A 2-player game for adaptive covering codes
Robert B. EllisTexas A&M
coauthors:Vadim Ponomarenko, Trinity University
Catherine Yan, Texas A&M
A football pool problem
Round 1 Round 2 Round 3 Round 4 Round 5
Bet 1 W W W W W
Bet 2 L W W W W
Bet 3 W L W W W
Bet 4 W W L L L
Bet 5 L L W L L
Bet 6 L L L W L
Bet 7 L L L L W
Payoff: a bet with · 1 bad predictionQuestion. Min # bets to guarantee a payoff?
Ans.=7
Covering code formulation11111
11101 11011 1011111110
11100 11010 11001 10110 10101 10011
10100 10010 1000111000
10000
01111
01101 01011 0011101110
01100 01010 01001 00110 00101 00011
00100 00010 0000101000
00000
W!1, L!0
Equivalent questionWhat is the minimum number of radius 1 Hamming balls needed to cover the hypercube Q5?
1111110111
1100001111
001000001000001
C=
Sparse history of covering code density
An adaptive football pool problem
Round 1 Round 2 Round 3 Round 4 Round 5
Bet 1 W
Bet 2 W
Bet 3 W
Bet 4 L
Bet 5 L
Bet 6 L
Actual
Payoff: a bet with · 1 bad predictionQuestion. Min # bets to guarantee a payoff?
Ans.=6
Rou
nd 1
Bets $ adaptive Hamming ballsA “radius 1 bet” with predictions on 5 rounds can pay off in 6 ways:
Root 1 1 0 1 0 All predictions correct
Child 1 0 * * * * 1st prediction incorrect
Child 2 1 0 * * * 2nd prediction incorrect
Child 3 1 1 1 * * 3rd prediction incorrect
Child 4 1 1 0 0 * 4th prediction incorrect
Child 5 1 1 0 1 1 5th prediction incorrect
Rou
nd 2
Rou
nd 3
Rou
nd 4
Rou
nd 5
A fixed choice in {0,1} for each “*” yields an adaptive Hamming ball of radius 1.
Strategy tree for adaptive betting
W/1 L/0
W/1 L/0 W/1 L/0
Paths to leaves containing 1:11111 Root (0 incorrect predictions)00101 Child 1 (1 incorrect prediction)10101 Child 2 11001 Child 3 11101 Child 4 11110 Child 5 (1 incorrect prediction)
11011 10111
11100 11010 10110 10011
10100 10010 1000111000
10000
01111
01101 01011 0011101110
01100 01010 01001 00110
11111
1110111110
11001 10101
00101 00011
00100 00010 0000101000
00000
Adaptive covering code reformulation
Definition. An adaptive (q,k)-code is a set of adaptive Hamming balls of radius k which cover the hypercube Qq.
Theorem (E-P-Y). There exists a winning betting strategy for the q-round game with · k payoff-threshold
iff there exists an adaptive (q,k)-code.
Definition. Fk*(q) = minimum size adaptive (q,k)-code
= minimum #bets for a winning betting strategy in q-rounds with · k payoff-threshold
The (x,q,k)*-game reformulation
Players: Paul and Carole
Parameters: q (#rounds), k, (x0,x1,…,xk), a nonneg. int. vec.
Initial state: x=(x0,x1,…,xk)
Game play:
At an intermediate state x=(x0,x1,…,xk), a round consists of:
a vector a=(a0,a1,…,ak), where 0 · ai · xi,
chosen by Paul,
and next state W(x,a)=(a0, a1+x0-a0, …, ak+xk-1-ak-1) or
L(x,a)=(x0-a0, x1-a1+a0, …, xk-ak+ak-1)
chosen by Carole.
Determination of winner:
After q rounds, Paul wins if the state vector is nonzero.
Otherwise, Carole wins.
The Berlekamp weight function
Restated Theorem (E-P-Y). Paul can win the ((x0,x1,…,xk),q,k)*-game iff there is a covering of Qq with xi adaptive Hamming balls of radius (k-i).
Corollary. Fk*(q) = min size of an adaptive (q,k)-code
= min n such that Paul can always win
the ((n,0,…,0),q,k)-game.
Definition (Berlekamp weight function).
Intuition: when q rounds remain, the size of an adaptive Hamming ball of radius k is .
Conservation of weight lemma
Lower bound by probabilistic strategy
Upper bound: A counterexample
10 6 9 7 7 9
3-weight of possible next states
W L
Upper bound: Perfect balancing
16 (4-weight)
8 (3-weight)
4
2
1
Upper bound: A balancing theorem
Upper bound: Main theorem
Upper bound: Stage I, x! y’
Upper bound: Stages I (con’t) & II
Upper bound: Stage III and conclusion
Exact result for k=1
Exact result for k=2
Linear relaxation and a random walkIf Paul is allowed to choose entries of a to be real rather than integer, then a=x/2 makes the weight imbalance 0.
Example: ((n,0,0,0),q,3)*-game and random walk on the integers:
Future directions•Efficient Algorithmic implementations of encoding/decoding using adaptive covering codes
•Generalizations of the game to k a function of n
•Generalization to an arbitrary communication channel(Carole has t possible responses, and certain responses eliminate Paul’s vector entirely)
•Pullback of a directed random walk on the integers with weighted transition probabilities
•Generalization of the game to a general weighted, directed graph
•Comparison of game to similar processes such as chip-firing and the Propp machine via discrepancy analysis
[email protected] http://www.math.tamu.edu/~rellis/
[email protected] http://www.trinity.edu/~vadim/
[email protected] http://www.math.tamu.edu/~cyan/