# Recreational Math Games and Grap by fjzhangweiqun

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Other Applications of
Graphs and Networks

Jo Ellis-Monaghan
St. Michaels College
Colchester, VT 05439
e-mail: jellis-monaghan@smcvt.edu
Graphs and Networks
A Graph or Network is a set of vertices (dots) with edges (lines)
connecting them.

A                              A
A                                      B                         B
B

A multiple edge

C        D                C
D                                                        D           C

A loop

Two vertices are adjacent if there is a line between them. The vertices A and
B above are adjacent because the edge AB is between them. An edge is
incident to each of the vertices which are its end points.

The degree of a vertex is the number of edges sticking out from it.
The Kevin Bacon Game
or
6 Degrees of separation
Bacon      # of             Connery    # of
Number     People           Number     people
0           1              0           1
1         1766             1         2216
2       141840             2        204269
3       385670             3        330591
4        93598             4        32857
5         7304             5         2948
6         920              6          409
http://www.spub.ksu.edu/issues/v100/FA/n069/fea-        7         115              7          46
making-bacon-fuqua.html
8          61              8           8
Kevin Bacon is not even                            actors: 631275
Average Connery
Number: 2.706
among the top 1000 most                            Weighted total of linkable
actors: 1860181
connected actors in Hollywood                      Average Bacon number:        Data from The Oracle of
Bacon at UVA
(1222th).                                          2.947
The small world phenomenon

Stanley Milgram sent a
series of traceable letters
from people in the
Midwest to one of two
destinations in Boston.
The letters could be sent
only to someone whom
the current holder knew
by first name. Milgram
kept track of the letters
and found a median
thus supporting the
notion of "six degrees of
http://mathforum.org/mam/04/poster.html   separation."
Social Networks
•Stock Ownership (2001 NY
Stock Exchange)
•Children‘s Social Network
•Social Network of Sexual
Contacts

http://mathforum.org/mam/04/poster.html
Infrastructure and Robustness
Scale Free

Number of vertices
Vertex degree
JetBlue

Distributed

Number of vertices

Vertex degree
MapQuest
Maximal Matchings in Bipartite Graphs

A Bipartite Graph

Start at an
unmatched
vertex on the left                           End at an unmatched
vertex on the right

Find an alternating path

A maximal                         Switch matching to
matching!                         nonmatching and vice
versa
Conflict Scheduling

A                                              A

E                          B                  E                            B

D                 C                            D                  C
Draw edges between classes       Color so that adjacent vertices have different colors.
with conflicting times           Minimum number of colors = minimum required
classrooms.
Conflict Scheduling
Frequency Assignment                       Register Allocation
Assign frequencies to mobile radios and    Assign variables to hardware registers
other users of the electromagnetic         during program execution. Variables
spectrum. Two customers that are           conflict with each other if one is used both
sufficiently close must be assigned        before and after the other within a short
different frequencies, while those that    period of time (for instance, within a
are distant can share frequencies.         subroutine). Minimize the use of non-
Minimize the number of frequencies.        register memory.

 Vertices: users of mobile radios         Vertices: the different variables
 Edges: between users whose               Edges: between variables which conflict
frequencies might interfere                 with each other
 Colors: assignments of different         Colors: assignment of registers
frequencies

Need at least as many frequencies as the   Need at least as many registers as the
minimum number of colors required!         minimum number of colors required!
Rectilinear pattern recognition
joint work with J. Cohn (IBM), R. Snapp and D. Nardi (UVM)
IBM‘s objective is to check a chip‘s design and find all occurrences of a
simple pattern to:
– Find possible error spots
– Check for already patented segments
– Locate particular devices for updating

The Haystack

The Needle…
Pre-Processing
BEGIN    /* GULP2A CALLED ON THU FEB 21 15:08:23 2002 */
EQUIV 1 1000 MICRON +X,+Y
MSGPER -1000000 -1000000 1000000 1000000 0 0
LEVEL PC
LEVEL RX
(Raw data format)

PGON N RX 1467923 780300 1468180 780300 1468180 780600 +
1469020 780600 1469020 780300 1469181 780300 1469181 +
781710 1469020 781710 1469020 781400 1468180 781400 +
1468180 781710 1467923 781710
PGON N PC 1468500 782100 1468300 782100 1468300 781700 +
1468260 781700 1468260 780300 1468500 780300 1468500 +
780500 1468380 780500 1468380 781500 1468500 781500
RECT N PC 1467800 780345 1503 298
ENDMSG

Two different layers/rectangles
are combined into one
layer that contains three shapes;
one rectangle (purple)
and two polygons (red and blue)

Algorithm is cutting edge, and not currently used for this application in industry.
Linear time subgraph search for
target

Both target pattern and entire chip are encoded like this, with the vertices
also holding geometric information about the shape they represent. Then
we do a depth-first search for the target subgraph. The addition information
in the vertices reduces the search to linear time, while the entire chip
encoding is theoretically N2 in the number of faces, but practically NlogN.
Netlist Layout
How do we convert this…

… into this?
Netlist
A set S of vertices ( the pins) hundreds of thousands.
A partition P1 of the pins (the gates) 2 to 1000 pins per gate, average of
A partition P2 of the pins (the wires) again 2 to 1000 pins per wire,
A maximum permitted delay between pairs of pins.

Example

Gate      Pin             Wire
The Wires
The Wiring Space

Placement layer-         Vias (vertical
gates/pins go here       connectors)

Horizontal wiring      Vertical wiring
layer
Up to 12 or so layers
layer
The general idea
Place the pins so that pins are in
their gates on the placement
layer with non-overlapping
gates.

Place the wires in the wiring
space so that the delay
constrains on pairs of pins
are met, where delay is
proportional to minimum
distance within the wiring,
and via delay is negligible
Lots of ProbLems….
Identify Congestion
B    D   G
 Identify dense substructures from the netlist
 Develop a congestion ‗metric‘                    A    F

C    E   H
Congested area                              Congested
area

What often happens                      What would be good
Automate Wiring Small Configurations

Some are easy to place and route
Simple left to right logic
No / few loops (circuits)
Uniform, low fan-out
Statistical models work

Some are very difficult
E.g. ‗Crossbar Switches‘
Many loops (circuits)
Non-uniform fan-out
Statistical models don‘t work
Biomolecular constructions

Nano-Origami: Scientists At
Scripps Research Create
Single, Clonable Strand Of
DNA That Folds Into An
Octahedron
A group of scientists at The
Scripps Research Institute has
designed, constructed, and imaged
a single strand of DNA that
spontaneously folds into a highly
rigid, nanoscale octahedron that is
several million times smaller than
the length of a standard ruler and
about the size of several other
common biological structures, such
as a small virus or a cellular
ribosome.
http://www.sciencedaily.com/releases/2004/02/040
212082529.htm
Assuring cohesion
A problem from biomolecular computing—physically constructing
graphs by ‗zipping together‘ single strands of DNA

(not allowed)
N. Jonoska, N.
Saito, ‘02
DNA sequencing

It is very hard in
off‘ the sequence of
a long strand of
researchers probe
AGGCTC
AGGCT                            for ―snippets‖ of a
GGCTC
fixed length, and
CTACT
TCTAC
The problem then
becomes
reconstructing the
original long strand
of DNA from the
CTCTA                      TTCTA
set of snippets.
Enumerating the reconstructions

This leads to a directed graph with the same number of in-arrows as out
arrows at each vertex.

The number of reconstructions is then equal to the number of paths
through the graph that traverse all the edges in the direction of their
arrows.
Conquering the crazy cubes

The cubes from my puzzle are represented below.

B                                        R

G     W      R       G                  G      W   G       B

W                                        B
2
1

G                                        R

R     W      R       B                 R       W   B       G

R          3                             W       4
Build the Model

We will model each cube with a multigraph.
The vertices will correspond to the four colors and we connect the
corresponding vertices u and v if there is a pair of opposite faces colored u
and v.

1                                        2

3                                       4
Build the Model

Now construct a single multigraph with 4 vertices and the 12 edges,
labeling each edge by the cube associated with it.

3                 4
R                          W
3
2
1
1 4
1               2                4       2

1
3
B                         2 G

2
3               4
Characterize a Solution

Suppose the puzzle has a
3            4
solution. How would it be
represented on the final                            3
multigraph?                                             2
1
1 4
4   2

1
3
2

One subgraph will represent the front and back of the tower               2
and a second subgraph will represent the sides of the tower.

Using an edge in a subgraph corresponds to a positioning of the
cube (either front/back or sides).
Characterize Front/Back and Left/Right
subgraphs

What are the restrictions on the subgraphs and how do they
relate to the solution?

1. Uses all four vertices.                        (all four colors)

2. Must contain four edges, one from each cube.   (orient each cube)

3. No edge can be used more than once.            (can‘t use same
orientation twice)

4.   Each vertex must be of degree 2.             (use that color twice,
one front & one back,
or one left & one right)
Two Subgraphs satisfying the conditions

4
Here are two such subgraphs:                                 3
3
21
2                    14
4
1
3    2

2                            3                                                   2

4                 1           2               4
3                            1

Front/Back                      Sides

Now, stack the cubes using these faces as the front/back and sides.
Since each edge represents an orientation, label the edges to determine the
orientation.
Build the solution stack

4
Now, stack the cubes using these faces as the front/back                 3
3
and sides.                                                                                   21
2                    14
4
1
3    2

b        b                         r                                                         2
f                 f                            3
2                         l               r   l
4                      1             2                           4
b         3 f     b                 r                   r
l
f                                   l     1

Front/Back                              Sides
? Is there another solution?
? Is it possible to find a set of subgraphs that use the loops ?
All subgraphs with 4 edges and
4 vertices of degree 2
Only six types of subgraphs meet the solution requirements:

―square cycle‖           ―crossed cycle‖      ―loop and triangle‖

―four loops‖          ―two loops and a C2‖      ―two C2‘s‖
“Same”
Crossed Cycles

Loop and a Triangle
“Same” Arrangements
Two double loops

Three loops
So, look for these types of subgraphs

―square cycle‖                   ―crossed cycle‖              ―loop and triangle‖

―four loops‖                   ―two loops and a C2‖              ―two C2‘s‖

Crazy Cubes/Instant Insanity slides modified from: