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Distance and Routing Labeling Schemes in Graphs Feodor F. Dragan Computer Science Department Kent State University

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Talk is based on Known results for trees –Kannan&Naor&Rudich’88, –Peleg’99 and ’00, –Thorup&Zwick’01, –Fraigniaud&Gavoille’01

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Traditional graph representation Store adjacency information in global data structure –Requires much storage –Node labels contain no information, serve only as “pointers” to global data structure

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Local graph representation IDEA: Store local pieces of information, i.e., –Associate label Label(v) with each node v, s.t. labels allow inferring adjacency information locally, without using additional memory and information Question: Can this be done compactly & efficiently?

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“Ancient” Example: “Ancient” Example: Hamming adjacency labeling Goal: Label each node v with m-bit label Label(v) s.t. Label(v), Label (u) allow deciding if u and v are adjacent: Question: Can this be done with short labels? [Breuer&Folkman’67]: For any n-node graph there exists a Hamming distance adjacency labeling with m=2n , T=4 -4, where = max node degree.

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Interest Revived: Interest Revived: Adjacency Labeling [Kannan&Naor&Rudich’88] What’s in label? Note: - Algorithm knows nothing beyond the labels (even, it does not know which graph they come from) Adjacency-labeling (for graph family ): Adjacency-labeling (for graph family F ): 1.Function Label labeling nodes of any graph in F with short distinct labels (compactness) 2.Efficient algorithm for deciding adjacency of two nodes given their labels (efficiency) (polylogarithmic labels polylogarithmic time)

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Adjacency Labeling on Trees [Kannan&Naor&Rudich’88] Labeling: 1. Arbitrarily root the tree and prelabel each node v with distinct integer I(v) from [1..n]. 2. Label root r by Label(r)=(I(r)). 3. Label each non-root node v with parent w by Label(v)=(I(v),I(w)). Labels contain bits

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Adjacency Labeling on Trees [Kannan&Naor&Rudich’88] Adjacency Decoder for u,v: Check if 1 st entry in Label(v) = 2 nd entry in Label(u) or vice versa. Constant time! Labeling: (with bits) 1. Arbitrarily root the tree and prelabel each node v with distinct integer I(v) from [1..n]. 2. Label root r by Label(r)=(I(r)). 3. Label each non-root node v with parent w by Label(v)=(I(v),I(w)).

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Extensions Question (compactness): Is this doable for all graphs with -bit labels? 1.Bounded arboricity graphs –Bounded degree –Bounded-genus graphs (e.a., planar graphs) 2.Intersection-based graphs (e.a., interval graphs) 3.c-decomposable graphs Labels contain bits

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Negative results Negative examples: Bipartite / chordal graphs Note: Graph class Note: Graph class F is adjacency-labelable using -bit labels each G from F is fully defined by bits # of represented graphs Therefore: Graph family F containing n-node graphs is not adjacency-labelable using -bit labels.

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Question: Can this be done for other types of information beyond adjacency? Distance In general graphs Distance Edge / node connectivity Routing Flow … In trees Ancestry Lowest common ancestor (LCA or NCA) Depth of NCA (separation level)

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Question: Can this be done for other types of information beyond adjacency? Distance In general graphs Distance Edge / node connectivity Routing Flow … In trees Ancestry Lowest common ancestor (LCA or NCA) Depth of NCA (separation level)

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Distance Labeling Scheme Distance Goal: Goal: Short labels that encode distances and distance decoder, an algorithm for inferring the distance between two nodes only from their labels (in time polynomial in the label length) …a method for storing and extracting distances in a “distributed” fashion… Labeling: v Label(v) Distance decoder: D (Label(v), Label(u)) dist(u,v)

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Simple examples Distance Enjoy -bit distance labeling schemes with constant-time decoder

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General graphs Distance Easy labeling: Label each node in n-node graph G on with all distances Hence, for the class of all n-node graphs Currently: Similar bounds apply for other large graph families (bipartite / chordal / … ) [Gavoille,&Peleg&Perennes&Raz ’ 01]

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Distance labeling on trees Distance labeling on trees [Peleg’99] Preprocessing: Arbitrarily label each node v of T with distinct integers I(v) from [1..n]

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Recursive partitioning and decomposition tree Tree separator: node v in n-node tree T, whose removal breaks T into subtrees of size at most n/2. Known fact: Every tree has a separator (can be found in linear time) T H

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Labeling T H Construct for H Depth of NCA labeling scheme ( -bit labels [Peleg’00] and constant (can be made) decision time) Then, v A(v) Hence, -bit labels

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Constant Time Distance Decoder T H Given Label(u) and Label(v) Function distance_decoder_trees(Label(u),Label(v)) Extract A(u), A(v) Use A(u), A(v) to find the depth k of NCA(u,v) in H 1 Output, by extracting appropriate entries from labels 2+2=4 (A(u),2,2,0)) (A(v),6,2,1,0))

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Lower bounds and Planar graphs [Gavoille,&Peleg&Perennes&Raz ’ 01]

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Routing Labeling Scheme Distance Goal: Goal: Short labels that encode the routing information and routing protocol, an algorithm for inferring port number of the first edge on a shortest path from source to destination, giving only labels and nothing else Labeling: v Label(v) Distance decoder: R (Label(v), Label(u)) port(v,u)

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General graphs Distance Easy labeling (full routing tables): Label each node in n-node graph G on with full routing information Hence, for the class of all n-node graphs one needs nlogn -bit labels Currently: Nothing better can be done for general graphs if we insist on routing via shortest paths

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Labeling trees with -bit labels T H Construct for H Depth of NCA labeling scheme ( -bit labels Then, v A(v) Hence, -bit labels

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Constant Time Routing Decision T H Given Label(v) and Label(u) Function routing_decision_trees(Label(u),Label(v)) Extract A(v), A(u) Use A(v), A(u) to find the depth k of NCA(v,u) in H 1 If k

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Best routing labeling scheme for trees and Planar graphs [Thorup&Zwick ’ 01] For family of n-node trees there is a routing labeling scheme with labels of size O(logn)-bits per node and constant time routing decision. [Fraigniaud&Gavoille ’ 01] For family of n-node trees there is a routing labeling scheme with labels of size -bits per node and constant time routing decision. [Gavoille&Hanusse ’ 99] For family of n-node planar graphs there is a routing labeling scheme with labels of size (8n +o(n))-bits per node.

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