t t t < n/3 n t = n/3 1 S yi

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2 t t t < n/3 n t = n/3 1 V = {p 1, p 2,, p n } n X D v Π V X D D X p i y i X Π y i y i S yi

3 y y, y?? p i, p j y j = y i D D

4 y y y?? t < n/3 t < c/2 c t c/2

5 G = (V, E) V E D D X Π D D D D (D, v), v V D D X R R D D G = (V, E) V E

7 t t t t t t

8 Z Z V V Z 2 V, Z Z, Z Z, Z Z

9 v v t t

10 t G D t t t t < n/3 t t t t t

11 t t t G D K(G, D) t < K(G, D)/2 t t < K(G, D) K(G, D)/2 1 t < K(G, D) X (G, D) K(G, D) X (G, D) t M(G, D, t) t t G D M(G, D, t) t + 1 t M(G, D, t)

12 T (G, D) = {t N M(G, D, t) t + 1} t (G, D) M(G, D, t) t G D t G, D t G = (V, E) N (v) v G G V (G) t A t v V, A N (v) t t G D A t t G, D t A G, D

13 v D v t D D X X D D X D v N (D) D D D v / N (D) D t(v) + 1 t(v) + 1 G D t (G, D) t t G D t t (G, D) X t < X t X 1 t t X (G, D) b v b D X (G, D) 2t + 1 G k 2t + 1 k 1

14 k D k k t + 1 k 1 k k k + 1 t + 1 X /2 1 t k k k k L k (G, D) G = (V, E) D V \ {D} = m i=1 L i, m N L 1 =N (D), L 2 ={v V \ L 1 : N (v) L 1 k} m 1 m 1 L m ={v V \ L j : N (v) L j k} j=1 k k G = (V, E) D V \ {D} = m i=1 L i, m N j=1 L 1 = N (D), v L i : N (v) i 1 j=1 L j k k k k k

15 k L V = m i=1 L i, m N u L h L L d 1 < d < h m N (u) d 1 j=1 L j k k k k L k L k k G D k N k G, D k G, D k k L k (G, D) k L V = m i=1 L i, m N 1 < d < h m u L h N (u) d 1 j=1 L j k L : V = L 1 L 2... {L d {u}} {L h \ {u}} L m = k k L k L k k k L : V = m i=1 L i v v L i { } d 1 i = d {1,..., m} N (v) L j k k j=1 m i=1 L i

16 k L k G D k L = {L 1,, L m }, L = {L 1,, L h} L 1 = L 1 i L i L i v, v L i v / L i v L i L i k K G D K(G, D) def. = {k N k L k (G, D)} G D t N t < K(G, D)/2 t 2t < K(G, D) (2t + 1) L 2t+1 (G, D) L 2t+1 (G, D) {L 1,..., L m } V V = m i=1 L i 1 i m v L i D i v L 1 = N (D) D u L i, 1 i h D v L h+1 h j=1 L j N (v) 2t + 1 t + 1 v D G D t K(G, D)/2 1 t < K(G, D)/2 t t t + 1 (G, D) (K(G, D) 1) (G, D)

17 D t + 1 players } 2t subsets v 1 v 2 v 2t K 2t } K(G, D) = t + 1 t

18 t D 2t 2 + 2t v 1,..., v 2t 2t N (D) t = K(G, D) 1 G v i {v 1,..., v 2t } M M = M A + M B M A : N (D) M B : B = {v 1,..., v 2t } \ {v i } T i = T N (D) N (v i ) D v i M A = N (D) N (v i ) \ T i = t + 1 T i v i B C B1 = {v B v t N (D) } C B2 = {v B v } C B = C B1 C B2 C B1 N (v j ) N (D), v j B C B 1 = T (N (D) \ N (v i)) = t T i T :t T :t C B2 t T i B N (v i ) N (D) v i t C B C B = C B1 C B2 C B1 + C B2 (t T i ) + (t T i ) = 2t 2 T i M B = 2t 1 C B = 2t 1 2t + 2 T i = 2 T i 1 M (1), (2), (3) M = M A + M B t + 1 T i + 2 T i 1 = = t + T i v i T i > 0 M t + 1 T i = 0 v i t + 1 N (D) (G, D)

19 t K(G, D)/2 1 K(G, D) 1 K(G, D) 1 G D t t G D t K(G, D) t t K(G, D) t s s (G, D) G L i i v L i i v t + 1 L 1,..., L i 1 v L i N (v) i 1 j=1 L j t + 1. (t + 1) G D (t + 1) G D t K(G, D) t K(G, D) G D t < X (G, D) G = (V, E) D σ = (v 1, v 2,...) V \ (N (D) D) δ(w i, v) v N (D) {v 1,..., v i 1 } σ i, j, 1 i < j V \ (N (D) D) δ(w i 1, v i ) δ(w i 1, v j )

20 X (G, D) = {δ(w i 1, v i ) i = 1, 2,...} X (G, D)/2 1 t < X (G, D). K(G, D) X (G, D) K(G, D) = X (G, D) σ = (v 1, v 2,...) {L 1 = N (D), L 2 = {v 1 }, L 3 = {v 2 },...} X (G, D) X (G, D) X (G, D) K(G, D) X (G, D) (1) t < X (G, D) K(G, D) (K(G, D) 1) K(G, D) X (G, D) X (G, D) < K(G, D) t < K(G, D) 1 K(G, D) X (G, D) K(G, D) m G D K(G, D) t X (G, D) t < X (G, D) (1)

21 m (G, D) (G, D, m) m m m L m (G, D) t K(G, D) K(G, D) < v V \(N (D) D) (v) = δ K(G, D) δ K(G, D)/2 1 t K(G, D) t t K(G, D) t /2 1 G K(G, D) = t + 1 t K(G, D) δ O( E ) O( E δ) X (G, D) O( V ( V + E )) <

22 t t T G = (V, E) G T = (V \ T, E ) G V \ T G D t t M(G, D, t) = K(G T : t T, D) G = (V, E) D t M(G, D, t) t + 1 ( ) M(G, D, t) t + 1 T V \ D t K(G T, D) t + 1 (t + 1) L t+1 (G T, D) = {L 1,..., L m } v t + 1 L t+1 (G T, D) v D ( ) t t T G T D m N L i = {v V \ T v i i {1,..., m} (L i ) m i=1 (t + 1) G T D L 1 = N (D) \ T L 2 = {v V \ T N (v) L 1 t + 1} t + 1 L k = {v V \ {T k 1 j=1 L j} : N (v) k 1 j=1 L j t + 1} L k+1 = {v V \ {T k L j } : N (v) j=1 k L j t + 1} k + 1 t + 1 T t j=1 T (G, D) = {t N M(G, D, t) t + 1}

23 T (G, D) t (G, D) = T (G, D) t M(G, D, t) t K(G T, D) t t = T (G, D) δ M(G, D, t) δ V \ (N (D) D) t G D t t A t A t G D t G, D t (G, D) M(G, D, t) = K(G T : t T, D) t t T G T (t + 1) (t + 1) V T = V \ (T {D}) L 1 = N G T (D), L i = {v V T \ i 1 j=1 L j : N G T (v) i 1 j=1 L j t + 1}, 2 i m

24 T H. D..... w A At most t neighbors in A B G A, B, T h N j h, L j = h i=1 L i V T h h N h 2 h = 1 G T D T h A = L i B = V T \ A (t + 1) i=1 w B, N G T (w) A t h i=1 L i V T B H = (N G T (w) A) H G T D B G A, B, T G G (u, v) E = {(u, v) u, v A T } H t G A T w B, N G T (w) H t t A t G D σ σ A w B σ G D D = 0 T

25 σ T σ G D D = 1 H σ H T, H G, G t H T D B G G σ, σ w B A t G D w 0 σ G σ G 1 A t t t G D G 1 G D t t t G D G t

26 G t < LP C(G, D) a t-locally resilient algorithm a t-locally resilient safe Ad-Hoc algorithm CPA is t-locally resilient (t T (G, D)) t < K(G, D)/2 t < X (G, D)/2 t < X (G, D)/2 t LP C(G, D) X (G, D) X (G, D) t t

27 n/3, n = V, D V T (G, D) T (G, D) D V M(G, D, t) n/3 ( ) t n/3 1, T (G, D) D V K(G, D) t t < n/3 n t = n/3 1

28 t < c/2 c t t t n > 3t t + 1

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