Tight Bounds for Graph Problems in Insertion

Streams

Xiaoming Sun and David P. WoodruffChinese Academy of Sciences and IBM Research-Almaden

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

Graph Streams• See a stream of edges e = {i,j} defining a graph G on n nodes

• Example problems: decide if G is connected, bipartite, or planar. Approximate the minimum cut size, etc.

• Most problems have an Ω(n) bit space lower bound

• Many problems have an n poly(log n) bit upper bound– A significant savings over a naïve bit upper bound

• Question: what is the optimal space complexity of these problems?– Is it Θ(n) bits? Θ(n log n) bits? Or n poly(log n) bits?

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?

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

Our Results Continued• Another popular model is the dynamic graph model, in which edges can be

inserted and deleted

• We show an Ω(n bit lower bound for approximating the minimum cut up to a constant factor

• Implies an Ω(n (log n) log W) bound for computing cut or spectral sparsifiers of a graph in dynamic graphs with edge weights bounded by W– Batson, Spielman, Srivastava give an O(n (log n + log log W)) bit sparsifier – Our result is the first dynamic graph stream lower bound larger than the size

of a sparsifier– Upper bounds in a dynamic stream are or (see work by Ahn, Guha,

McGregor and Kapralov, Lee, Musco, Musco, Sidford)

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

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 Technique1. 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

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”

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

𝜎∈ {0,1 }𝑛 log𝑛 𝑖∈¿

M

AugmentedPerm Problem

• Alice has log n permutations • Bob should output the i-th bit of with constant probability• Bob also knows

σ 1,… ,σ log n∈ {0,1 }n log n

M

Lower Bound for Perm • Uses information theory• Recall for random variables X, Y, the mutual information

• Let be a uniformly random permutation• By the chain rule,

=

=

,

where the inequality is Fano’s, and uses the correctness of the protocol• Setting the failure probability δ small enough gives an n log n lower bound• Similar argument gives lower bound for AugmentedPerm

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

Connectivity Reduction

σ

• σ specifies a random matching in a bipartite graph with n nodes in each part

• Bob is interested in the -th bit of for some k and , determined by i

• Graph is connected if and only if -th bit of is 1

𝜎 (𝑘 )

k

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

Lower Bound for Approximate MinCut

• Alice sends all duplicate edges to Bob (there are O(such edges whp)• Alice runs the streaming algorithm on the union of the log n matchings• Bob has i and knows • Bob deletes all edges in the graphs corresponding to • Bob is interested in the -th bit of for some k and , determined by i

σ 1 σ 2

1010

10

10

…

100100

100

10010 log𝑛

10 log𝑛

10 log𝑛

10 log𝑛

σ log n

Lower Bound for Approximate MinCut

• For a given node, it has j edges from matchings • The edges are weighted • If the the -th bit of is 0, minimum cut is at most • Otherwise, minimum cut value is at least

k

10 j

10 j

10 j

10 j

Open Questions

1. k-edge connectivity lower bound holds only for deterministic algorithms. For randomized we only get a kn lower bound – can this be improved?

2. k-vertex connectivity lower bound holds only for deterministic algorithms which can process multi-edges. Can these assumptions be removed?

3. For graph sparsifiers and minimum cut, our lower bound is only Ω(n while the upper bound is O(n n) – can this gap be closed?

4. For connectivity in dynamic streams, the upper bound is and our Ω(n log n) lower bound is all that is known – can this gap be closed?

Lower Bound for AugmentedPerm

• Let be independent and uniformly random permutations

• By the chain rule,

= , M) =

,

where the inequality is Fano’s, and uses the correctness of the protocol

• Setting the failure probability δ small enough gives an lower bound