Diameter computation

Small world

pierluigi.crescenzi@irif.fr

GM

#01

• The small average distance observed in the complex networks is referred as small world effect
• The average distance is constant (six)
• If the size of the network is \(n\), the diameter have at most the order of magnitude of \(\log(n)\)

By Ageev Andrew - Own Work based on Image:Map of USA showing state names.png

CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=22945395

Diameter computation

Small world

pierluigi.crescenzi@irif.fr

GM

#01

• The average distance in Facebook (\(721.1\)M nodes and \(68.7\)G edges) is 4.7 and the diameter is 41

• Average distance has been computed by applying HyperANF tool
• Diameter has been computed by applying \(i\)FUB.

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

Definitions

• Given an unweighted graph \(G = (V,E)\) (strongly) connected

  • Distance
    The distance \(d(u,v)\) is the number of edges along shortest path from \(u\) to \(v\)

  • Diameter
    \(D = \max_{u,v \in V} d(u,v)\)

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

Definitions: undirected graphs

• Eccentricity of a node \(u\): \(\mathrm{ecc}(u)=\max_{v\in V}d(u,v)\)

  • Diameter: \(D=\max_{u\in V}\mathrm{ecc}(u)\)

• Frontier \(F_i(u)\) of a node \(u\): set of nodes at distance \(i\)

  • Nodes at level \(i\) of BFS tree

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

Definitions: directed graphs

• Forward eccentricity of a node \(u\): \(\mathrm{ecc}_F(u)=\max_{v\in V}d(u,v)\)

• Backward eccentricity of a node \(u\): \(\mathrm{ecc}_B(u)=\max_{v\in V}d(v,u)\)

  • ​Diameter: \(D=\max_{u\in V}\{\mathrm{ecc}_F(u),\mathrm{ecc}_B(u)\}\)

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

Definitions: directed graphs

• Forward frontier \(F^F_i(u)\) of a node \(u\): nodes at level \(i\) of forward BFS tree from \(u\)

• Backward frontier \(F^B_i(u)\) of a node \(u\): nodes at level \(i\) of backward BFS tree to \(u\)

Forward BFS tree

Backward BFS tree

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

State of the art: positive results

• \(G=(V,E)\) with \(n=|V|\) and \(m=|E|\)
• Textbook algorithm
  • Perform \(n\) BFSes and return maximum eccentricity
    • A BFS from \(u\) returns all the distances from \(u\) and takes \(O(m)\) time
  • Complexity \(O(nm)\): too expensive
• Several other approaches for all pairs shortest path problem
  • \(O(n^{(3+\omega)/2}\log n)\) where \(\omega\) is the exponent of the matrix multiplication
  • Still too expensive
• Empirically finding lower bound \(L\) and upper bound \(U\)
  • That is, \(L \leq D \leq U\)
  • \(D\) found, when \(L=U\)

• Quadratic algorithms are not feasible
• Look for "hardest" quadratic time solvable problems
  • Approach similar to NP-completeness
  • Definition of specific reducibility
    • Preserving subquadratic solvability
• Hardness relative to complexity hypothesis
  • Similar to P vs NP
  • ​SETH: no algorithm solving k-SAT in subexponential time
    • Quadratic time solvable version (k-SAT*)

Diameter computation

Negative results

pierluigi.crescenzi@irif.fr

GM

#01

• Quasi-linear reducibility: \(\mathcal{P} \leq_{ql}\mathcal{Q}\)
  • \(I \mathrm{\ instance\ of\ }\mathcal{P} \rightarrow \Phi(I)\mathrm{\ instance\ of\ }\mathcal{Q}\)
    • \(\mathrm{Computable\ in\ time\ }\tilde{O}(|I|)\)
  • \(I \mathrm{\ and\ } \Phi(I) \mathrm{\ same\ output}\)
    • \(\mathrm{Linear\ time\ computable\ output\ mapping}\)
• Fact

Diameter computation

Negative results

pierluigi.crescenzi@irif.fr

GM

#01

• \(k\)-SAT*
  • Input
    • \(\mathrm{Two\ sets \ of\ } n \mathrm{\ variables\ }\{x_i\},\{y_i\}\)
    • \(\mathrm{Set\ of\ clauses\ } C\)
    • \(\mathrm{Possible\ assignments\ to \ } x_i\)
    • \(\mathrm{Possible\ assignments\ to \ } y_i\)
  • Output
    • \(\mathrm{\ True\ iff\ }C\mathrm{\ satisfiable}\)
• Fact

Diameter computation

Negative results

pierluigi.crescenzi@irif.fr

GM

#01

• The reduction web

Diameter computation

Negative results

pierluigi.crescenzi@irif.fr

GM

#01

• From \(k\)-two disjoint sets to diameter
  • Input
    • \(\mathrm{Set\ of\ items\ } X\)
    • \(\mathrm{Collection\ } C \mathrm{\ of\ subsets\ of\ } X\) with \(|X|\leq \log^k(|C|)\)
  • Output
    • \(\mathrm{\ True\ iff\ }C\mathrm{\ has\ two\ disjoint\ sets}\)
  • Reduction

Diameter computation

Negative results

pierluigi.crescenzi@irif.fr

GM

#01

• By using one BFS tree

Diameter computation

Lower and upper bounds: undirected graphs

pierluigi.crescenzi@irif.fr

GM

#01

• Lower bound: eccentricity (height of the BFS tree)

  • Example: 3

• Upper bound: twice the eccentricity

  • Example: 6 (every node can reach another node going to \(v_1\) by \(\leq 3\) edges and going to the destination by \(\leq 3\) edges)

  • Fact: \(x\in F_i(u)\) and \(y\in F_j(u)\) implies \(d(x,y)\leq i+j\)

• Bounds by sampling but very often \(L < D < U\)

  • In the example diameter is 4: \(d(v_{7},v_{8})=4\)

• By using one forward BFS (fBFS) tree and one backward BFS (bBFS) tree

Diameter computation

Lower and upper bounds: directed graphs

pierluigi.crescenzi@irif.fr

GM

#01

• Lower bound: maximum between \(\mathrm{ecc}_F(u)\) (height of the fBFS tree) and \(\mathrm{ecc}_B(u)\) (height of the bBFS tree)

  • Example: 5

• Upper bound: \(\mathrm{ecc}_F(u) + \mathrm{ecc}_B(u)\)

  • Example: 9 (every node can reach another node going to \(v_1\) by \(\leq 5\) edges and going to the destination by \(\leq 4\) edges)

  • Fact: \(x\in F_i^B(u)\) and \(y\in F_j^F(u)\) implies \(d(x,y)\leq i+j\)

• Bounds by sampling but very often \(L < D < U\)

  • In the example diameter is 7: \(d(v_{10},v_{12})=7\)

Diameter computation

Lower bounds: (undirected) 2-sweep

pierluigi.crescenzi@irif.fr

GM

#01

• Run a BFS from a (random) node \(r\): let \(a\) be the farthest node

• Run a BFS from \(a\): let \(b\) be the farthest node

• Return \(d(a,b)\)

• Experiments (\(r\) node of maximum degree)

Diameter computation

Lower bounds: (directed) 2-sweep

pierluigi.crescenzi@irif.fr

GM

#01

• Run a fBFS and a bBFS from a (random) node \(r\): let \(a_1\) and \(a_2\) be the farthest nodes

• Run a bBFS (fBFS) from \(a_1\) (\(a_2\)): let \(b_1\) (\(b_2\)) be the farthest node

• Return \(\max\{d(b_1,a_1),d(a_2,b_2)\}\)

• Experiments (\(r\) node of maximum in or out degree)

Diameter computation

Lower bounds: bad case for 2-sweep

pierluigi.crescenzi@irif.fr

GM

#01

• Modified grid with \(k\) rows and \(1+3k/2\) columns
• The algorithm can return \(k\)
• The diameter is \(3k/2\)

Diameter computation

Lower bounds: 4-sweep

pierluigi.crescenzi@irif.fr

GM

#01

• Apply 2-Sweep
• Pick the middle vertex of the returned path
• Apply again 2-Sweep

Diameter computation

Lower bounds: bad case for 4-sweep

pierluigi.crescenzi@irif.fr

GM

#01

• The diameter is \(\max\{ x-1, y, z-1 \}\)
• By choosing \(x\), \(y\), and \(z\) appropriately, diameter is equal to \(z-1\), while 4-sweep can return \(x-1>(z+1)/2\)

Diameter computation

Exact computation (undirected graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• The textbook algorithm runs a BFS for any node and return the maximum found eccentricity

• Idea

  • Perform the BFSes one after the other specifying the order in which they have to be executed

  • While doing this

    • Refine the lower bound (maximum eccentricity)

    • Upper bound eccentricities of remaining nodes

    • Stop when the remaining nodes cannot have eccentricity higher than our lower bound

  • Good order can be inferred looking at some properties of BFS trees

Diameter computation

Exact computation (undirected graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• Main observation

  • For any \(1\leq i< \mathrm{ecc}(u)\)  and \(1 \leq k < i\), and for any \(x\in F_{i-k}(u)\) such that \(\mathrm{ecc}(x)>2(i-1)\), there exists \(y\in F_j(u)\) such that \(d(x,y)=\mathrm{ecc}(x)\) with \(j \geq i\)

\(\mathrm{ecc}(x)>2(i-1)\Rightarrow\exists y[\mathrm{ecc}(y)\geq\mathrm{ecc}(x)]\)

Diameter computation

Exact computation (undirected graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• Main observation

  • For any \(1\leq i< \mathrm{ecc}(u)\)  and \(1 \leq k < i\), and for any \(x\in F_{i-k}(u)\) such that \(\mathrm{ecc}(x)>2(i-1)\), there exists \(y\in F_j(u)\) such that \(d(x,y)=\mathrm{ecc}(x)\) with \(j \geq i\)

• Proof

  • Since \(\mathrm{ecc}(x)>2(i-1)\), then there exists \(y_x\) whose distance from \(x\) is equal to \(\mathrm{ecc}(x)\) and, hence, greater than \(2(i-1)\)

  • If \(y_x\) was in \(F_j(u)\) with \(j < i\), then \[d(x,y_x)\leq (i-1)+(i-k)\leq 2\max\{i-1,i-k\} = 2(i-1)\]

  • Contradiction: hence, \(y_x\in F_j(u)\) with \(j\geq i\)

• Corollary: if \(lb\) is the maximum among all the eccentricities of the nodes in or below the level \(i\), then the eccentricities of all other nodes is bounded by \(\max\{lb,2(i-1)\}\)

Diameter computation

Exact computation (undirected graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• Corollary of the main observation

  • If \(lb\) is the maximum among all the eccentricities of the nodes in or below the level \(i\), then the eccentricities of all other nodes is bounded by \(\max\{lb,2(i-1)\}\)

• ​Notation: \(B_{i}(u)=\max_{v\in F_i(u)}\mathrm{ecc}(v)\)

• The algorithm (bottom-up)

  • Given a node \(u\) and its BFS tree

    • Set \(i=\mathrm{ecc}(u)\) and \(M=B_{i}(u)\)

    • If \(M > 2(i-1)\), then return \(M\), else set \(i=i-1\) and \(M=\max\{M,B_{i}(u)\}\) and repeat this step

Diameter computation

Exact computation (undirected graphs)

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

Exact computation (undirected graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• \(n\): number of nodes
• \(v\): (average) number of executed BFSes

Diameter computation

Exact computation (undirected graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• Bad case for iFUB
  • A cycle with \(n\) nodes (\(n\) odd) has diameter \(\frac{n-1}{2}\), and each  node has the same BFS tree
  • The loop stops the first time that \(2(i-1)<\frac{n-1}{2}\), that is, \(i<\frac{n+3}{4}\)
  • The total number of iterations is equal to \(\frac{n-1}{2}-\frac{n+3}{4}+2=\frac{n+3}{4}\)
  • The number of BFSes is \(\frac{n+3}{2}\)

Diameter computation

Exact computation (directed graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• Notation: \((X,Y,a,b)\in\{(B,F,x,y),(F,B,t,z)\}\)

• Main observation

  • For any \(1\leq i< \mathrm{ecc}_X(u)\)  and \(1 \leq k < i\), and for any \(a\in F^X_{i-k}(u)\) such that \(\mathrm{ecc}_Y(a)>2(i-1)\), there exists \(b\in F^Y_j(u)\) such that \(\mathrm{ecc}_X(b)\geq\mathrm{ecc}_X(a)\) with \(j \geq i\)

Diameter computation

Exact computation (directed graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• Notation: \((X,Y,a,b)\in\{(B,F,x,y),(F,B,t,z)\}\)

• Main observation

  • For any \(1\leq i< \mathrm{ecc}_X(u)\)  and \(1 \leq k < i\), and for any \(a\in F^X_{i-k}(u)\) such that \(\mathrm{ecc}_Y(a)>2(i-1)\), there exists \(b\in F^Y_j(u)\) such that \(\mathrm{ecc}_X(b)\geq\mathrm{ecc}_X(a)\) with \(j \geq i\)

• Corollary: if \(lb\) is the maximum among all the \(\mathrm{ecc}_B\) of nodes in or below the level \(i\) of the fBFS and among all the \(\mathrm{ecc}_F\) of nodes in or below the level \(i\) of the bBFS, then the eccentricities of all other nodes is bounded by \(\max\{lb,2(i-1)\}\)

• Further notation:

Diameter computation

Exact computation (directed graphs)

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

Exact computation (directed graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• Example

• \(i=lb=\max\{\mathrm{ecc}_F(v_1), \mathrm{ecc}_B(v_1)\} = \max\{4, 5\} = 5\), and \(ub=2i = 10\)

• Since \(ub>lb\), the algorithm enters the while loop with \(i=5\) 

• \(B_5^F(u)=0\) (since \(5>\mathrm{ecc}_F(u)\)) and \(B_5^B(u)=\mathrm{ecc}_F(v_{10})=7\): since \(7 < 8 = 2(i-1)\), the algorithm enters the else branch and set \(lb\) equal to 7 and \(ub\) equal to 8

• Since \(ub>lb\), the algorithm enters the while loop with \(i=5\)

• \(B_4^F(u)=\mathrm{ecc}_B(v_{10})=6\) and \(B_4^B(u)=\mathrm{ecc}_F(v_{8})=6\): since \(\max\{lb, B_4^B(u), B_4^F(u)\} = 7 > 6 = 2(i-1)\), the algorithm enters the if branch and returns the value 7 which is the correct diameter value

Diameter computation

Exact computation (directed graphs)

pierluigi.crescenzi@irif.fr

GM

#01

Diameter computation

Exact computation (directed graphs)

pierluigi.crescenzi@irif.fr

GM

#01

• For any graph with more than 10000 nodes, diFUB  performs \(0.001\%n\) visits instead of \(n\)

Diameter computation

Final considerations

pierluigi.crescenzi@irif.fr

GM

#01

• Suitable properties of the starting node \(u\)

  • (1) \(u\) has to be the node with minimum eccentricity, called radius \(R\)

  • (2) Constant number of nodes in \(F_{\mathrm{ecc}(u)}(u)\)

• If you are able to infer the node \(u\) such that (1) and \(R=D/2\) you will stop after one iteration

  • High degree node is very often a good choice

  • If the lower bound path returned by 2-sweep is tight and \(R=D/2\), the node in the middle of this path make us stop after one iteration

• Almost always in real-world graphs \(R=D/2\) (the minimum possible, maximum heterogeneity) and (2) is true if \(u\) is central

Diameter computation

Final considerations

pierluigi.crescenzi@irif.fr

GM

#01

• diFUB can be generalized to weighted graphs

  • Using Dijkstra algorithm instead of BFS and sorting the nodes according to their distance from \(u\)

  • It works well, but not for road networks

• Further optimization allow us to do better than this and to compute also the diameter of weakly connected graphs

• It is possible to prove that for some graph random generation models (fixing the power law distribution) the number of BFSes is almost constant

Diameter computation

Back to theory

pierluigi.crescenzi@irif.fr

GM

#01

• Average case complexity
  • Very hard and technical
  • Many models
  • Are models realistic?
  • Which properties are used?

• Axiomatic framework
  • Define properties
  • Deduce probabilistic analyses from the properties
  • Prove that random graphs satisfy the properties
  • Show empirically that real-world graphs satisfy the properties

Diameter computation

Back to theory

pierluigi.crescenzi@irif.fr

GM

#01

• The models
  • Erdös-Renyi model: connect each pair of nodes with probability p
    • Not realistic (all nodes are "equal")
    • Heuristics are not efficient on this model
  • Random graph with prescribed degree distribution w(u)
    • Configuration model
      • Each node u has w(u) half-edges
      • Half-edges paired at random
    • Rank-1 Inhomogeneous Random Graphs (e.g. Chung-Lu)
      • Edge between u and v exists with probability w(u)w(v)/M
        • M: sum of all w(u)
• Power law degree distribution\[|\{v\in V:\mathrm{deg}(v)=d\}| \approx nd^{-\beta})\]

Diameter computation

Back to theory

pierluigi.crescenzi@irif.fr

GM

#01

• Some definitions
  • \(\gamma^l(s)=|\{v\in V:d(s,v)=l\}|\)
  • \(\tau_s(n^x) = \min\{l:\gamma^l(s)> n^x\}\)
  • \(T(d \rightarrow n^x) = \mathrm{\ avg}_{\mathrm{deg}(s)=d}\tau_s(n^x)\)

...

Diameter computation

Back to theory

pierluigi.crescenzi@irif.fr

GM

#01

• Property 1 \[|\{s\in V:\tau_s(n^x)\geq T(\mathrm{deg}(s)\rightarrow n^x)+l\}|\approx\frac{n}{c^l}\]

Diameter computation

Back to theory

pierluigi.crescenzi@irif.fr

GM

#01

• Property 2 \[d(s,t) \approx \tau_s(n^x)+\tau_t(n^{1-x})-1\]
  

Diameter computation

Back to theory

pierluigi.crescenzi@irif.fr

GM

#01

• Estimate of the diameter

By properties 2 and 3

By property 1

Hence

Hence

Hence

Diameter computation

Back to theory

pierluigi.crescenzi@irif.fr

GM

#01

• Sampling versus 2-sweep
• Sampling returns the eccentricity of a random node s

• 2-sweep
  • Returns the eccentricity of the node farthest from random node s