative Algorithm

• Iterate the double sweep lower bound and highest
  degree tree upper bound until the difference
  between the best bounds obtained is lower than or
  equal to a given threshold value
• Multiple choices for this threshold value
• Depending factors the graph considered, the
  desired quality of the bounds, or even set the
  threshold to be a given precision (e.g. D-D/Dltp)
• All heuristics have a T(m) time complexity, and a
  T(mn) space complexity.
• Does the tree upper bound eventually converge to
  the exact diameter?

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Possibly Unmatching Upper Bound

• No guarantee of obtaining the exact diameter as
  all the tree upper bounds may be strictly larger
  than D
• E.g. if G is a cycle of n vertices, its diameter
  is n/2 and the tree upper bound is n-1 which ever
  vertex one starts from
• Is there an algorithm that provides more matching
  upper bounds?

D 5
D 3
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Outline
Introduction
R98943086 ???

• R98943090 ???
• R98943088 ???

Previous Work

• R99921040 ???
• R99942061 ???
• R98943086 ???

Finding the Diameter in Real-World Graphs
F96943167 ??? F97943070 ???
Other Related Topics
Conclusion and Future Work
R98921072 ? ?
23
The Fringe Algorithm

• Fringe method is used to improve the upper bound
  U and possibly match the lower bound L obtained
  by the double sweep method

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The Fringe Algorithm

• An unweighted, undirected and connected graph G(
  V, E )
• For any vertex
• Tu denotes an unordered BFS-tree
• Eccentricity ecc(u) is the height of Tu
• gt 2 ecc(u) ? diam(G)

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The Fringe Algorithm

• Proof 2 ecc(u) ? diam(G)
• gt ecc(u) ? diam(G)/2
• 1) if ecc(u) lt diam(G)/2, diam(G) d(a,b)
• d(u,v) lt diam(G)/2, for all
• then d(u,a)ltdiam(G)/2
• d(u,b)ltdiam(G)/2
• gt d(u,a)d(u,b)lt d(a,b)
• contradiction!!!
• ? 2 ecc(u) ? diam(G)

diameter
b
a
u
diameter
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The Fringe Algorithm

• Tu denotes an unordered BFS-tree
• Tu is a subgraph of G
• , , ,
  
• gt
• let , so

diam (Tu )
U
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The Fringe Algorithm

• The fringe of u, denote F(u), as the set of
  vertices such that

U
F(U) 3
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The Fringe Algorithm

U
B(u) max ecc(A), ecc(B), ecc(C)
A
A
B
C
B
C
BFS(A) gtecc(A)
BFS(B) gtecc(B)
BFS(C) gtecc(C)
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The Fringe Algorithm

• The fringe of u, denote F(u), as the set of
  vertices such that

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The Fringe Algorithm

• Lemma. U(u) ?D, where D is the diameter of G

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The Fringe Algorithm

• Case 1 F(u) 1 gt
• Case 2 F(u) gt 1 , B(u)2ecc(u)
• gt
• Case 3 F(u) gt 1 , B(u)2ecc(u)-1
• gt
• Case 4 F(u) gt 1 , B(u)lt2ecc(u)-1
• gt

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The Fringe Algorithm

• Case 1 F(u) 1

U
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The Fringe Algorithm

• Case 2 F(u) gt 1 , B(u)2ecc(u)
• ecc(u) 3 , diam(Tu) 6
• diameter upper bound 6
• B(u) provides lower bound
• gt if B(u) 2 ecc(u)
• ? diameter diam(Tu)

U
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The Fringe Algorithm

• Case 3 F(u) gt 1 , B(u)2ecc(u)-1
• Non-leave node
• upper bound 2ecc(u)-2
• Leave node
• upper bound 2ecc(u)
• if B(u) 2ecc(u)-1
• gt diameter 2ecc(u)-1

U
d(a,u) ? ecc(u)-1
d(b,u) ? ecc(u)-1
a
b
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The Fringe Algorithm

• Case 4 F(u) gt 1 , B(u)lt2ecc(u)-1
• Non-leave node
• upper bound 2ecc(u)-2
• Leave node
• upper bound 2ecc(u)
• if B(u) lt 2ecc(u)-1
• gt diameter ? 2ecc(u)-2

U
d(a,u) ? ecc(u)-1
d(b,u) ? ecc(u)-1
a
b
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The Fringe Algorithm

• The fringe algorithm correctly computes an upper
  bound for the diameter of the input graph G,
  using at most F(u)3 BFS.

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The Fringe Algorithm

• Let r,a,and b be the vertices identified by
  double sweep(using two BFSes)
• Find the vertex u that is halfway along the path
  connecting a and b inside the BFS-tree Ta
• Compute the BFS-tree Tu and its eccentricity
  ecc(u)
• If F(u)gt1,find the BFS-tree Tz for each
  and compute B(u)
• If B(u)2ecc(u)-1,return 2ecc(u)-1
• If B(u)lt2ecc(u)-1,return 2ecc(u)-2
• Return the diameter(Tu)

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Example(1/2)
x1 xp
When number of P is large !! We choose X1 as r
choose A ,B, x1 as b y1-gtA 4 y1-gtB 4 y1-gtx1 4
diameter 4
B
DS x1-gtA 3 x1-gtB 3 x1-gty1 4 Choose y1 as a
Diameter6
row3
B
A
Wrong !!!
y1
column6
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Example(2/2)
x1 xp
II. Find a vertex u that is
halfway along the path connecting a
and b
Case 1 III. ecc(u) 4 F(u)gt1
B(u)6
Case 2 IV.B(u)2ecc(u) 6 (23)
return 2ecc(u) diameter 6
Case 2 III. ecc(u) 3 F(u)gt1 B(u)6

Case 1 IV.B(u)lt2ecc(u)-1 6 lt (24) -1
return 2ecc(u)-2 diameter 6

• Fringe
• I. Use DS to find
• a and b
• x1 as a
• y1 as b

row3
y1
column6
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A Bad Case for Fringe
r
a
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A Bad Case for Fringe
b
u
a
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A Bad Case for Fringe

• Ecc(u) 3
• B(u) 3
• B(u) lt 2ecc(u) 1(5)
• return 2ecc(u) 2(4)
• Real diameter 3
• ? Fringe fail !!!

u
F(u)
43
Experimental Results (1/2)

• Implemented in C on a 2.93Ghz Linux workstation
  with 24 GB memory
• 44 real-word graphs are tested
• each with 4000 50 million nodes, 20000 3000
  million edges
• Real diameter is found by exhaustive search to
  check the obtained upper bounds

Approaches Results (44 in total) Results (44 in total)
Approaches Matches Failures
fub 37 7
mtub 13 31
hdtub 10 34
rtub 7 37
43
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Experimental Results (2/2)

• The proposed method generates the tightest upper
  bound for the 7 mismatches, compared with the
  approaches in previous work

Benchmarks D fub mtub hdtub rtub
CAH2 18 20 20 20 20
CITP 26 28 30 29 31
DBLP 22 24 24 24 25
P2PG 11 14 15 14 15
ROA1 865 987 987 1047 988
ROA2 794 803 803 873 832
ROA3 1064 1079 1079 1166 1128
44
45
Outline
Introduction
R98943086 ???

• R98943090 ???
• R98943088 ???

Previous Work

• R99921040 ???
• R99942061 ???
• R98943086 ???

Finding the Diameter in Real-World Graphs
F96943167 ??? F97943070 ???
Other Related Topics
Conclusion and Future Work
R98921072 ? ?
46
Finding the Diameter on Weighted Graphs

• Consider a large complete graph with edge weight
  be 1 except for only one edge
• The eccentricities of most points are 1
• However, the diameter of the graph is larger than
  1
• The fringe algorithm may not efficiently find
  tight diameter bounds for weighted graphs

1
1
1
1
1
1.5
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Minimum Diameter Spanning Trees

• Minimum diameter spanning tree (MDST) problem
• Given a graph G(V,E) with edge weight
• Find a spanning tree T for G such that
• is minimized

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1
1
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2
2
4
1
1
Diameter3
Diameter5
MDST
48
Outline
Introduction
R98943086 ???

• R98943090 ???
• R98943088 ???

Previous Work

• R99921040 ???
• R99942061 ???
• R98943086 ???

Finding the Diameter in Real-World Graphs
Other Related Topics
Geometric MDST
F97943070 ???
MDST
F96943167 ???
Conclusion and Future Work
R98921072 ? ?
49
Geometric MDST

• Geometric MDST (GMDST)
• Given a set of n points in the Euclidean space,
  find a spanning tree connecting these points so
  that the length of its diameter is minimum
• GMDST corresponds to finding an MDST on a
  complete graph with edge weight being the
  Euclidean distance between two points

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Monopolar and Dipolar

• A spanning tree is said to be monopolar if there
  exists a point (called monopole) s.t. all
  remaining points are connected to it
• A spanning tree is said to be dipolar if there
  exists two points (called dipole) s.t. all
  remaining points are connected to one of the two
  points in the dipole

dipole
monopole
A dipolar spanning tree
A monopolar spanning tree
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GMDST with a Simple Topology

• Theorem
• There exists a GMDST of a set S of n points
  which is either monopolar (n?3) or dipolar (n?4)

All monopolar spanning trees of the 4 points
4 dipolar spanning trees of the 4 points
52
Center Edge

• An edge (ai-1,ai) is a center edge of a path
  P(a0,a1,,ak) if
• is minimized

dist(ai, ak)
dist(a0, ai-1)
ak
a0
ai
ai-1
a1
ak-1
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Lemma

• Lemma
• Let (ai-1,ai) be a center edge of a path
  P(a0,a1,,ak), then
• (1) and
• (2)

dist(ai-1, ak)
dist(a0, ai-1)
ak
a0
ei-1
ei-2
ai
ai-1
ak-1
a1
ai-2
Otherwise, the center edge is not (ai-1, ai)
B
If A gt B max A, B-ei-1 gt max A-ei-2, B
A
ak
a0
ei-1
ei-2
ai
ak-1
a1
ai-1
ai-2
54
Proof of the Theorem

• Theorem There exists a GMDST of a set S of n
  points which is either monopolar or dipolar
• Proof
• Case 1 Given any optimal GMDST T with a diameter
  composed of only two edges, i.e., D(T) (a0, a1,
  a2) of size DT, a monopolar spanning tree T can
  be constructed with the same diameter

a1
a1
Optimal T
a0
a0
T
a2
a2
v
v
u
u
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Proof of the Theorem (contd)

• Case 2 Given any optimal GMDST T with diameter
  D(T) (a0,a1,,ak) of size DT, k ?3. A dipolar
  spanning tree T can be constructed with the
  same diameter
• Let (ai-1,ai) be the center edge of D(T)
• Connect all points in the subtree Ti-1 to ai-1,
  and connect all points in the subtree Ti to ai

Ti-1
Optimal T
Ti
T
a1
a1
ak-1
ak-1
a0
a0
ai
ai
ak
ak
ai-1
ai-1
Center edge
56
Proof of the Theorem (contd)

• For any point pair u and v, if the two points are
  in different subtrees, their distance is
  obviously less than DT
• If u and v are in the same subtree

Ti-1
Optimal T
Ti
T
a1
a1
ak-1
ak-1
a0
a0
ai
ai
ak
ak
ai-1
ai-1
v
v
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Finding a Geometric MDST

• Theorem There exists a GMDST of a set S of n
  points which is either monopolar or dipolar
• By enumerating all monopolar and dipolar spanning
  trees of a set of given points, an optimal GMDST
  can be found
• The enumeration process can be done in ?(n3)

58
Outline
Introduction
R98943086 ???

• R98943090 ???
• R98943088 ???

Previous Work

• R99921040 ???
• R99942061 ???
• R98943086 ???

Finding the Diameter in Real-World Graphs
Other Related Topics
Geometric MDST
F97943070 ???
MDST
F96943167 ???
Conclusion and Future Work
R98921072 ? ?
59
Introduction

• Ho et al., SIAMJ91 solved the geometric MDST
  in O(n3)
• Actually, the general MDST problem is identical
  to the absolute 1-center problem (A1CP)
• The absolute 1-center problem

Let
Find xx such that F(x) is minimized
59
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Equivalence of A1CP and MDST

• Theorem

SPT(x) is minimum diameter spanning tree
x is absolute 1-center (continuum set)
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The Proof of Equivalence

• Proof idea
• Considering metric space solution with continuum
  set SPT(y)
• Diameter of SPT(x) equals to that of SPT(y)
• As SPT(y) is minimum, the MDST is solved
• Continuum set
• Let the graph be rectifiable
• Refer interior points on an edge by their
  distances from the two nodes

3
5
10
7
5
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Property of Continuum Set

• For given tree T, diameter D(T) equals to
  2?FT(y)
• y is the absolute 1-center of T

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Proof

• Assume that z is the absolute 1-center of G
• By following the property of continuum set,
• Since the tree is the shortest path tree rooted
  at z,
• Since z is the absolute 1-center of G,
• For any tree Ti rooted at u,
• It implies that, for any spanning tree Tj ,

63
63
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Conclusion

• The concepts of monopolar and dipolar Ho et al.,
  SIAMJ91 are exactly the same as the proved
  result
• By using all pairs shortest distance Fredman and
  Tarjan, JACM87, the A1CP can be solved in O(mn
  n2 log n)

dipole
monopole
absolute 1-center
64
65
Outline
Introduction
R98943086 ???

• R98943090 ???
• R98943088 ???

Previous Work

• R99921040 ???
• R99942061 ???
• R98943086 ???

Finding the Diameter in Real-World Graphs
F96943167 ??? F97943070 ???
Other Related Topics
Conclusion and Future Work
R98921072 ? ?
66
Conclusion

• In todays presentation, we have
• Introduce the difference between finding the
  diameter on a tree and finding the diameter on a
  general graph
• Give some naïve algorithms for finding the
  diameter on a graph
• Present the double sweep algorithm introduced in
  the previous work
• Present the fringe algorithm which extends the
  double sweep algorithm
• Compare the double sweep algorithm and the fringe
  algorithm

67
Conclusion (contd)

• Besides, we further
• Identify the difference between finding the
  diameter on an unweighted graph and finding the
  diameter on a weighted graph
• Present two algorithms that find minimum diameter
  spanning trees on weighted graphs

68
Future Work

• Another topic related to the design methodology
  for directed graphs with minimum diameter is
  interesting as well
• Design to minimize diameter on building-block
  network, Makoto Imase and Masaki Itoh
• A design for directed graphs with minimum
  diamter, Makoto Imase and Masaki Itoh
• Given nodes and the upper bounds of in- and
  out-degree, design a directed graph s.t. the
  diameter is minimized

n 9, d 2
69
Future Work (contd)

• How to find the diameter (or find the tight upper
  and lower bounds) of a weighted graph is still an
  opening problem

3
2
1
1
3
1
1
1
1
3
2
1.5
5
5
3
2
4
Diameter 1.5
Diameter 9