# Tight Bounds for Graph Problems in Insertion Streams PowerPoint Presentation, PPT - DocSlides

2018-03-01 33K 33 0 0

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Xiaoming. Sun and David P. Woodruff. Chinese Academy of Sciences and IBM Research-. Almaden. Streaming Models. Long sequence of items appear one-by-one. numbers, points, edges, …. (usually) . adversarially. ID: 640128

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Tight Bounds for Graph Problems in Insertion Streams

Xiaoming

Sun and David P. Woodruff

Chinese Academy of Sciences and IBM Research-

Almaden

Slide2Streaming Models

Long sequence of items appear one-by-one

numbers, points, edges, …

(usually) adversarially orderedone pass over the streamGoal: approximate a function of the underlying streamuse small amount of space (in bits)Efficiency: usually necessary for algorithms to be both randomized and approximate

…

2

1

1

3

7

3

4

Slide3Graph Streams

See a stream of edges e = {

i,j

} defining a graph G on n nodesExample problems: decide if G is connected, bipartite, or planar. Approximate the minimum cut size, etc.Most problems have an Ω(n) bit space lower boundMany problems have an n poly(log n) bit upper boundA significant savings over a naïve bit upper boundQuestion: 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?

Slide4Comparison to Other Problems

Many asymptotically tight bounds are known in streaming

approximating distinct elements, entropy, norms

linear regression, approximate matrix productFor 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?

Slide5Our Results

T

ight lower bounds for a number of graph problems

Connectivity Ω(n log n) bitsPlanarity and H-minor freeness Ω(n log n) bitsBipartiteness Ω(n log n) bitsCycle-freeness, Eulerian testing, testing if a sparse graph has bounded diameter, finding a minimum spanning tree all require Ω(n log n) bitsk-Edge ConnectivityAny deterministic algorithm requires Ω(nk log n) bitsk-Vertex ConnectivityAny deterministic algorithm allowing multi-edges requires Ω(nk log n) bits

Slide6Our 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 factorImplies 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 WBatson,

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 sparsifierUpper bounds in a dynamic stream are or (see work by Ahn,

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

Slide7Talk Outline

Permutation-Based Communication Complexity Problems

Tight lower bound for Graph Connectivity

Lower bound for approximating the minimum cut in dynamic graph streams

Slide8Streaming Lower Bounds via Communication Complexity

a

2

{0,1}

nCreate stream s(a)

b 2

{0,1}nCreate stream s(b)

Lower Bound Technique1. Run Streaming Alg on s(a), transmit state of Alg(s(a)) to Bob2. 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

Slide9Connectivity

There is an

Ω

(n log n) bit lower bound for deterministic protocols for connectivity (Dowling and Wilson)The randomized communication complexity is not knownWe show the randomized 1-way communication complexity is Ω(n log n) bitsTo do so, we introduce a few “permutation-problems”

Slide10Perm 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 BobBob should output the i-th bit of σ with probability > 2/3

M

Slide11AugmentedPerm Problem

Alice has log n permutations

Bob should output the

i-th bit of with constant probabilityBob also knows

M

Slide12Lower 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 protocolSetting the failure probability δ small enough gives an n log n lower boundSimilar argument gives lower bound for AugmentedPerm

Slide13Talk Outline

Permutation-Based Communication Complexity Problems

Tight lower bound for Graph Connectivity

Lower bound for approximating the minimum cut in dynamic graph streams

Slide14Connectivity Reduction

σ

σ

specifies a

random matching

in a bipartite graph with n nodes in each partBob is interested in the -th bit of for some k and , determined by iGraph is connected if and only if -th bit of

is 1 k

Slide15Talk Outline

Permutation-Based Communication Complexity Problems

Tight lower bound for Graph Connectivity

Lower bound for approximating the minimum cut in dynamic graph streams

Slide16Lower 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 matchingsBob 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

10

101010…100

100100100

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

Open Questions

k-edge connectivity lower bound holds only for deterministic algorithms. For randomized we only get a

kn

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

Slide19Lower 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