Approximating Average Parameters of Graphs

1. Approximating Average Parameters of Graphs Oded Goldreich, Weizmann Institute Dana Ron, Tel Aviv University

2. The Type of Problems we Consider Letf be a “natural” function defined on graphs. The domain of f : vertices/pairs of vertices/etc. The goal: Estimate the average value of f . The means:(1) Queries to f ;(2) Queries to the graph:a.Neighbor queries (who is j’th neighbor of v?)b.Vertex-pair queries (are uand v neighbors?) Questions of interest:(1) Can we do this much more efficiently as compared to general functions?(2) How do different queries influence the complexity?

3. deg(v) distv(u) dist(u,v) The Particular Problems we Study Problem1:Estimating the average degree in a graph (first considered by Feige (STOC04)) Problem2:Estimating the average distanceto a given vertex in a graph Problem3:Estimating the average distancebetween pairs of vertices in a graph

4. deg(v) Our Results Problem1: Estimating average degree in a (simple) graph G=(V,E) , |V|=n, |E|=m, dG=2m/n UB: Can obtain (1+)-approximation in time n1/2 poly(log n / )by usingneighbor queries (only). LB: A (1+)-approximation requires ((n / )1/2)queries (when allowed all types of queries). Compare to Feige:(2+)-approx in similar complexity using degree queries only (queries to f); (2-o(1))-approx requires (n)degree queries. Note: Can improvewhenavg degree is high: (n/dG)1/2 instead of n1/2 with matching lower bound.

5. Our Results Cont’ Problem2: Estimating the average distanceto a vertexProblem3: Estimating the average distancebetween verticesG=(V,E) , |V|=n, |E|=m, dG= avg distance UB: Can obtain (1+)-approx in time (n/dG)1/2 poly(log n / )by usingdistance queries (only). LB: A (1+)-approximation requires ((n /( dG))1/2)queries when allowed all types of queries.If allow only neighbor queries: (m) necessary forconstant approximation. Note (non sublinear): Can obtain (1+)-approx using O(m n1/2 poly(1/))neighbor queries only. When graph not too dense ( m=o(n3/2) ) save as compared to computing exact/approx for all pairs.

6. Our Results: Summary • Common to problems: • Can obtain (1+)-approximation in sublinear-time. • Dependence on n: n1/2; dependence on (1/  ): polynomial. • Running time improves as avg increases. • Matching lower bounds in terms of dependence on n/avg • Differences between problems: • In case of avg degree: neighbor queries help(to improve quality of estimate), cando without degree queries (to f) • In case of avg distance: neighbor queriesdo not help,cannot do without distance queries (to f)

7. deg(v) Estimating Average Degree G=(V,E) , |V|=n, |E|=m, dG=2m/n ≥ 1 Idea for algorithm inspired by [Kaufman, Krivelevich, R]’s procedure for selecting an edge “almost uniformly” Ingredient 1: consider partition of all graph vertices into buckets: In bucket Bi vertices v s.t. (1+)i-1 < deg(v) ≤ (1+)i( =/8 , O((log n)/) buckets ) Suppose for every Bi had estimatebis.t. bi= |Bi|(1  /8) .(1/n)  ibi  (1+)i= (1 )  dG(*) How to get bi? By sampling. Difficulty: if |Bi| is small (<< n1/2) then necessary sample is too large ( (|Bi|/n)-1 >> n1/2 ). Ingredient 2: ignore small Bi ‘s, take sum in (*) over sufficiently large bi‘s (Bi ‘s). ( For large Bi ‘s get biby samplingn1/2 vertices )

8. Large buckets Small buckets Counted twice Counted once G=(V,E) , |V|=n, |E|=m, dG=2m/n ≥ 1 Estimating Average Degree Cont’ Reminders:Bi= { v : (1+)i-1 < deg(v) ≤ (1+)i }( =/8 )bi= |Bi|(1 /8) for |Bi| > (( n)1/2/ 4(log n)/)) Consider sum:(1/n) i: Bi largebi  (1+)i(**) clearly:≤ (1+ )dG i.e., small overestimate. By how much can we underestimate? Sum of degrees = 2 num of edges. Not Counted Total not counted ≤ ( n)/2. All others, at least once. Hence, sum in (**)≥ dG/ (2 +)

9. Small buckets Large buckets Not Counted (few) Counted twice Counted once G=(V,E) , |V|=n, |E|=m, dG=2m/n ≥ 1 Estimating Average Degree Cont’ Recap: Can get factor-(2+ ) approx using n1/2 poly((log n)/)degree queries only (alternative to [Feige]). Recall [Feige]:cannot do (much) better using degree queries only. Ingredient 3: Estimate num of edges counted once and compensate for them.

10. Bi Bi Small buckets Small buckets Large buckets Large buckets G=(V,E) , |V|=n, |E|=m, dG=2m/n ≥ 1 Estimating Average Degree Cont’ Ingredient 3: Estimate num of edges counted once and compensate. For each large Biestimate num of edges btwn Bi and small buckets. Implementation: Let Si be vertices sampled in Bi . For each uSiselect random neighbor v . Let i be frac of v’s in small buckets. Estimate is: (1/n)i: Bi largebi  (1+ i) (1+)i W.h.p estimate is (1 )  dG

11. G=(V,E) , |V|=n, |E|=m, dG=2m/n (≥ 1) Estimating Average Degree Summary • Average Degree Approximation Algorithm: • Unif. and indep. select K=O(n1/2 poly((log n)/)) vertices. Let Sbe (multi-)set of selected vertices. • For i=0,…,log1+n , let Si=SBi . • Let L = { i : |Si|/K ≥ (/(8n))1/2 / (log1+n +1) } • For each i  L, every u  Si, select random neighborv of u. Let i be frac. of rand. neighbors in small buckets. • Output (1/K) iL|Si|  (1+ i)  (1+)i Note 1: If have l.b.  , dG≥, then : O((n/)1/2 poly((log n)/))(sufficient to get good bi for|Bi| > ((n)1/2/ 4(log n)/)) ) Note 2: Can get same complexitywithout knowing l.b. (Can search for l.b.because alg does never overestimates)

12. G1 : G2 : k verticesclique n-k verticesd-regular k verticesd-regular n-k verticesd-regular G=(V,E) , |V|=n, |E|=m, dG=2m/n (≥ 1) Estimating Average Degree L.B. Thm. For any n, d [2 ,o(n) ],  ((1/(dn)) , o(n/d) ) distinguishing between avg. deg. d and avg degree (1+ )d requires ((n/( d))1/2) queries (all types allowed) For k  ( d n)1/2 consider (random labelings of) graphs: To distinguish must select vertex in small component dG2= (1+ )d dG1= d

13. Estimating Average Distance Problem2: Estimating the average distanceto a vertexProblem3: Estimating the average distancebetween verticesG=(V,E) , |V|=n, |E|=m, dG= avg distance UB: Can obtain (1+)-approx in time (n/dG)1/2 poly(log n / )by usingdistance queries (only). LB: A (1+)-approximation requires ((n /( dG))1/2)queries when allowed all types of queries.If allow only neighbor queries: (m) necessary forconstant approximation. Note (non sublinear): Can obtain (1+)-approx using O(m n1/2 poly(1/))neighbor queries only. When graph not too dense ( m=o(n3/2) ) save as compared to computing exact/approx for all pairs.

14. Estimating Average Distance Algorithms: For both problems simply take sample of vertices / pairs of vertices and compute average over sample. • Analysis: Uses Chebyshev’s inequality – prove small variance. • Sketch for Problem 3 (avg. dist. btwn pairs): • Let dmax be max distance btwn pairs; • For i=0,…,dmax let pi be fraction of pairs at distance i; • Let  be distributed according to pi: E[]=dG. • We show that E[2] = O(n1/2 E[]2).Core of proof: showing that E[]= (d2max /n). Reason: if some pair are far, then many pairs are far. w For i=1,…, d/3 , d=dmax,  wdist(w,vi)+dist(w,vd-i)≥d/3 . . . . . . . . . vi v1 v0 vd-i vd

15. w1 w1 w2 w2 vt+1 . . . . . . . . vt+2 . s . v1 vt vk v1 s vk wn-k-1 wn-k-1 G=(V,E) , |V|=n, |E|=m, dG= avg dist to s Estimating Average Distance L.B. (for Problem 2 – avg. dist. to vertex s) Thm. For any n, d [2 ,o(n) ],  ((1/(dn)) , o(n/d) ) distinguishing between avg. dist. d and avg dist (1+ )d requires ((n/( d))1/2) queries (all types allowed) Construction for  >1/4: For k (2(1+ )(d n))1/2 andt  ((1+ )1/2-1/2)(2 d n)1/2 consider (rand labelings of) graphs: G2(two-sided broom graph): G1(broom graph): To distinguish must select/reach right-side edge dG1= (1+ ) d dG2<d

16. Summary Study estimation of average value of “natural” functions on graphs: average degree and average distance. Give sublinear algorithms (dependence on n is n1/2) and roughly matching lower bounds. Different problems exhibit different behavior in terms of the “power” of the queries: queries to the estimated functions vs. queries to the structure of the graph.