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SKIP GRAPHS Slides adapted from the original slides by James Aspnes Gauri Shah

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2 Skip List [Pugh ’90] Data structure based on a linked list. AGJMRW HEAD TAIL Each node linked at higher level with probability 1/2. Level 0 AJM Level 1 J Level 2

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3 Skip List [Pugh ’90] Probabilistic alternative to balanced trees. L0 = linked list of ALL nodes ordered by key. Each element of Li appears in L.i+1 with prob p. Higher levels denote express lanes. We will only consider p=1/2

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4 Searching in a skip list AGJMRW HEADTAIL AJ J Search for key ‘R’ M Time for search: O(log n) on average. On average, constant number of pointers per node. Level 0 Level 1 Level 2 -- ++ success failure

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5 Constant number of pointers? Total number of pointers = 2.n + 2. n/2 + 2. n/4 + 2. n/8 + … = 4.n So, average number of pointers per node = 4

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6 Skip lists for P2P? Heavily loaded top-level nodes. Easily susceptible to random failures. Lacks redundancy. Disadvantages Advantages O(log n) expected search time. Retains locality. Dynamic node additions/deletions.

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7 Look back at DHT v2 HASH v1 v4 v3 v1 v2 v3 v4 Keys Nodes Actual Route Physical Link Virtual Link Virtual Route PHYSICAL NETWORK VIRTUAL OVERLAY NETWORK

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8 AdvantagesDisadvantages Load balancing. Decentralization. O(log n) space and search time. O(log 2 n) insert and delete time [search for (log n) neighbors]. Tolerance of random faults. SKIP GRAPHS No locality properties. No tolerance to adversarial faults. No self-stabilization (?). No optimization wrt. geography.

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9 A Skip Graph A 001 J M 011 G 100 W 101 R 110 Level 1 G R W AJM 000001011 101 110 100 Level 2 A G JMRW 001 011100110101 Level 0 Membership vectors Link at level i to nodes with matching prefix of length i. Think of a tree of skip lists that share lower layers.

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10 Properties of skip graphs 1.Efficient Searching. 2.Eficient node insertions & deletions. 3.Independence from system size. 4.Locality and range queries.

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11 Searching: avg. O (log n) Same performance as DHTs. AJM GWR Level 1 G R W AJM Level 2 AGJMRW Level 0 Restricting to the lists containing the starting element of the search, we get a skip list.

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12 Node Insertion – 1 A 001 M 011 G 100 W 101 R 110 Level 1 G R W A M 000011 101 110 100 Level 2 A GM R W 001011100110101 Level 0 J 001 Starting at buddy node, find nearest key at level 0. Basically a range query looking for key closest to new key. Takes O(log n) on average. buddy new node

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13 Node Insertion - 2 At each level i, find nearest node with matching prefix of membership vector of length i+1. A 001 M 011 G 100 W 101 R 110 Level 1 G R W A M 000011 101 110 100 Level 2 A GM R W 001011100110101 Level 0 J 001 J J Total time for insertion: O(log n) DHTs take: O(log 2 n)

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14 Independent of system size No need to know size of keyspace or number of nodes. E Z 10 E Z J insert Level 0 Level 1 E Z 10 E Z J 0 J 0001 E ZJ Level 0 Level 1 Level 2 Old nodes extend membership vector as required with arrivals. DHTs require knowledge of keyspace size initially.

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15 Locality and range queries Find key F. Find largest key < x. Find least key > x. Find all keys in interval [D..O]. Initial node insertion at level 0. D F A I D F A I L O S

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16 Applications of locality news:10/29 e.g. find latest news from yesterday. find largest key < news:10/29. news:10/27news:10/28news:10/26news:10/25 Level 0 DHTs cannot do this easily as hashing destroys locality. e.g. find any copy of some Britney Spears song. britney05britney03britney04britney02britney01 Level 0 Data Replication Version Control

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17 So far... Decentralization. Locality properties. O(log n) space per node. O(log n) search, insert, and delete time. Independent of system size. Coming up... Load balancing. Tolerance to faults. Self-stabilization. Random faults. Adversarial faults.

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18 Load balancing Interested in average load on a node u. i.e. the number of searches from source s to destination t that use node u. Theorem: Let dist (u, t) = d. Then the probability that a search from s to t passes through u is < 2/(d+1). where V = {nodes v: u <= v <= t} and |V| = d+1.

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19 Nodes u Skip list restriction Level 0 Level 1 Level 2 Node u is on the search path from s to t only if it is in the skip list formed from the lists of s at each level. s

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20 Tallest nodes Node u is on the search path from s to t only if it is in T = the set of k tallest nodes in [u..t]. u u t s u is not on path. t u u s u u is on path. Pr [u T] = Pr[|T|=k] k/(d+1) = E[|T|]/(d+1). k=1 d+1 Heights independent of position, so distances are symmetric.

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