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

2. The Type of Problems we Consider Let f 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 u and 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. The Particular Problems we Study Problem1: Estimating the average degree in a graph (first considered by Feige (STOC04)) Problem2: Estimating the average distance to a given vertex in a graph Problem3: Estimating the average distance between pairs of vertices in a graph deg(v) dist v (u) dist(u,v)

4. Our Results Problem1: Estimating average degree in a (simple) graph G=(V,E), |V|=n, |E|=m, d G =2m/n deg(v) UB: Can obtain (1+  )-approximation in time n 1/2 poly(log n /  ) by using neighbor 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 improve when avg degree is high: (n/d G ) 1/2 instead of n 1/2 with matching lower bound.

5. Our Results Cont’ Problem2: Estimating the average distance to a vertex Problem3: Estimating the average distance between vertices G=(V,E), |V|=n, |E|=m, d G = avg distance UB: Can obtain (1+  )-approx in time (n/d G ) 1/2 poly(log n /  ) by using distance queries (only). LB: A (1+)-approximation requires  ( ( n /(  d G ) ) 1/2 ) queries when allowed all types of queries. If allow only neighbor queries:  (m) necessary for constant approximation. Note (non sublinear): Can obtain (1+)-approx using O(m n 1/2 poly(1/)) neighbor queries only. When graph not too dense ( m=o(n 3/2 ) ) 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: n 1/2 ; 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), can do without degree queries (to f) In case of avg distance: neighbor queries do not help, cannot do without distance queries (to f)

7. Estimating Average Degree G=(V,E), |V|=n, |E|=m, d G =2m/n ≥ 1 deg(v) 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 B i vertices v s.t. (1+) i-1 < deg(v) ≤ (1+) i ( =  /8, O ( (log n)/ ) buckets ) Suppose for every B i had estimate b i s.t. b i = |Bi|(1   /8). (1/n)   i b i  (1+) i = (1   )  d G (*) How to get b i ? By sampling. Difficulty: if |B i | is small ( > n 1/2 ). Ingredient 2: ignore small B i ‘s, take sum in (*) over sufficiently large b i ‘s (B i ‘s). ( For large B i ‘s get b i by sampling  n 1/2 vertices )

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

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

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

11. Estimating Average Degree Summary G=(V,E), |V|=n, |E|=m, d G =2m/n (≥ 1) Average Degree Approximation Algorithm: 1.Unif. and indep. select K=O ( n 1/2 poly((log n)/  ) ) vertices. Let S be (multi-)set of selected vertices. 2.For i=0,…,log 1+ n, let S i =S  B i. 3.Let L = { i : |S i |/K ≥ (  /(8n)) 1/2 / (log 1+ n +1) } 4.For each i  L, every u  S i, select random neighbor v of u. Let  i be frac. of rand. neighbors in small buckets. 5.Output (1/K)   iL |S i |  (1+  i )  (1+  ) i Note 1: If have l.b., d G ≥, then : O((n/ ) 1/2 poly((log n)/  )) (sufficient to get good b i for |B i | > ((  n) 1/2 / 4(log n)/  )) ) Note 2: Can get same complexity without knowing l.b. (Can search for l.b. because alg does never overestimates)

12. Estimating Average Degree L.B. G=(V,E), |V|=n, |E|=m, d G =2m/n (≥ 1) 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: d G 1 = d k vertices clique G 2 : n-k vertices d-regular d G 2 = (1+ ) d To distinguish must select vertex in small component n-k vertices d-regular G 1 : k vertices d-regular

13. Estimating Average Distance Problem2: Estimating the average distance to a vertex Problem3: Estimating the average distance between vertices G=(V,E), |V|=n, |E|=m, d G = avg distance UB: Can obtain (1+  )-approx in time (n/d G ) 1/2 poly(log n /  ) by using distance queries (only). LB: A (1+)-approximation requires  ( ( n /(  d G ) ) 1/2 ) queries when allowed all types of queries. If allow only neighbor queries:  (m) necessary for constant approximation. Note (non sublinear): Can obtain (1+)-approx using O(m n 1/2 poly(1/)) neighbor queries only. When graph not too dense ( m=o(n 3/2 ) ) 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 d max be max distance btwn pairs;  For i=0,…,d max let p i be fraction of pairs at distance i;  Let  be distributed according to p i : E[  ]=d G.  We show that E[  2 ] = O(n 1/2 E[  ] 2 ). Core of proof: showing that E[  ]=  (d 2 max /n). Reason: if some pair are far, then many pairs are far. v0v0 v1v1 vdvd... vivi v d -i w For i=1,…, d/3, d=d max,  w dist(w,v i )+dist(w,v d-i )≥d/3

15. Estimating Average Distance L.B. (for Problem 2 – avg. dist. to vertex s) G=(V,E), |V|=n, |E|=m, d G = avg dist to 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 and t  ((1+  ) 1/2 -  1/2 )(2 d n) 1/2 consider (rand labelings of) graphs: d G 1 = (1+  ) d G 2 (two-sided broom graph): dG2 < ddG2 < d To distinguish must select/reach right-side edge G 1 (broom graph): vkvk s... v1v1 w1w1 w2w2 w n-k-1.. s... v1v1 vtvt w1w1 w2w2 w n-k-1.. v t+1 v t+2 vkvk

16. Summary Study estimation of average value of “natural” functions on graphs: average degree and average distance. Give sublinear algorithms (dependence on n is n 1/2 ) 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.