new empirical study of alternative traffic equilibrium algorithms
DESCRIPTION
New Empirical Study of Alternative Traffic Equilibrium Algorithms. Zhong Zhou & Matthew Martimo Citilabs. Outline. Background & Motivation New Assignment Algorithms in Cube Voyager Empirical Studies Conclusions. Background & Motivation. Traffic Assignment Problems. - PowerPoint PPT PresentationTRANSCRIPT
Discover CubeNew Empirical Study of Alternative Traffic Equilibrium
Algorithms
Zhong Zhou & Matthew Martimo
*
*
Outline
Empirical Studies
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Background & Motivation
*
*
Traffic Assignment Problems
Traffic Assignment is a process of allocating the given origin-destination (OD) trip table to the transportation network under certain rules
User Equilibrium (UE) Principle: “No traveler can improve his or her travel cost by unilaterally changing routes” (Wardrop, 1952)
(All of the used paths have equal and minimum travel times; all of the unused paths have equal or higher travel times)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Frank-Wolfe Algorithm
Basic Idea:
Find a new solution by using line search
Advantage:
The linearized subproblem can be efficiently solved by AON assignment
Memory efficiency (only link variables need to be stored)
Disadvantage:
Slow convergence near the optimal point, take long time to reach high precision
Zig-zagging effect that means the flow may not stable
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Basic Idea:
Find a new solution by using line search
Frank-Wolfe Algorithm
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
High Level of Convergence is Important
A Relative Gap of 0.01 % (0.0001) is required to assure that the assignment is sufficiently converged to achieve stable link flows. (Boyce, et al., 2004)
Traditional FW suffers from slow converge speed to desired precision level ( e.g., relative gap < 0.0001)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Overview of Different Traffic Assignment Algorithms
Link-based Algorithm
Path-based Algorithm
Origin-based Algorithm
Decision variable
Link flow
Path flow
Information
Minimum
Richest
Convergence speed
Fast converge at the beginning, but maybe hard to reach high accuracy
Fast
Fast
Main categories
Frank-Wolfe algorithm (Frank and Wolfe, 1956) Improved FW algorithms (Powell and Sheffi, 1982; Fukushima, 1984;Weintraub et al., 1985;LeBlanc et al., 1985; Lupi, 1986; Florian et al., 1987;Arezki, 1987;Arezki and Van Vliet, 1990;Daneva, 2003) Restricted Simplicial Decomposition (RSD) algorithm (Hearn et al., 1985) Nonlinear Simplicial Decomposition (NSD) (Larsson and Patriksson, 1997)
Disaggregate Simplicial Decomposition (DSD) algorithms (Larsson and Patriksson, 1992) Gradient Projection (GP) algorithm (Jayakrishnan et al., 1994;Sun et al., 1996; Chen et al., 2002) Gradient Projection method (Rosen , 1960; Florian et al., 2008)
Origin-based Algorithm (OBA) (Bar-Gera, 1999,2002) Improved Origin-based Algorithm (NOB) (Nie, 2008) Algorithm B (Dial, 1999, 2006) Traffic Assignment by Paired Alternative S egments (TAPAS) (Bar-Gera and Luzon, 2007)
Observations
The latest Conjugate/Bi-conjugate FW algorithms appear comparable with origin-based algorithm in speed and accuracy
Gradient Projection algorithm shows more advantages than DSD in small to medium size networks
Algorithm B and TAPAS show better performance than Bar-Gera’s origin-based algorithm
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Assignment Algorithms in CUBE Voyager
Los Angeles - Discover Cube Workshop
*
*
Improved Frank-Wolfe algorithms
Improved FWs
Powell and Sheffi (1982)
Fukushima(1984); Arezki (1987)
Point of sight is a convex combination of previous FW points (normally 2 points) Point of sight is a convex combination of previous FW points, such that search direction of current iteration is orthogonal to the previous FW direction
PARTAN (Parallel Tangent) (Leblanc et al., 1985; Florian et al., 1987; Arezki and Vliet, 1990)
Extra (PARTAN) step is taken between the FW iteration steps (2 line searches)
Lupi (1986)
Current search direction is a convex combination of the FW direction and previous search direction, such that the current and previous search direction are orthogonal
Conjugate/Bi-Conjugate FW (Daneva, 2003)
Point of sight is a convex combination of previous CFW point(s) and current FW point, such that the search direction is conjugate to the previous search direction(s) (i.e., previous (Bi-)CFW direction)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Link-based Assignment Algorithms in Voyager
Basic Idea:
Advantages
Only one line search step has to be performed in order to find the new solution (same as FW)
At each iteration, only three (four) vectors in memory to find a new conjugate search direction
Conjugate FW & Bi-Conjugate FW (Daneva, 2003)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Good Features
Able to keep consistence with existing practice
Full functionality as that provided by the traditional FW assignment procedure without need to modify anything (network, input data etc.)
Multiple user classes, Turning penalties, Junction Modeling
Select link analysis and similar analysis
Distributed computing, Etc.
Maintain ‘proportionality’ very well in select link analysis based on our preliminary tests
New Link-based Assignment Algorithms in Voyager (Cont.)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Path-based Assignment Algorithms in Voyager
Firstly introduced to transportation field by Jayakrishnan et al. (1994).
Based on Goldstein-Levitin-Polyak gradient projection method formulated by Bertsekas (1976) for general nonlinear multi-commodity problem
Extensively used in computer communication networks for optimal flow routing
Basic Idea
In contrast to FW, which finds auxiliary solutions that are vertices (extreme points) of the feasible region, GP uses a transformed objective function and makes successive moves in the direction of negative gradient, scaled by the approximation of the second derivative Hessian
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Path-based Assignment Algorithms in Voyager
Feasibility
The memory restriction for tracking the paths has been relaxed considerably in recent years due to rapid advances in computing environment
Advantages
Unique Link Flow Solution
Select Link, Select Zone, …
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Empirical Studies
*
*
Testing Environments
Computing Platform
64 bit Intel Platform with Vista 64
Two Xeon E5335 2GHz Quad Core Processors and 8GB of RAM
Chicago Regional Network
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Los Angeles - Discover Cube Workshop
*
*
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
# of Iteration
CPU Time
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Effect of Distributed Computing
2 Cores
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Effect of Distributed Computing
2 Cores
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Convergence Analysis on the New Path Based Algorithm
Los Angeles - Discover Cube Workshop
*
*
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
# of Iteration
CPU Time
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Select Link Analysis
Using North Ave Bridge
L=8032-8037 && L=8037-8752 && L=8752-8753 && L=8753-6380 && L=6380-6389 && L=6389-10344
Not using North Ave Bridge
L=8032-8749 && L=8749-8750 && L=8750-8751 && L=8751-8994 && L=8994-8993 && L=8993-10344
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Conjugate FW with Relative Gap = 1e-4
Number of OD
OD Flow Proportion
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Flow Distribution
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Proportionality under Relative Gap 1e-4
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Conclusions
*
*
Conclusions
Converge faster to small relative gap than traditional FW algorithm
Memory efficiency (require similar memory as FW algorithm)
Consistent with existing practice
Keep all available abilities as that provided by FW algorithm (select link analysis, distributed computing, etc.)
Maintain ‘proportionality’ in select link analysis based on our preliminary tests
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Conclusions (Cont.)
Converge much quickly to desired precision level than FW algorithms
Loss of detail and proportionality in results
More research and enhancement are undergoing, and more tests are needed
Will be available soon in new release of Cube Voyager
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Acknowledgements
Professor David Boyce, Hillel Bar-Gera and Yu Nie
for their research and helpful discussions!
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
*
*
Zhong Zhou & Matthew Martimo
*
*
Outline
Empirical Studies
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Background & Motivation
*
*
Traffic Assignment Problems
Traffic Assignment is a process of allocating the given origin-destination (OD) trip table to the transportation network under certain rules
User Equilibrium (UE) Principle: “No traveler can improve his or her travel cost by unilaterally changing routes” (Wardrop, 1952)
(All of the used paths have equal and minimum travel times; all of the unused paths have equal or higher travel times)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Frank-Wolfe Algorithm
Basic Idea:
Find a new solution by using line search
Advantage:
The linearized subproblem can be efficiently solved by AON assignment
Memory efficiency (only link variables need to be stored)
Disadvantage:
Slow convergence near the optimal point, take long time to reach high precision
Zig-zagging effect that means the flow may not stable
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Basic Idea:
Find a new solution by using line search
Frank-Wolfe Algorithm
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
High Level of Convergence is Important
A Relative Gap of 0.01 % (0.0001) is required to assure that the assignment is sufficiently converged to achieve stable link flows. (Boyce, et al., 2004)
Traditional FW suffers from slow converge speed to desired precision level ( e.g., relative gap < 0.0001)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Overview of Different Traffic Assignment Algorithms
Link-based Algorithm
Path-based Algorithm
Origin-based Algorithm
Decision variable
Link flow
Path flow
Information
Minimum
Richest
Convergence speed
Fast converge at the beginning, but maybe hard to reach high accuracy
Fast
Fast
Main categories
Frank-Wolfe algorithm (Frank and Wolfe, 1956) Improved FW algorithms (Powell and Sheffi, 1982; Fukushima, 1984;Weintraub et al., 1985;LeBlanc et al., 1985; Lupi, 1986; Florian et al., 1987;Arezki, 1987;Arezki and Van Vliet, 1990;Daneva, 2003) Restricted Simplicial Decomposition (RSD) algorithm (Hearn et al., 1985) Nonlinear Simplicial Decomposition (NSD) (Larsson and Patriksson, 1997)
Disaggregate Simplicial Decomposition (DSD) algorithms (Larsson and Patriksson, 1992) Gradient Projection (GP) algorithm (Jayakrishnan et al., 1994;Sun et al., 1996; Chen et al., 2002) Gradient Projection method (Rosen , 1960; Florian et al., 2008)
Origin-based Algorithm (OBA) (Bar-Gera, 1999,2002) Improved Origin-based Algorithm (NOB) (Nie, 2008) Algorithm B (Dial, 1999, 2006) Traffic Assignment by Paired Alternative S egments (TAPAS) (Bar-Gera and Luzon, 2007)
Observations
The latest Conjugate/Bi-conjugate FW algorithms appear comparable with origin-based algorithm in speed and accuracy
Gradient Projection algorithm shows more advantages than DSD in small to medium size networks
Algorithm B and TAPAS show better performance than Bar-Gera’s origin-based algorithm
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Assignment Algorithms in CUBE Voyager
Los Angeles - Discover Cube Workshop
*
*
Improved Frank-Wolfe algorithms
Improved FWs
Powell and Sheffi (1982)
Fukushima(1984); Arezki (1987)
Point of sight is a convex combination of previous FW points (normally 2 points) Point of sight is a convex combination of previous FW points, such that search direction of current iteration is orthogonal to the previous FW direction
PARTAN (Parallel Tangent) (Leblanc et al., 1985; Florian et al., 1987; Arezki and Vliet, 1990)
Extra (PARTAN) step is taken between the FW iteration steps (2 line searches)
Lupi (1986)
Current search direction is a convex combination of the FW direction and previous search direction, such that the current and previous search direction are orthogonal
Conjugate/Bi-Conjugate FW (Daneva, 2003)
Point of sight is a convex combination of previous CFW point(s) and current FW point, such that the search direction is conjugate to the previous search direction(s) (i.e., previous (Bi-)CFW direction)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Link-based Assignment Algorithms in Voyager
Basic Idea:
Advantages
Only one line search step has to be performed in order to find the new solution (same as FW)
At each iteration, only three (four) vectors in memory to find a new conjugate search direction
Conjugate FW & Bi-Conjugate FW (Daneva, 2003)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Good Features
Able to keep consistence with existing practice
Full functionality as that provided by the traditional FW assignment procedure without need to modify anything (network, input data etc.)
Multiple user classes, Turning penalties, Junction Modeling
Select link analysis and similar analysis
Distributed computing, Etc.
Maintain ‘proportionality’ very well in select link analysis based on our preliminary tests
New Link-based Assignment Algorithms in Voyager (Cont.)
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Path-based Assignment Algorithms in Voyager
Firstly introduced to transportation field by Jayakrishnan et al. (1994).
Based on Goldstein-Levitin-Polyak gradient projection method formulated by Bertsekas (1976) for general nonlinear multi-commodity problem
Extensively used in computer communication networks for optimal flow routing
Basic Idea
In contrast to FW, which finds auxiliary solutions that are vertices (extreme points) of the feasible region, GP uses a transformed objective function and makes successive moves in the direction of negative gradient, scaled by the approximation of the second derivative Hessian
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Path-based Assignment Algorithms in Voyager
Feasibility
The memory restriction for tracking the paths has been relaxed considerably in recent years due to rapid advances in computing environment
Advantages
Unique Link Flow Solution
Select Link, Select Zone, …
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
New Empirical Studies
*
*
Testing Environments
Computing Platform
64 bit Intel Platform with Vista 64
Two Xeon E5335 2GHz Quad Core Processors and 8GB of RAM
Chicago Regional Network
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Los Angeles - Discover Cube Workshop
*
*
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
# of Iteration
CPU Time
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Effect of Distributed Computing
2 Cores
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Effect of Distributed Computing
2 Cores
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Convergence Analysis on the New Path Based Algorithm
Los Angeles - Discover Cube Workshop
*
*
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
# of Iteration
CPU Time
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Select Link Analysis
Using North Ave Bridge
L=8032-8037 && L=8037-8752 && L=8752-8753 && L=8753-6380 && L=6380-6389 && L=6389-10344
Not using North Ave Bridge
L=8032-8749 && L=8749-8750 && L=8750-8751 && L=8751-8994 && L=8994-8993 && L=8993-10344
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Conjugate FW with Relative Gap = 1e-4
Number of OD
OD Flow Proportion
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Flow Distribution
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Proportionality under Relative Gap 1e-4
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Conclusions
*
*
Conclusions
Converge faster to small relative gap than traditional FW algorithm
Memory efficiency (require similar memory as FW algorithm)
Consistent with existing practice
Keep all available abilities as that provided by FW algorithm (select link analysis, distributed computing, etc.)
Maintain ‘proportionality’ in select link analysis based on our preliminary tests
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Conclusions (Cont.)
Converge much quickly to desired precision level than FW algorithms
Loss of detail and proportionality in results
More research and enhancement are undergoing, and more tests are needed
Will be available soon in new release of Cube Voyager
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
Acknowledgements
Professor David Boyce, Hillel Bar-Gera and Yu Nie
for their research and helpful discussions!
Los Angeles - Discover Cube Workshop
*
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
*
*
*