An identity for dual versions of a chip-moving game Robert B. Ellis April 8 th , 2011 ISMAA 2011, North Central College Joint work with Ruoran Wang

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

2

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

3

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

4

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.

5

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)

6

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?

7

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

11

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

13

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

15

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

16

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

17

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

23

9-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

24

11

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

41

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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