Classification and reduction of Yutsis graphs





Project description




Dries Van Dyck

Veerle Fack

GYutsis Advanced Applet

The GYutsis advanced applet generates a summation formula over products of 6j-coefficients for a general angular momenta recoupling coefficient (or 3nj-coefficients). For problems upto 15 j's, corresponding with 6 initial angular momenta, the generated formulas are confirmed to be optimal.

In the field labeled "Summation Formula:" the formula corresponding with the current state of the graph is shown. The applet is divided by a slider in the middle, allowing the user to redistribute the space between the upper and lower part of the applet. We refer to the lower part as the advanced panel.

Defining a problem and generating the summation formula

In the input field, indicated by the label "Braket:", one can enter a general recoupling coefficient in its mathematical standard format. Once a problem is succesfully defined the buttons become active. By pressing the "Reduce"-button the graph will be completely reduced to a triangular delta.

When the applet initially starts, a problem is already presented in the input field, but not entered yet so that the user can edit or delete it in order to define his own problem. In the examples menu, one can select some special cases, which are filled in but not entered yet, allowing the user to edit them. The cases are the following:

  • W9j: the definition of the Wigner 6-j symbol.
  • Cb6: smallest problem for which the heavier heuristics perform better than the Edge Cost Heuristic.
  • C8: problem delivering a cubic cage of girth 8.
  • C9,7: problem for which the Cycle Count Heuristic performs best.
  • C9,12: problem for which the More Smaller/Less Bigger Heuristic performs 3 6-j symbols better than the Cycle Count Heuristic.

All features described above can also be selected from the menus.

Changing the heuristic of the algorithm

When no triangles or bubbles are available in the Yutsis graph, the algorithm used to generate the summation formula delegates the task of selecting an operation to an heuristic. Three heuristics are available:
Edge Cost Heuristic
this is the most simple and fastest heuristic: a cost is associated with each edge, equal to the difference in length of the two smallest cycles in which the edge participates. The cost of a cycle is defined as the minimum cost of its edges. This heuristic will interchange the edge with minimum cost out of the cycle with minimum cost. When two cycles have the same cost, the cycle for which the minimum edge cost is most reached is preferred, and if this is equal, the cycle with minimum total cost is choosen.

For small problems (upto 22 j's) this heuristic suffices, for higher problems the heavier heuristics provide shorter formulae.

More Smaller/Less Bigger Heuristic
this is the default heuristic and for most cases the best choice. This heuristic considers all possible interchanges making a girth cycle smaller. An interchange is preferred over an other if it makes more cycles of length l smaller, or if equal, makes less bigger. This criterium is repeatedly used for rising l starting at the girth until a difference found.
Cycle Count Heuristic
this heuristic also considers all possible interchanges making a girth cycle smaller, but prefers an interchange over an other if it it results in a graph with more cycles of length l. Again this criterium is used for rising l, starting at the girth minus 1, until a difference is found.

This heuristic yields shorter formulae than the Edge Cost Heuristic. For some cases it also performs better than the More Smaller/Less Bigger Heuristic, but these cases are rare.

Changing the format of the formula

The summation formula can be generated in four different output formats:
a compact, human readable format. This is the default.
for the popular typesetting system LaTeX. By default a macro is used to represent the Wigner 6-j symbol. This can be turned of by clicking on the "Use macros" checkbox under the Output-menu. Afterwards the formula will be outputted in plain LaTeX.
for the popular computer algebra package Maple. Macros are used to represent the Kronecker delta symbol, the triangular symbol and the Wigner 6-j symbol (they are in fact Maple functions).
for the Maple package Racah (package especially for Racah algebra).
Default macros for the LaTeX and Maple format can be printed to the "User"-pane by pressing the "Print Macros"-button in the advanced panel or by selecting "Generate Macros" from the output-menu.

Advanced Features

The advanced panel has a tabbed pane with three panels (left) and a row of buttons with user operations (right):
  • The "User"-pane contains all the ouput generated by the user.
  • In the "Operations"-pane all graph operations are logged. These operations can be:
    1. compound operations: removing/formatting a triangle, removing/formatting a bubble or formatting the graph as a triangular delta,
    2. elementary operations: performing an interchange, removing nodes, inverting node signs or inverting edge directions.
    By formatting we mean the process of locally inverting node signs and edge directions in order to get the subgraph in the correct form for applying a rule.
  • In the "Rules"-pane all applied rules are logged from an algorithmic point of view: if we apply an interchange to reduce a square to a triangle, the square is shown as "Best Cycle" and the edge wich is interchanged out of the square as "best edge". From the graph operations point of view this will be an interchange on the "best edge".

The user operations are:

Print Graph
Outputs the graph in the following format:
-0 | j1:-3 j2:-4 j5:+2 
-1 | j3:-3 j4:-4 j6:+2 
-2 | j5:-0 j6:-1 j7:+5 
+3 | j1:+0 j3:+1 j8:-5 
+4 | j2:+0 j4:+1 j9:-5 
+5 | j8:+3 j9:+4 j7:-2 
The first line contains the size of the problem n, for a general recoupling coefficient with n+1 leaves, corresponding with a cubic graph of 2n nodes and 3n edges. The next 2n lines contain the couplings of the graph:
<nodesign><node>| <edge><edge><edge>
with <edge>: <label>:<direction><endpoint>, where direction is '+' for an incoming edge and '-' for a leaving edge.
Prints all the girth cycles of the graph.
Best Cycle
Prints the cycle the heuristic marks as the best cycle together with its best edge (if the cycle is bigger than a triangle).
Performs one step in the reduction process, i.e. applies one rule.
Print Macros
Prints default macros for the selected output format (only for LaTeX and Maple). In the case of Maple, these macros contain Maple functions for the Kronecker delta symbol, the triangular symbol and the Wigner 6-j symbol.

Each of the panes can be cleared by pressing the "clear"-button.

All features described above can also be selected from the menus.

Mathematical Standard Format

The mathematical standard format, e.g.
is the most known format for general recoupling coefficients. Note that this problem is the same as the one used in the "print graph"-example.

It is not needed to specify the intermediate angular momenta:
is also accepted. In addition the root label can be dropped too:
The intermediate angular momenta/root will get labels of the form t<number>, with <number> starting at 1. Note that in this case it is forbidden to use labels of this form.

GYutsis Applet

For people only interested in the generated formula a more basic version, limiting the features to formula generation, can be found at:
Ofcourse the user is still able to choose the used heuristic and the output format of the generated summation formula.