An identity for dual versions of a chip-moving game Robert B. Ellis April 8 th , 2011 ISMAA 2011,...
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An identity for dual versions of a chip-moving game
Robert B. Ellis
April 8th, 2011ISMAA 2011, North Central College
Joint work with Ruoran Wang
Motivation I: Binary Search
S
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a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
Is x>a5? Yes.
a6 a7 a8 a9 a10
Is x>a7? No.
a6 a7
Is x>a6? …
Motivation I: Binary Search
Search question: which half of surviving list might x be in?
f(M)=d lg M e rounds to search length M list
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a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
Is x>a5? Yes.
a6 a7 a8 a9 a10
Is x>a7? No.
a6 a7
Is x>a6? …
Motivation I: Binary Search on Z>=0
Redisplay binary search as on Z with e=0. Go a couple of rounds Straight reformulation, no difference
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Motivation I: Binary Search with Errors
Let e>=0 and assume up to e responses are erroneous We can’t be sure to have found x unless other candidates
have e+1 “no” votes.
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Motivation II: Random Walk on Z>=0
M chips at origin. Each round, at each position, half of the chips stay in place and half move to the right.
A (good) search algorithm is a discretization of this random walk.
Our search algorithm from now on: number chips left-to-right 1,…,M; split chips into odds and evens
Define P*(n,e), K*(n,e)
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Game tree and tabular data
A (5,1) game tree, M=4 chips for P* tree, 3 chips for K* tree. Plus implication for P* and K*. Maybe tables?
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Outline of Talk
Coding theory overview– Packing (error-correcting) & covering codes– Coding as a 2-player game– Liar game and pathological liar game
Diffusion processes on Z– Simple random walk (linear machine)– Liar machine– Pathological liar game, alternating question strategy
Improved pathological liar game bound– Reduction to liar machine– Discrepancy analysis of liar machine versus linear machine
Concluding remarks
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Coding Theory Overview
Codewords:fixed-length strings from a finite alphabet
Primary uses: Error-correction for transmission in the presence of noiseCompression of data with or without loss
Viewpoints:Packings and coverings of Hamming balls in the hypercube2-player perfect information games
Applications:Cell phones, compact disks, deep-space communication
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Coding Theory Overview
Codewords:fixed-length strings from a finite alphabet
Primary uses: Error-correction for transmission in the presence of noiseCompression of data with or without loss
Viewpoints:Packings and coverings of Hamming balls in the hypercube2-player perfect information games
Applications:Cell phones, compact disks, deep-space communication
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Transmit blocks of length n
Noise changes≤ e bits per block(||||1 ≤ e)
Repetition code 111, 000– length: n = 3 – e = 1– information rate: 1/3
Coding Theory: (n,e)-Codes
x1…xn
(x1+1)…(xn+ n)
110 010 000
000
101
000 111111
Received:
Decoded:
blockwise majority vote
Richard Hamming
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0010011
3 errors: incorrect decoding
Coding Theory – A Hamming (7,1)-Code
1 0 0 0 1 1 1 0 1 1 0 1 1 0
0 1 0 0 0 1 1 0 1 0 1 1 0 1
0 0 1 0 1 0 1 0 0 1 1 0 1 1
0 0 0 1 1 1 0 1 1 1 0 0 0 1
0 0 0 0 0 0 0 1 1 0 1 0 1 0
1 1 0 0 1 0 0 1 0 1 1 1 0 0
1 0 1 0 0 1 0 0 1 1 1 0 0 0
1 0 0 1 0 0 1 1 1 1 1 1 1 1
Length n=7, corrects e=1 error
1001011
received
decoded
1001001
1 error: correct decoding
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A Repetition Code as a Packing
(3,1)-code: 111, 000
Pairwise distance = 3 1 error can be corrected
The M codewords of an(n,e)-code correspond toa packing of Hamming ballsof radius e in the n-cube
110 011101
111
000
010 001100
000
010 001100
110 011101
111
A packing of 2 radius-1 Hamming balls
in the 3-cube
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A (5,1)-Packing Code as a 2-Player Game
(5,1)-code: 11111, 10100, 01010, 00001
0What is the 5th bit?1What is the 4th bit?0What is the 3rd bit?0What is the 2nd bit?0What is the 1st bit?
CarolePaul 11111
00001
1010001010
0 1 >1# errors
11111 0000110100 01010
01111 00100 00010 0001100100
01010
000100001000010
00001000010000111111 10100 01010 00001
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Covering Codes
Covering is the companion problem to packing
Packing: (n,e)-code
Covering: (n,R)-code
lengthpacking radius
covering radius
110 011101
111
000
010 001100
000
010 001100
110 011101
111
(3,1)-packing code and(3,1)-covering code
“perfect code”11111
00001
1010001010
11111
11000
0111110111 00001
0010000010
(5,1)-packing code (5,1)-covering code
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Optimal Length 5 Packing & Covering Codes
0100101100
01110 01101
00100
11100
01000
11110 11101 01111
00000
0101011000 10100 00110 00101
10110 10011
1000110010
11011
00011
10111
000010001010000
11111
10101 00111010111100111010
01110 01101
0100101100
00100
11100
01000
11110 11101 01111
00000
0101011000 10100 00110 00101
10110 10011
1000110010
11011
00011
10111
000010001010000
11111
10101 00111010111100111010
(5,1)-packing code
(5,1)-covering code
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Sphere bound:
A (5,1)-Covering Code as a Football Pool
WLLLLBet 7
LWLLLBet 6
LLWLLBet 5
LLLWWBet 4
WWWLWBet 3
WWWWLBet 2
WWWWWBet 1
Round 5Round 4Round 3Round 2Round 1
Payoff: a bet with ≤ 1 bad predictionQuestion. Min # bets to guarantee a payoff? Ans.=7
00100
01111
11000
10111
00001
00010
11111
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Codes with Feedback (Adaptive Codes)
FeedbackNoiseless, delay-less report of actual received bits
Improves the number of decodable messagesE.g., from 20 to 28 messages for an (8,1)-code
sender receiver
Noise
Noiseless FeedbackElwyn Berlekamp
1, 0, 1, 1, 0 1, 1, 1, 1, 0
1, 1, 1, 1, 0
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A (5,1)-Adaptive Packing Code as a 2-Player Liar Game
A
D
BC
0 1 >1# liesYIs the message C?
NIs the message D?
NIs the message B?
NIs the message A or C?
YIs the message C or D?
CarolePaul
00101
Message
Originalencoding
Adaptedencoding
A B C D
01110 0101011000 10011
1**** 1****11*** 10*** 10*** 1000*101** 100**1000* 1000010001
Y $ 1, N $ 0
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A (5,1)-Adaptive Covering Code as a Football Pool
LWLLWCarole
LBet 6
LBet 5
LBet 4
WBet 3 W
L
L
WWBet 2
L
W
W
W
W
W
L
L
WWBet 1
Round 5Round 4Round 3Round 2Round 1
Payoff: a bet with ≤ 1 bad predictionQuestion. Min # bets to guarantee a payoff?
Ans.=6
Bet 3
Bet 6
Bet 4Bet 5
0 1 >1# bad
predictions(# lies)
Bet 2Bet 1
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Optimal (5,1)-Codes21
Code type Optimal size
(5,1)-code 4
(5,1)-adaptive code 4
Sphere bound 5 1/3 (= 25/(5+1) )
(5,1)-adaptive covering code 6
(5,1)-covering code 7
Adaptive Codes: Results and Questions22
Sizes of optimal adaptive packing codes
• Binary, fixed e ≥ sphere bound - ce (Spencer `92)
• Binary, e=1,2,3 =sphere bound - O(1), exact solutions (Pelc; Guzicki; Deppe)
• Q-ary, e=1 =sphere bound - c(q,e), exact solution (Aigner `96)• Q-ary, e linear unknown if rate meets Hamming bound for all e. (Ahlswede,
C. Deppe, and V. Lebedev)
Sizes of optimal adaptive covering codes
• Binary, fixed e · sphere bound + Ce Binary, e=1,2 =sphere bound + O(1), exact solution (Ellis, Ponomarenko, Yan `05)
Near-perfect adaptive codes• Q-ary, symmetric or “balanced”, e=1 exact solution (Ellis `04+)• General channel, fixed e asymptotic first term (Ellis, Nyman `09)
Outline of Talk
Coding theory overview– Packing (error-correcting) & covering codes– Coding as a 2-player game– Liar game and pathological liar game
Diffusion processes on Z– Simple random walk (linear machine)– Liar machine– Pathological liar game, alternating question strategy
Improved pathological liar game bound– Reduction to liar machine– Discrepancy analysis of liar machine versus linear machine
Concluding remarks
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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
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Linear Machine on Z
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Linear Machine on Z
5.5 5.5
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Linear Machine on Z
2.75 5.5 2.75
Time-evolution: 11 £ binomial distribution of {-1,+1} coin flips
Liar Machine on Z
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
11 chips
t=0
• Approximates linear machine• Preserves indivisibility of chips
Liar Machine on Z
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=1
Liar Machine on Z
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=2
Liar Machine on Z
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=3
Liar Machine on Z
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=4
Liar Machine on Z
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=5
Liar Machine on Z
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=6
Liar Machine on Z
Liar machine time-stepNumber chips left-to-right 1,2,3,…Move odd chips right, even chips left(Reassign numbers every time-step)
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Height of linear machine at t=7l1-distance: 5.80l∞-distance: 0.98
t=7
Discrepancy for Two Discretizations
Liar machine: round-offs spatially balanced
Rotor-router model/Propp machine: round-offs temporally balanced
The liar machine has poorer discrepancy… but provides bounds to the pathological liar game.
Proof of Liar Machine Pointwise Discrepancy
The Liar Game as a Diffusion Process
A priori: M=#chips, n=#rounds, e=max #liesInitial configuration: f0 = M ¢ 0
Each round, obtain ft+1 from ft by: (1) Paul 2-colors the chips(2) Carole moves one color class left, the other right
Final configuration: fn
Winning conditionsOriginal variant (Berlekamp, Rényi, Ulam)
Pathological variant (Ellis, Yan)
Pathological Liar Game Bounds
Fix n, e. Define M*(n,e) = minimum M such that Paul can win the pathological liar game with parameters M,n,e.
Sphere Bound
(E,P,Y `05) For fixed e, M*(n,e) · sphere bound + Ce
(Delsarte,Piret `86) For e/n 2 (0,1/2), M*(n,e) · sphere bound ¢ n ln 2 .
(C,E `09+) For e/n 2 (0,1/2), using the liar machine,M*(n,e) = sphere bound ¢ .
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Liar Machine vs. (6,1)-Pathological Liar Game39
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
9 chips
9 chips
t=0
disqualified
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
40
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=1
disqualified
Liar Machine vs. (6,1)-Pathological Liar Game
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
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9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=2
disqualified
Liar Machine vs. (6,1)-Pathological Liar Game
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Liar Machine vs. (6,1)-Pathological Liar Game42
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=3
disqualified
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Liar Machine vs. (6,1)-Pathological Liar Game43
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=4
disqualified
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Liar Machine vs. (6,1)-Pathological Liar Game44
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=5
disqualified
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Liar Machine vs. (6,1)-Pathological Liar Game45
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
t=6
disqualified
No chips survive: Paul loses
Comparison of Processes46
Process Optimal #chips
Linear machine 9 1/7
(6,1)-Pathological liar game 10
(6,1)-Liar machine 12
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
(6,1)-Liar machine started with 12 chips after 6 rounds
disqualified
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Loss from Liar Machine Reduction47
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8t=3
disqualified
9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8disqualified
Paul’s optimal 2-coloring:
Reduction to Liar Machine
Saving One Chip in the Liar Machine49
Summary: Pathological Liar Game Theorem
Further Exploration
Tighten the discrepancy analysis for the special case of initial chip configuration f0=M 0.
Generalize from binary questions to q-ary questions, q ¸ 2.
Improve analysis of the original liar game from Spencer and Winkler `92; solve the optimal rate of q-ary adaptive block codes for all fractional error rates.
Prove general pointwise and interval discrepancy theorems for various discretizations of random walks.
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Reading List
This paper: Linearly bounded liars, adaptive covering codes, and deterministic random walks, preprint (see homepage).
The liar machine– Joel Spencer and Peter Winkler. Three thresholds for a liar.
Combin. Probab. Comput.1(1):81-93, 1992. The pathological liar game
– Robert Ellis, Vadim Ponomarenko, and Catherine Yan. The Renyi-Ulam pathological liar game with a fixed number of lies. J. Combin. Theory Ser. A 112(2):328-336, 2005.
Discrepancy of deterministic random walks– Joshua Cooper and Joel Spencer, Simulating a Random Walk
with Constant Error, Combin. Probab. Comput. 15 (2006), no. 06, 815-822.
– Joshua Cooper, Benjamin Doerr, Joel Spencer, and Gabor Tardos. Deterministic random walks on the integers. European J. Combin., 28(8):2072-2090, 2007.
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