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Routing Complexity of Faulty Networks Omer Angel Itai Benjamini Eran Ofek Udi Wieder The Weizmann Institute of Science

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Routing in a Faulty Network Node u knows the topology of the graph. Can choose a path to node v. Each link survives independently with probability p. u has partial knowledge on the topology of the graph. How many links (edges) should u probe before a path to v is found (if a path exists). u v G:G: Gp:Gp:

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Routing in a Faulty Network Local Router – an algorithm which: Starts at node u. Probes edges which it has reached. Outputs a path to v. Local Routing Complexity of A (with respect to u, v ): The random variable counting the number of probed edges until a path is found (given that a path exists). Interesting when is bounded away from 0. Efficiency: a local algorithm is efficient if its complexity is polynomial in the diameter of the largest component of G p. u v P r [ u » v ]

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Routing in a Faulty Network The existence of short paths does not guarantee the ability of finding them. A cycle with a random matching has diameter O ( log n ) [BC88]. Finding a path requires time [Kleinberg00]. On the other hand: The Small World Phenomenon… Our perspective: fault tolerance of networks. Study the effect of random failures on routing. Related to percolation theory – studies the effect of random failures on connectivity. ( n 1 = 2 ) O ( l ogn )

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Outline The Hypercube Lower bound: if local routing is not efficient. Tight upper bound: if. For short paths exist but are hard to find. The Mesh Tight upper and lower bounds. Whenever short paths exist (as a function of p ), they can be found. The importance of the locality assumption Local and non local routers may have exponential gap. Another example: tight analysis of G n, p. p << 1 p n p >> 1 p n 1 n < p < 1 p n

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The Faulty Hypercube – Some History – The n -dimensional hypercube in which each edge fails independently with probability 1 - p. If then w.h.p. is connected [Burtin77]. Disconnected w.h.p. when. If then w.h.p. H n can emulate H n with constant slowdown [HLN85] (considered node failures). Implicit: local routing in is possible. If then w.h.p contains a giant component [AKS82]. Sharpened by [BKL92],[BSH+04]. Diameter of giant component is. Short paths exist. When all components are of size O ( n ) w.h.p. H p n 1 ¡ p H p n p < 1 2 p > 1 2 p > 1 2 H p n H p n p > ( 1 + ² ) n ¡ 1 H p n po l y ( n ) p < ( 1 ¡ ² ) n ¡ 1

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The Faulty Hypercube Question: What probabilities in the range allow local routing (inside the g.c.) with complexity polynomial in n ? Graph is connected. Emulation (and routing) possible No giant component Threshold for constant distortion embedding of H n in [AB03] 1 n < p < n 1 p n p = 1 p = H p n

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Local Routing Phase Transition Let 0 < < ½. Lower bound (for ): Any local routing algorithm makes at least queries w.h.p.. Short paths exist but cannot be efficiently found! Upper bound (for ): There exists a local routing algorithm that finds a path between u, v in poly ( n ) time with high probability. 2 ( n ¯ ) p = n ¡ 1 = 2 ¡ ¯ p = n ¡ 1 = 2 + ¯

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The Faulty Hypercube Graph is connected. Emulation (and routing) possible No giant compone nt Threshold for constant distortion embedding of H n in [AB03] Local routing in poly ( n ) queries No efficient local routing H p n p = 0 p = 1 1 n 1 p n 1 2

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Lemma: Assume V. Denote: -– v is connected to u inside S. Q – the number of queries of a local router from u to v. For each e crossing the cut. The Lower Bound Lemma V = S [ ¹ S, v 2 S S ¹ S vu e f ( u S » v ) g ( S ; ¹ S ), P r [( v S » e )] < ³ 8 t P r [ Q < t ] · t ³ + P r [( u S » v )]

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The Lower Bound Lemma – Simple Example Lemma: Assume V. Denote: -– v is connected to u inside S. Q – the number of queries of a local router from u to v. For each e crossing the cut. Double Tree (0< p <1): S = the bottom tree,. Lemma implies: for, V = S [ ¹ S, v 2 S v u S ³ = p n f ( u S » v ) g ( S ; ¹ S ), P r [( v S » e )] < ³ 8 t P r [ Q < t ] · t ³ + P r [( u S » v )] t = ² 1 p n P r [ Q < t ] · t p n = ²

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The Double Tree – u, v are connected Double Binary Tree – 2 depth n trees joined at their leaves. A path u ~ v exists iff there is a leaf w and mirroring paths. The event { u ~v}is tantamount to a branching process with p 2. Path exists with constant probability, when p is a constant >. u v u » v u » w ; v » w f u » v g 1 p 2

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Lower Bound Lemma proof – Relaxed Model If, the algorithm stops successfully (complexity = 0). When a cut edge is probed, its entire component in S is given to the algorithm for free. If this component contains v the algorithm stops successfully. S ¹ S v ( u S » v )

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Assume: For each probed edge e i entering S : Lower Bound Lemma - Proof u v C 0 C 1 C 2 C i u 2 S P r [( e i » v ) 2 S j C 0 ;:::; C i ¡ 1 ] · ³ · t ³ S ¹ S P r [ Q < t ] · P r [ Q < t j ( u S ¿ v )] + P r [( u S » v )]

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Hyper Cube - Lower Bound Fix: (almost surely ). For any two vertices u, v, any local routing algorithm (almost surely) makes at least queries to find a path between u, v. p = n ¡ 1 = 2 ¡ ¯ ( u » v ) n ¡ ( n ¯ = 2 )

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Fix: (almost surely ). Claim: #of paths s of length is at most. s Applying the Lemma to the Hypercube ( v » x ) 2 S ` + 2 k n k ` 2 k `! p = n ¡ 1 = 2 ¡ ¯ ( u » v ) v x y e ¹ S S ` = n ¯ = 2 8 e 2 S £ ¹ S : P r [( v S » e )] < ³ ) P r [ Q < t ] · t ³ + P r [( u S » v )] · 2 n ¡ ( n ¯ = 2 ) ³ = P r [( v S » x )] · 1 X k = 0 p l + 2 k n k l 2 k l!

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Lemma: #of paths s inside S of length is at most. s Proof: Let A k be the set of such paths of length. A 0 = l ! There exists a mapping between A k and A k -1 that maps at most A k -paths into one A k -1 -path. A path is a list of coordinate changes: n possible coordinates and possible indices in the path. Applying the Lemma to the Hypercube ` + 2 k n k ` 2 k `! j A k j · n ` 2 j A k ¡ 1 j = ) j A k j · n k ` 2 k `! ` + 2 k ` + 1 A 0 = `! n ¢ ` 2 ¡ ` ¢ ( v » x )

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Fix: (almost surely ). Claim: #of paths s of length is at most. s Applying the Lemma to the Hypercube ( v » x ) 2 S ` + 2 k n k ` 2 k `! p = n ¡ 1 = 2 ¡ ¯ ( u » v ) v x y e ¹ S S ` = n ¯ = 2 ³ = P r [( v » x ) 2 S ] · 1 X k = 0 p l + 2 k n k l 2 k l! 8 e 2 S £ ¹ S : P r [( v S » e )] < ³ · 2 n ¡ ( n ¯ = 2 ) = o ( 1 ) ) P r [ Q < t ] · t ³ + P r [( u S » v )]

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Claim: for any vertex of distance m from v : Proof sketch: #paths inside S of length m +2 k is at most. n k ` 2 k m ! v u S ` = n ¯ = 2 [ = n ¡ m ¯ = 2 ] u 2 S P r [( u S » v )] = o ( 1 ) ( v » u ) P r [( u S » v )] · 1 X k = 0 p m + 2 k n k l 2 k m ! = o ( 1 )

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Hyper Cube So far we have shown: if, then #queries made by any local algorithm is exponential. We will now show: a local algorithm which (almost surely) makes only poly( n ) queries for. p << 1 p n p >> 1 p n

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The Hypercube – Efficient Algorithm for We observe that the embedding of [AB03]: For any adjacent u, v in H n : with probability u, v are mapped to themselves and their distance in is at most. The Algorithm: Fix a shortest path in H n :. With high probability all nodes are mapped to themselves. Any two adjacent vertices in the above path are at distance from each other in. Exhaustively search balls around x i until x i + 1 is found. Requires at most probes. The algorithm does not know the embedding. p = n ¡ 1 = 2 + ¯ ` = ` ( ¯ ) u = x 0 ; x 1 ;:::; x k = v ` n ` 1 ¡ e ¡ ( p n ) H p n H p n

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The Mesh M d We will show: An efficient local algorithm for the mesh.

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The Infinite Mesh M d M - Each edge fails with probability. For each dimension d there exists such that: If then contains one infinite component with prob. If then with prob. all components of are finite. The value of is not always known:. and decreasing. For finite meshes: translates to high probability bounds on the existence of giant components. M d p 1 ¡ p 1 1 p c p d c M d p p < p d c p > p d c M d p p 2 c = 1 2 p d c = ( 1 + o ( 1 )) 1 2 d

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Routing in the Faulty Mesh Theorem: let u, v be two vertices at distance k in M d. Assuming, there exists a routing algorithm which finds a path using O ( k ) probes in expectation. The Algorithm – similar to the hypercube algorithm: Fix a shortest path –. Once in x i – exhaustively search inside increasing balls around x i until x j ( j>i ) is found. Assuming the algorithm will output a path. u » v u » vu = x 0 ; x 1 ;:::; x k = v

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Proof Outline x 1 x i + 1 ¿ x k = v P r [ ¿ > a ] < e ¡ ca x i 1 X a = 1 ( 2 a ) d e ¡ ca = O ( 1 ) u = x 0 Claim: Let x i be a vertex in the shortest path. Its potential contribution is expected to be O(1). Show:

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Let x i be a vertex in the shortest path and in the giant component: [AP96] The next vertex in g.c. is not likely to be far: Let d, D be the metrics before and after the percolation. [AP96]: There is such that for any : Proof – Bounding : x i x i + 1 a > ½ ¢ d ( x ; y ) u = x 0 x k = v l x i + l + 1 P r [ l > a ] < e ¡ ( a ) P r [( D ( x ; y ) > a ) ^ ( x » y )] < e ¡ ( a )

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Local Routing vs. Oracle Rounting Oracle Routing: The algorithm may probe any edge of the graph (even edges it did not reach). Oracle Routing adds power: there are graphs in which there is a noticeable gap between Oracle and Local Routing complexities. Double binary tree – exponential gap. - polynomial gap. G n ; c = n

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Double Binary Tree Find a mirror path by querying simultaneously from both sides (using DFS). Equivalent to finding a path from a root to a leaf in a super-critical tree. Bad branches are expected to have constant size. u v p 2 > 1 2

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G n, c / n Lower Bound for Local Routing Lower bound: Any local algorithm almost surely needs ( n 2 / c 2 ) queries. Proof Sketch: After k queries the algorithm reveals roughly kp vertices. Any new revealed vertex has probability p to be connected to v. total probability of connection to v after k queries is kp 2 (=o(1) for k = o( n 2 / c 2 ) ).

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G n, c / n Oracle Routing using O( n 3/2 ) queries Grow a ( n 1/2 ) size component around each of u, v. Roughly n 3/2 / c queries are needed. Almost surely there is an edge between C u, C v (and only O( n ) queries are needed to find it). Remark: the above algorithm is optimal up to constant factors. C u C v u v

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Summary connectivity Efficient oracle routing Efficient local routing Gap: double binary tree p 2 > ½. Gap: in G n, p for p = c / n. Oracle router needs O( n 3/2 ) queries but diameter is poly(log n ). Gap: Hyper-cube 1/ n < p < n -1/2

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