Approximating the MST Weight in Sublinear Time

1. Approximating the MST Weight in Sublinear Time Bernard Chazelle (Princeton) Ronitt Rubinfeld (NEC) Luca Trevisan (U.C. Berkeley)

2. Sublinear Time Algorithms • Make sense for problems on very large data sets • Go contrary to common intuition that “an algorithm must be given at least enough time to read all the input” • Must be probabilistic • Must be approximate

3. Approximation • For decision problems:the output is the correct answer either for the given input, or at least for some other input “close” to it.(Property Testing) • For optimization problems:the output is a number that is close to the cost of the optimal solution for the given input.(There is not enough time to construct a solution)

4. Previous Examples • The cost of the max cut in a graph with n nodes and cn2 edges can be approximated to within a factor e in time 2poly(1/ec).(Goldreich, Goldwasser, Ron) • Other results for “dense” instances of optimization problems, for low-rank approximation of matrices, . . . • No results (that we know of) for problems on bounded-degree graphs.

5. Our Result • Given a connected weighted graph G, with maximum degree d and with weights in the range {1, . . . , w}, • we can compute the weight of the minimum spanning tree of G to within a factor of e in time O(dwe-2log w/e); • we also prove that it is necessary to look at W(dwe-2) entries in the representation of G. (We assume that G is represented using adjacency lists)

6. Main Intuition • Suppose all weights are 1 or 2 • Then the MST weight is equal to n – 2 + # of conn. comp. induced by weight-1 edges weight 1 weight 2 MST connected components Induced by weight-1 edges

7. Algorithm

8. Algorithm for weights in {1,2} • To approximate the MST weight to within a multiplicative factor (1+e) it’s enough to approximate c1 to within an additive factor en (c1:= # of connected components induced by weight-1 edges) • To approximate c1 we use ideas from Goldreich-Ron (property testing of connectivity) • The algorithm runs in time O(de-2loge-1)

9. Approximating # of connected components • Given a graph G of max degree d with n nodes we want to compute c, the number of connected components of G up to an additive error en. • For every vertex u, definenu := 1 / size of component of u • Thenc = Sunu • And if we callau:= max {nu, e } • Thenc = Suauen

10. Wrapping up the analysis • Can estimate summation of au using sampling • Once we pick a vertex u at random, the value au can be computed in time O(d/e) • We need to pick O(1/e2) vertices, so we get running time O(d/e3)

11. Algorithm CC-APPROX(e) Repeat O(1/e2) times pick a random vertex v do a BFS from v, stopping after 2/e steps b:= 1/ number of visited vertices return (average of the values b) * n

12. Improved Algorithm CC-APPROX(e, W) Repeat O(1/e2) times pick a random vertex v do first step of a BSF from v b:=0; t:=1 (*) flip a coin If heads, and visited <W nodes so far t:=2*t continue BSF until ends or t nodes are visited if BSF ends, b:= 2#random coins / nodes visited else go to (*) return (average of the values b) * n • Inner procedure takes average O(dlog W) time

13. Analysis • Main idea: if v is in a component of size c<W, then b is zero with prob. ~(1 – 1/c) and ~1 with probability ~1/c. The average of b is 1/c. • Setting W:=2/e we get • each time, the average of b is within e/2 from the average over v of nv(that is, (# conn. comp.)/n) • Repeating O(1/e2) times, the probability of deviating by another factor e/2 is bounded by a constant • The average running time is O(de-2logW), that is O(de-2log e-1).

14. General Weights • Generalize argument for weight 1 and 2. • Let ci = # of connected components induced by edges of weight at most i • Then the MST weight is n – w + Si=1,. . ., w-1ci

15. Final Algorithm • For j=1,. . ., w-1, call CC-APPROX(e,2w/e) on the subgraph of G obtained by removing edges of cost >j • Get ai, an approximation of ci • Return n – w + Si=1,. . ., w-1ai • Average answer is within en/2 from cost of MST, and variance is bounded • Total running time O(dwe-2log w/e)

16. Extensions • Low average degree • Non-integer weights

17. Lower Bound

18. Abstract sampling problem • Fix p,e • Define two binary distributions A,B • Pr[A=1] = p, Pr[A=0]=1-p • Pr[B=1] = p+ ep, Pr[B=0]=1-p-ep • Distinguishing A from B with constant probability requires W(1/pe2) samples

19. Reduction • Fix p = 1/w • We consider two distributions of weights over a cycle of length n • In distribution G, for each edge we sample from A; if A=0 the edge gets weight 1, otherwise it gets weight w • In distribution H, same with B • H and G are likely to have MST costs that differ by about en • To distinguish them we need to look at W(w/e2) edge weights

20. Higher Degree • Sample from G or H as before, • also add d-1 forward edges of weight w+1 from each vertex • randomly permute names of vertices • Now, on average, reading t edge weights gives us t/d samples from A or B, so t=W(dw/e2)

21. Conclusions • A plausibility result showing that approximation for a standard graph problem in bounded degree (and sparse) graphs can be achieved in time independent of number of vertices • Use of approximate cost without solution? • More problems? • The trivial Max SAT approximation algorithms can be implemented in constant time, and give (an implicit representation of) a solution • Non-trivial Max SAT approximation? (say, 3/4) • Something really useful?