A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in...
-
Upload
kory-wells -
Category
Documents
-
view
220 -
download
0
Transcript of A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in...
![Page 1: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/1.jpg)
A Retrograde Approximation Algorithm for One Player Can’t Stop
James Glenn Loyola College in Maryland
Haw-ren Fang University of Maryland, College Park
Clyde Kruskal University of Maryland, College Park
![Page 2: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/2.jpg)
Can’t Stop Equipment 2 to 4 players 4 6-sided dice Board with
columns 2-12 Colored markers
for each column 3 neutral markers Goal: advance to
top of 3 columns78
910
1112
65
43
2
![Page 3: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/3.jpg)
Game Play
Roll 4 dice Split into 2 pairs Advance neutral markers in columns
for pair totals Roll again or stop Turn ends if no way to use pair totals
![Page 4: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/4.jpg)
Example
78
910
1112
65
43
2
![Page 5: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/5.jpg)
Position Graph
Similar to Backgammon or Yahtzee Bipartite (V1, V2, E) (v2, v1) in E correspond to player
choices (v1, v2) in E correspond to outcomes of
random event
![Page 6: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/6.jpg)
Can’t Stop Graph
![Page 7: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/7.jpg)
Comparison to Other Games
Solitaire Yahtzee No cycles
Backgammon General: long cycles Bearing off: only short cycles
Can’t Stop Cycles long but only within one turn components form a DAG
![Page 8: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/8.jpg)
Can’t Stop Graph
anchor
anchors of other components
![Page 9: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/9.jpg)
Retrograde Analysis Topologically sort graph Compute position value for each
vertex Start with final states Work back towards start state Player choice: compute max/min over
outgoing edges Random event: weighted average Can’t Stop: retrograde analysis on
components
![Page 10: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/10.jpg)
Symbolic Analysis
f(B)=f(E) f(C)=min(f(E), f(F)) f(D)=f(A) f(A)=p1f(E)
+p2•min(f(E),f(F))
+p3f(A)A
B C D
E F
p1p2
p3
![Page 11: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/11.jpg)
Symbolic Analysis for Can’t Stop f(E), f(F) given f(H)=f(F) f(G)=f(A) f(D)=p3f(G)+p4f(H) f(C)=min(f(E),f(F)) f(B)=min(f(D),f(E)) f(A)=1+p1f(B)+p2f(C)=
1+p1min(p3f(A)+p4f(F),f(E))
+p2min(f(E), f(F))A
B C
E F
D
G H
p1 p2
p3 p4
![Page 12: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/12.jpg)
Numerical Analysis Make a copy of anchor component is now a DAG Guess value of f(A’) f(A) is a function of f(A’) Want fixed point
Function is piecewise linear and continuous
Fast convergence from Newton’s method
A
B C
E F
D
G H
p1 p2
p3 p4
A’
![Page 13: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/13.jpg)
Retrograde Analysis for Can’t Stop
f(v) = 0 for all final states v For each non-final anchor in reverse
order of topological sort Make a copy of the anchor Topologically sort the anchor’s
component Apply Newton’s method to find value of
anchor; using retrograde analysis on each iteration
![Page 14: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/14.jpg)
Results
Official game: estimated 3000 years
Dice Size Graph Size Time to Solve Optimal Turns
2 1 225 0.166 sec 1.298
2 2 1,936 0.405 sec 1.347
2 3 9,025 0.601 sec 1.400
3 1 64,372 1.70 sec 1.480
3 2 787,600 5.05 sec 1.722
3 3 4,934,006 23.3 sec 1.890
4 1 20,802,843 5 min 2.187
4 2 289,091,584 59 min 2.454
4 3 2,104,663,011 6 hr 2.700
5 1 7,105,015,062 2.8 days 2.791
![Page 15: A Retrograde Approximation Algorithm for One Player Can’t Stop James Glenn Loyola College in Maryland Haw-ren Fang University of Maryland, College Park.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bf701a28abf838c7dab8/html5/thumbnails/15.jpg)
Future Work
Optimizations to Algorithm Better initial estimate Shortcuts when evaluating components
Distributed Algorithm Analysis of 2-player Can’t Stop