Recreational Math Games and Graph Theory by ert554898


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

                               A                              A
     A                                      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
                           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        7         115              7          46
                                                        8          61              8           8
                                                   Total number of linkable
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
Maximal Matchings in Bipartite Graphs

                     A Bipartite Graph

                                          Start with any matching

   Start at an
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
              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
                                          chain length of about six,
                                          thus supporting the
                                          notion of "six degrees of   separation."
Social Networks
•Stock Ownership (2001 NY
        Stock Exchange)
•Children’s Social Network
•Social Network of Sexual
Infrastructure and Robustness
             Scale Free

                       Number of vertices
                                                        Vertex degree


                                   Number of vertices

                                                         Vertex degree
           Rolling Blackouts inAugust 2003
     Some Networks are more robust than others.
           But how do we measure this?
                       A network modeled by a graph
                (electrical, communication, transportation)

    Question: If each edge operates independently with probability p, what
           is the probability that the whole network is functional?



     A functional network                   A dysfunctional network
(can get from any vertex to any              (vertices s and t can’t
other along functioning edges)                  communicate)
         Deletion and Contraction is a Natural
          Reduction for Network Reliability

If an edge is working (this happens
with probability p), it’s as thought the
two vertices were “touching”—i.e.
just contract the edge:

If an edge is not working (this
happens with probability 1-p), it
might as well not be there—i.e. just
delete it:

     Thus, if R(G;p) is the reliability of the network G where all edges
  function with a probability of p, and e is not a bridge nor a loop, then

                     R(G;p) =(1-p)R(G-e;p) + p R(G/e;p)
                                Reliability Example

Note that if every edge of the network is a bridge (i.e. the network is a disjoint
union of trees), then R(G;p) = (p)E, where E is the number of edges.
Also note that R(loop;p) = 1

                      (1-p)           + p

                = (1-p)p2 + p (1-p)   +pp        = (1-p)p2 + p(1-p)p + p2

   So R(G;p) = 3p2- 2p3 gives the probability that the network is functioning.
                             E.g. R(G; .5)=.5625

   Bothersome question: Does the order in which the edges are deleted and
                           contracted matter?
                             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
                   Coloring Algorithm
The Chromatic Polynomial counts the ways to vertex color a graph:
C(G, n ) = # proper vertex colorings of G in n colors.

             G          G\e                               G-e

                   +                     =

  Recursively: Let e be an edge of G . Then,
  CG; n  C(G  e; n)  CG \ e; n               C; n  n

         =                 -             =      n(n-1)2 +     +

                 = n(n-1)2 +n(n-1) + 0 = n2 (n-1)
                     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

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!
                      The Ising Model

Consider a sheet of Metal:

 It has the property that at low temperatures it is magnetized, but as the
 temperature increases, the magnetism “melts away”.
 We would like to model this behavior. We make some simplifying
 assumptions to do so—

 •The individual atoms have a “spin”, i.e., they act like little bar
 magnets, and can either point up (a spin of +1), or down (a spin of –1).
 •Neighboring atoms with different spins have an interaction energy,
 which we will assume is constant.
 •The atoms are arranged in a regular lattice.
At low temperature “coalescing” states are more probable and there is
non-zero magnetization
As the temperature rises, the states become more random, and the
magnetization “melts away”

Applet by Peter Young at

                      Magnetization =                       1
                                                            N    si           , Energy =  N  si s j

                              where N is the number of lattice points.
                                     Critcal Temperature is
                                                           2 ln(1  2)
                         Lattice and Hamiltonian

 A choice of spins at each point gives what is called a “state” of the lattice:

The Hamiltonian (total energy) of a state w is                    
                                                   H  w    f si , s j   

 where the sum is over all adjacent points, and f is 0 if the spins are the
 same and 1 if they are different.
 H(w) is just the total number of edges in the state with different spins on
 their endpoints.
                                A Little Thermodynamics

                                                                         e   H ( w)
                                                                                       H  w
The probability of a state occurring is:
                                                                                e
                                                                  all states w
                    1                                                                             23
        Here          , where T is the temperature and k is the Boltzman constant 1.38  10           joules/Kelvin.

  The numerator is easy. The denominator, called the partition function is the
  interesting (hard) piece.

                        It has a deletion-contraction reduction!
  Let                                                  H  w
                   P  G;                    e                . Then
                                  all states w

                                                                    
                   P  G;    P  G  e;    e   1 P  G / e;  
            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…
  BEGIN    /* GULP2A CALLED ON THU FEB 21 15:08:23 2002 */
  EQUIV 1 1000 MICRON +X,+Y
  MSGPER -1000000 -1000000 1000000 1000000 0 0
  HEADER GYMGL1 'OUTPUT 2002/02/21/14/47/12/cohn'
                                                               (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

                                                               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

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
(joint work with J. Cohn, A. Dean, P. Gutwin, J. Lewis, G. Pangborn)

                               How do we convert this…

             … into this?
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
   about 3.5.
A partition P2 of the pins (the wires) again 2 to 1000 pins per wire,
   average of about 3.5.
A maximum permitted delay between pairs of pins.


             Gate      Pin             Wire
The Wires
                                 The Wiring Space

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

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

          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

    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
Random layout   Spring embedded layout
             Biomolecular constructions

                                                   Nano-Origami: Scientists At
                                                   Scripps Research Create
                                                   Single, Clonable Strand Of
                                                   DNA That Folds Into An
                                                   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
    DNA Strands Forming a Cube
             Assuring cohesion
A problem from biomolecular computing—physically constructing
       graphs by ‘zipping together’ single strands of DNA

                                      (not allowed)
                                                      N. Jonoska, N.
                                                      Saito, ’02
                   A Characterization

A theorem of C. Thomassen specifies precisely when a graph may be
constructed from a single strand of DNA, and theorems of Hongbing and
Zhu to characterize graphs that require at least m strands of DNA in their

Theorem: A graph G may be constructed from a single strand of DNA if
and only if G is connected, has no vertex of degree 1, and has a spanning
tree T such that every connected component of G – E(T) has an even
number of edges or a vertex v with degree greater than 3.
    L. M. Adleman, Molecular Computation of
 Solutions to Combinatorial Problems. Science,
        266 (5187) Nov. 11 (1994) 1021-1024.

    Oriented Walk Double Covering and Bidirectional
                   Double Tracing
                                      Fan Hongbing, Xuding Zhu, 1998

“The authors of this paper came across the problem of bidirectional
double tracing by considering the so called “garbage collecting”
problem, where a garbage collecting truck needs to traverse each
side of every street exactly once, making as few U-turns
(retractions) as possible.”
            DNA sequencing
            (joint work with I. Sarmiento)

                                             It is very hard in
                                             general to “read
                                             off’ the sequence of
                                             a long strand of
                                             DNA. Instead,
                                             researchers probe
  AGGCT                                      for “snippets” of a
                                             fixed length, and
                                             read those.
                                             The problem then
                                             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
         Graph Polynomials Encode
             the Enumeration

A very fancy polynomial, the interlace polynomial,
of Arratia, Bollobás, and Sorkin ,2000, encodes the
number of ways to reassemble the original strand of

It is related, with a lot of work, to the contraction-
deletion approach of the Chromatic and Reliability
The interlace polynomial
is computed, not on the               a
                                                   The “snippet” graph
“snippet” graph, but on                        b
an associated circle
                           d                        c


             c             b
                                           a            c

         d                     a

                                          d             b
             c             d

     A chord diagram               The associated circle graph
                 Pendant Duplicate Graphs
             Effect of adding a pendant vertex or duplicating a vertex
         a                c
v’                                                                          v'

                              Adding a
         v                b   pendant vertex         v
                              to v.
     a                c                                                 v'

     v                        the vertex v.         v
                      b                                  v'

    A set of subsequences of DNA permits
exactly two reconstructions iff the circle graph
    associated to any Eulerian circuit of the
 ‘snippet’ graph is a pendant-duplicate graph.

  Side note to the cognesci: Pendant-duplicate graphs
  correspond to series-parallel graphs via a medial graph
  construction, so the two reconstructions is actually a new
  interpretation of the beta invariant.

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