Locality In Distributed Graph Algorithms Nathan Linial

Motivation: The Distributed Model Each node of the undirected graph 𝐺=(𝑉,𝐸) is occupied by a processor. Computation is completely synchronous and reliable. At each time unit a processor may pass messages to each of its neighbors, and message size is unrestricted. Any computations carried out by individual processors take one time unit and are not restricted in any way. Symmetry breaking: G may have symmetries. Hence, it’s almost impossible to distributively-compute many functions by anonymous processors. The solution: The use of IDs: a mapping from the set of vertices V to the positive integers. At time zero, the processor occupying a node G knows the ID of that node.

Motivation: The Time Complexity Computing local data in a processor Transferring data between processors Takes time O(1) Takes much more time than computing local data

Motivation: Radius Of The Neighborhood Our goal: To minimize the computation time (time units) by utilizing the locally available data: t v t

Introduction Definition 1: K-coloring for a graph G is a function 𝜑 :𝑉 →{1,…,𝑘} such that: for every 𝑢,𝑣 ∈𝐸 : 𝜑 𝑢 ≠ 𝜑(𝑣)

Example 2-coloring for a labeled 2n-cycle 5 4 6 3 7 2 8 1

Example 2-coloring for a labeled 2n-cycle 5 4 6 3 7 2 8 1

Example 2-coloring for a labeled 2n-cycle 5 4 6 3 7 2 8 1

Example 2-coloring for a labeled 2n-cycle 5 4 6 3 7 2 8 1

Example 2-coloring for a labeled 2n-cycle 5 4 6 3 7 2 8 1

Example 2-coloring for a labeled 2n-cycle 5 4 6 Time-units = Ω(𝑛) 3 7 8 1

Introduction 3-coloring for a cycle: Our goal: to prove that 3-coloring for a n-cycle takes Ω log ∗ 𝑛 time units

Introduction Definition 2: MIS (Maximal Independent Set) MIS is an independent set S, such that for each 𝑣∈𝑉/𝑆 , the set 𝑆∪{𝑣} is not an independent set.

Example MIS? YES!

Example MIS? YES!

Example MIS? Remark: In a cycle, there can be a sequence of at most 2 vertices which are not included in MIS NO!

Introduction Remark: Given a n-cycle with a clockwise orientation Given an algorithm that finds a MIS Then, in one more time step the cycle may be 3-colored

Introduction Remark: Given a n-cycle with a clockwise orientation Given an algorithm that finds a MIS Then, in one more time step the cycle may be 3-colored

Definitions Definition 3: The graph 𝐵 𝑡,𝑛 : ( 𝑥 1 , …, 𝑥 2𝑡+1 ) Vertices are all vectors ( 𝑥 1 , 𝑥 2 , …, 𝑥 2𝑡+1 ) where the 𝑥 𝑖 are mutually distinct integers from {1, …, 𝑛} Edges are given by: where 𝑦≠2𝑡+1 ( 𝑥 1 , …, 𝑥 2𝑡+1 ) (𝑦, 𝑥 1 , …, 𝑥 2𝑡 ) (?, 𝑥 1 , …, 𝑥 2𝑡 ) ( 𝑥 2 , …, 𝑥 2𝑡+1 , ?) (?, 𝑥 1 , …, 𝑥 2𝑡 ) ( 𝑥 2 , …, 𝑥 2𝑡+1 , ?) ( 𝑥 1 , …, 𝑥 2𝑡+1 ) (?, 𝑥 1 , …, 𝑥 2𝑡 ) ( 𝑥 2 , …, 𝑥 2𝑡+1 , ?)

Example 6 5 7 4 8 3 𝐵 2, 9 will include: 9 2 1 (1, 2,3,4,5) (2,3,4,5,6) (3,4,5,6,7) (2,4,7,6,9) (4,7,6,9,8) Remark: since there exists a 3-coloring for n-cycle in t time units, there is 3-coloring for 𝐵 𝑡,𝑛 , too!

Definitions Definition 4: the digraph 𝐷 𝑠,𝑛 : ( 𝑎 1 , … 𝑎 𝑠 ) Vertices are all the vectors ( 𝑎 1 , …, 𝑎 𝑠 ) such that 1≤ 𝑎 1 < 𝑎 2 < … < 𝑎 𝑠 ≤𝑛 Edges are given by: where 𝑎 1 < 𝑎 𝑠 <𝑏≤𝑛 ( 𝑎 1 , … 𝑎 𝑠 ) ( 𝑎 2 , …, 𝑎 𝑠 , 𝑏) Remark: 𝐷 2𝑡+1, 𝑛 is a subgraph of 𝐵 𝑡,𝑛 . Therefore, a proper coloring of 𝐵 𝑡,𝑛 is also a proper coloring for 𝐷 2𝑡+1, 𝑛 , i.e. 𝜒 𝐵 𝑡,𝑛 ≥ 𝜒( 𝐷 2𝑡+1, 𝑛 )

Definitions Definition 5: the dilinegraph : Given a graph H = (V,E) The dilinegraph DL(H) is directed graph, whose set of vertices is E, and (u,w) is an edge if ℎ𝑒𝑎 𝑑 𝐻 𝑢 =𝑡𝑎𝑖 𝑙 𝐻 (𝑤)

Example H: DL(H):

Example (1,2,3) (2,3,4) (3,4,5) H: (1,2,3,4) (2,3,4,5) DL(H):

Proposition 2.1 Proposition 2.1 : 𝐷 1, 𝑛 is the following graph: for each vertex with label i, the out-neighbors are all vertices with a label bigger than i. 𝐷 𝑠+1 , 𝑛 =𝐷𝐿( 𝐷 𝑠 , 𝑛 )

Proposition 2.2 Proposition 2.2 : For a directed graph G, 𝜒 𝐷𝐿 𝐺 ≥ log 𝜒(𝐺)

Proposition 2.2 : Proof Let 𝜓 be a proper k-coloring for DL(G) , we will show that there is a proper 2 𝑘 −𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 for G. Since 𝜓 is a k-coloring for DL(G) , it is also a “k-coloring” for E(G), in the following way: It is a mapping 𝜓 :𝐸 𝐺 →{1, …, 𝑘} such that if 𝑢,𝑤 ∈𝐸(𝐺) and ℎ𝑒𝑎𝑑 𝑢 =𝑡𝑎𝑖𝑙(𝑤) , then 𝜓 𝑢 ≠𝜓 𝑤 Define for each vertex 𝑥∈𝑉 : 𝑐 𝑥 = 𝜓 𝑢 𝑥=𝑡𝑎𝑖𝑙 𝑢 } Remark: Since 𝜓 is proper for DL(G), 𝑐 is proper for G. We’ve found a proper 2 𝑘 −𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 for G. 𝜒 𝐺 ≤ 2 𝜒 𝐷𝐿 𝐺 𝜒 𝐷𝐿 𝐺 ≥ log 𝜒(𝐺)

The Main Theorem The main theorem: Proof: A 3-coloring for a labeled n-cycle requires Ω log ∗ 𝑛 time units. Proof: Suppose it takes t units of time to 3-color a n-cycle , we should prove 𝑡= Ω log ∗ 𝑛 . 3≥𝜒 𝐵 𝑡,𝑛 ≥ 𝜒 𝐷 2𝑡+1, 𝑛 ≥ log 𝜒 𝐷 2𝑡, 𝑛 ≥ log log 𝜒 𝐷 2𝑡−1, 𝑛 ≥ …≥ log 2𝑡 𝜒 𝐷 1, 𝑛 = log 2𝑡 𝑛 . log 2𝑡 𝑛 ≤3 2𝑡+1 ≥ log ∗ 𝑛 𝑡=Ω log ∗ 𝑛

Summary For First Part We proved that 3-coloring for a labeled n-cycle takes Ω log ∗ 𝑛 time units . Cole and Vishkin proved that 3-coloring for a labeled n-cycle requires O log ∗ 𝑛 time units. Conclusion: 3-coloring for a labeled n-cycle takes θ log ∗ 𝑛 time units .

The t-Neighborhood Graph Given a graph 𝐺=(𝑉,𝐸) of order 𝑛, and 𝑡≥1, the 𝑡 −𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟ℎ𝑜𝑜𝑑 𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 𝐺, 𝑁 𝑡 (𝐺) is constructed as follows: For every 𝑥∈𝑉 let 𝑆 𝑡 (𝑥) be a subgraph of 𝐺 spanned by those vertices 𝑦 whise distance from 𝑥 is at most 𝑡. For every 𝑥 consider all the 𝑛-labelings of 𝑆 𝑡 𝑥 . Every such labeling Ψ is a node in 𝑁 𝑡 (𝐺).

Example 𝑮: For 𝑛=9 and the following 𝐺: 𝑁 1 (𝐺) will include: 1 2 3 7 8 1 2 3 6 6 3 6 4 5 2 1 9 1 9 4 5

The t-Neighborhood Graph Cont. Let Ψ 1 : 𝑉 𝑡 𝑥 →{1, …, 𝑛} and Ψ 2 : 𝑉 𝑡 𝑦 →{1, …, 𝑛} be two of these vertices of 𝑁 𝑡 (𝐺). They are taken to be neighbors in 𝑁 𝑡 (𝐺) if [𝑥,𝑦]∈𝐸(𝐺) and there is a labeling Φ :𝑉 𝐺 → 1, …, 𝑛 such that: Φ | 𝑆 𝑡 𝑥 = Ψ 1 and Φ | 𝑆 𝑡 𝑦 = Ψ 2

Example 𝑮: For 𝑛=9 and the following 𝐺: 𝑁 1 (𝐺) will include: 1 2 3 7 8 1 2 3 6 6 3 6 4 5 2 1 9 1 9 4 5

Proposition 2.3 Proposition 2.3 : Neighborhood graphs have the following properties: 𝜒 𝑁 𝑡 𝐺 is the least number of colors with which G may be colored distributively on time t. 𝜒 𝑁 𝑡 𝐺 =𝜒 𝐺 for 𝑡≥𝑑𝑖𝑎𝑚(𝐺). 𝜒 𝑁 𝑡 𝐺 is non-increasing with t. For 𝐺= 𝐶 𝑛 , the graph 𝐵 𝑡,𝑛 is obtained from 𝑁 𝑡 ( 𝐶 𝑛 ) by identifying vertices in 𝑁 𝑡 ( 𝐶 𝑛 ) with identical sets of neighbors. In particular, 𝜒 𝐵 𝑡,𝑛 = 𝜒 𝑁 𝑡 𝐺

An Algorithm For 𝑂 Δ 2 −Coloring That Takes 𝑂 log ∗ 𝑛 Units Of Time

Lemma 4.1 Lemma 4.1: For integers 𝑛>Δ , there is a family 𝐽 of n subsets of {1, …,5 Δ 2 log 𝑛 } such that if 𝐹 0 , …, 𝐹 Δ ∈𝐽 , then 𝐹 0 ⊈ 1 Δ 𝐹 𝑖

Lemma 4.1 : Proof Let 𝑚=5 Δ 2 log 𝑛 . Consider a random collection 𝐽 of n subsets of {1, …, 𝑚} as follows: For 1≤𝑥≤𝑚 and 1≤𝑖≤𝑛 , let Pr 𝑥∈ 𝑆 𝑖 = 1 Δ . All the decisions on whether 𝑥∈ 𝑆 𝑖 are made independently. We want to prove that there is a selection of such a family for which the lemma holds. Given 𝐹 0 , …, 𝐹 Δ ∈𝐽 , 1≤𝑥≤𝑚 , the probability that 𝑥∈ 𝐹 0 ∖( 1 Δ 𝐹 𝑖 ) is : 1 Δ 1− 1 Δ Δ ≥ 1 4Δ

Lemma 4.1 : Proof Cont. Therefore, the probability that 𝐹 0 ⊆ 1 Δ 𝐹 𝑖 is at most: 1− 1 4Δ 𝑚 . The number of ways for choosing 𝐹 0 , …, 𝐹 Δ is: Δ+1 𝑛 Δ+1 . We want to prove the existence for such a family 𝐽 which satisfies the lemma. Therefore, we require: 1− 1 4Δ 𝑚 Δ+1 𝑛 Δ+1 <1 . This holds when 𝑚≥5 Δ 2 log 𝑛 .

Theorem 4.1 Theorem 4.1 : Let G be a graph of order n and largest degree Δ . It is possible to color G with 5 Δ 2 log 𝑛 colors in one unit of time distributively. Equivalently: 𝜒 𝑁 1 𝐺 ≤5 Δ 2 log 𝑛 .

Theorem 4.1 : Proof Fix a family 𝐽={ 𝐹 1 , …, 𝐹 𝑛 } as in lemma 4.1 . Let 𝑚=5 Δ 2 log 𝑛 . We want to color each vertex with label i , with a color x such that 𝑥∈ 𝐹 𝑖 . Let 𝑖∈{1, …, 𝑛} be the label of a specific vertex. Let 𝑗 1 , …, 𝑗 𝑑 be the neighbors of that vertex , where 𝑑≤Δ . Since: 𝐹 𝑖 ⊈ 𝜈=1 𝑑 𝐹 𝑗 𝜈 , There is a 1≤𝑥≤𝑚 such that 𝑥∈ 𝐹 𝑖 ∖ 𝜈=1 𝑑 𝐹 𝑗 𝜈 The color of this vertex is x . we do this process for each vertex, and obtain a proper m-coloring.

Corollary 4.1 Corollary 4.1 : Proof: Let G be a graph whose vertices are properly colored with 𝑘 colors and whose largest degree is Δ . It is possible to color G with 5 Δ 2 log 𝑘 in one unit of time distributively. Proof: The proof is similar to theorem 4.1, but with replacing n with k: Instead of considering the label of a specific vertex, we consider its color. The same proof holds since G is properly k-colored.

Corollary 4.1 Cont. Let’s iterate corollary 4.1 log ∗ 𝑛 times : 5 Δ 2 log (5 Δ 2 log 𝑛 ) 5 Δ 2 log (5 Δ 2 log (5 Δ 2 log 𝑛 ) ) After log ∗ 𝑛 units of time, we obtain: 10 Δ 2 log Δ =𝑂 Δ 3 −𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 We want to reduce a 𝑂 Δ 3 −𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 to 𝑂 Δ 2 −𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 .

Lemma 4.2 Lemma 4.2 : We demand Δ≤ (𝑞−1)/2 . Let q be a prime power. Then, there is a collection 𝐽 of 𝑞 3 subsets of 1, …, 𝑞 2 such that if 𝐹 0 , … 𝐹 (𝑞−1)/2 ∈𝐽 then 𝐹 0 ⊈ 1 (𝑞−1)/2 𝐹 𝑖 . We demand Δ≤ (𝑞−1)/2 . Select q to be the smallest prime power with 𝑞≥2Δ+1 . There is certainly one with : 4Δ+1≥𝑞≥2Δ+1 . This construction transforms an 𝑂 Δ 3 −𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 to an 𝑂 Δ 2 −𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 as in proof of theorem 4.1 .

Theorem 4.2 Theorem 4.2 : Let G be a labeled graph of order n with largest degree Δ . Then in time O( log ∗ 𝑛 ) it is possible to color G with 𝑂 Δ 2 colors in a distributive synchronous algorithm.

Thank You!