Xiaoming Sun and David P Woodruff Chinese Academy of Sciences and IBM Research Almaden Streaming Models Long sequence of items appear onebyone numbers points edges usually adversarially ID: 640128
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Slide1
Tight Bounds for Graph Problems in Insertion Streams
Xiaoming
Sun and David P. Woodruff
Chinese Academy of Sciences and IBM Research-
AlmadenSlide2
Streaming 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
4Slide3
Graph 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? Slide4
Comparison 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?Slide5
Our 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) bitsSlide6
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 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) Slide7
Talk Outline
Permutation-Based Communication Complexity Problems
Tight lower bound for Graph Connectivity
Lower bound for approximating the minimum cut in dynamic graph streamsSlide8
Streaming 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 Slide9
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 knownWe show the randomized 1-way communication complexity is Ω(n log n) bitsTo do so, we introduce a few “permutation-problems”Slide10
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 BobBob should output the i-th bit of σ with probability > 2/3
MSlide11
AugmentedPerm Problem
Alice has log n permutations
Bob should output the
i-th bit of with constant probabilityBob also knows
MSlide12
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 protocolSetting the failure probability δ small enough gives an n log n lower boundSimilar argument gives lower bound for AugmentedPerm Slide13
Talk Outline
Permutation-Based Communication Complexity Problems
Tight lower bound for Graph Connectivity
Lower bound for approximating the minimum cut in dynamic graph streamsSlide14
Connectivity 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 kSlide15
Talk Outline
Permutation-Based Communication Complexity Problems
Tight lower bound for Graph Connectivity
Lower bound for approximating the minimum cut in dynamic graph streamsSlide16
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 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
Slide17
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
Slide18
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? Slide19
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