1. Tight Bounds for Graph Problems in Insertion Streams Xiaoming Sun and David P. Woodruff Chinese Academy of Sciences and IBM Research-Almaden

2. Streaming Models Long sequence of items appear one-by-one –numbers, points, edges, … –(usually) adversarially ordered –one pass over the stream Goal: approximate a function of the underlying stream –use small amount of space (in bits) Efficiency: usually necessary for algorithms to be both randomized and approximate … 2113734

3. Graph Streams

4. Comparison to Other Problems Many asymptotically tight bounds are known in streaming –approximating distinct elements, entropy, norms –linear regression, approximate matrix product For basic graph questions, such as connectivity: –O(n log n) bit upper bound (maintain a spanning forest) –Ω(n) bit lower bound (reduction from set disjointness) Could it be that there is an O(n) bit upper bound using a clever, possibly adaptive hashing scheme to store the edge identities?

5. Our Results Tight lower bounds for a number of graph problems –Connectivity Ω(n log n) bits –Planarity and H-minor freeness Ω(n log n) bits –Bipartiteness Ω(n log n) bits –Cycle-freeness, Eulerian testing, testing if a sparse graph has bounded diameter, finding a minimum spanning tree all require Ω(n log n) bits k-Edge Connectivity –Any deterministic algorithm requires Ω(nk log n) bits k-Vertex Connectivity –Any deterministic algorithm allowing multi-edges requires Ω(nk log n) bits

6. Our Results Continued

7. Talk Outline 1.Permutation-Based Communication Complexity Problems 2.Tight lower bound for Graph Connectivity 3.Lower bound for approximating the minimum cut in dynamic graph streams

8. Streaming Lower Bounds via Communication Complexity a 2 {0,1} n Create stream s(a) b 2 {0,1} n Create stream s(b) Lower Bound Technique 1. Run Streaming Alg on s(a), transmit state of Alg(s(a)) to Bob 2. Bob computes Alg(s(a), s(b)) 3. If Bob solves g(a,b), space complexity of Alg at least the 1- way communication complexity of g

9. Connectivity There is an Ω(n log n) bit lower bound for deterministic protocols for connectivity (Dowling and Wilson) The randomized communication complexity is not known We show the randomized 1-way communication complexity is Ω(n log n) bits To do so, we introduce a few “permutation-problems”

10. Perm Problem Alice is given a permutation σ of 1, 2, …, n represented as a redundant (n log n)-bit vector σ(1), σ(2), …, σ(n) Bob is given an index i in [n log n ] = {1, 2, …, n log n} Alice sends a single message M to Bob Bob should output the i-th bit of σ with probability > 2/3 M

11. AugmentedPerm Problem M

12. Lower Bound for Perm

13. Talk Outline 1.Permutation-Based Communication Complexity Problems 2.Tight lower bound for Graph Connectivity 3.Lower bound for approximating the minimum cut in dynamic graph streams

14. Connectivity Reduction σ k

15. Talk Outline 1.Permutation-Based Communication Complexity Problems 2.Tight lower bound for Graph Connectivity 3.Lower bound for approximating the minimum cut in dynamic graph streams

16. Lower Bound for Approximate MinCut 10 … 100

17. Lower Bound for Approximate MinCut k

18. Open Questions

19. Lower Bound for AugmentedPerm