massive data streams in graph theory and computational geometry ph.d. dissertation defense jian...

Massive Data Streams in Graph Theory and

Computational Geometry

Ph.D. Dissertation Defense Jian Zhang

Advisor: Joan Feigenbaum

Committee: Ravi Kannan Avi Silberschatz Sampath Kannan

(UPenn)Support: NSF grants 0105337 and 0331548

June 15, 2005 J. Zhang - Ph.D. Dissertation Defense 2

Talk Outline

Streaming computational model

Overview of results

Approximate graph distances in the streaming model

Future research directions


Data Streams A data stream is a sequence of data elements:

a1, a2, …, an . Stream of stock prices Stream of IP packets

Data elements have different forms in different applications. Scalar value Tuple

The semantics of the data elements are also different in different applications.


Streaming Computational Model

Sequential access to the input stream Order of data elements in the stream is not controlled by the

algorithm and may be adversarial.

Algorithms may perform pre- or post-processing without access to the data stream.

Working Space

STREAM


Features of Streaming Algorithms

Small working space compared to the stream length n Polylog n n

Small number of passes over the stream One pass Constant number of passes

Fast per-data-element processing time


Sliding-Window Model

A variation of streaming

Data stream is a time series and may be infinite.

Consider the n most recent data elements.

As time progresses, new data elements arrive, and old data elements expire.

The deletion of old data elements is implicit.


Why Streaming ?

Data streams occur in real systems. IP-traffic flow

Need to distinguish the working space from the data storage. Storage devices: large capacity but slow access Working space: small capacity but fast random access We want to restrict random access to the mass

storage but still see every element of the input set at least once.


Earlier Work on Streaming

Despite the restrictions of the model, a lot can be done, e.g.: Lp norms [FKSV02, Indyk00] histograms [GKS01] clustering [GMMO00]

Much of the work focuses on computing statistics.

Often the working-space size is restricted to polylog space.


Talk Outline


Overview of results




Dissertation Contributions

Investigate important problem domains. Computational geometry problems Graph problems

Show the importance of a more relaxed model. Sublinear space instead of polylog space Multiple passes

There are problems that are provably hard in the restricted model but feasible in the more relaxed model.


Results on Geometric Problems (1)

Exact computation is hard using sublinear space. Computing the exact Diameter, Closest Pair, or

Convex Hull requires (n) bits of space, where n is the number of points in the stream.

Approximation is feasible. We give a one-pass, ε-approximation, streaming

algorithm for diameter. The algorithm needs storage for O(1/ε) points and processes each point in O(log(1/ε)) time.

[ Feigenbaum-S. Kannan-Zhang ]


Results on Geometric Problems (2)

We give an ε-approximation algorithm that maintains the diameter in the sliding-window model.

The algorithm uses O(1/ε log3n logR) bits of space, where R is the largest diameter attained in any window. The amortized processing time for each point is O(logn).

We show that is (1/ε logn logR) space is required for such an approximation.


Graph Stream

Consider undirected graph: G=(V,E)

V = {v1, v2, …, vn} E = {e1, e2, …, em}

A graph stream is a sequence of edges in E.

Edges arrive in arbitrary order in the stream. More general than adjacency matrices or adjacency lists

(4,5) (2,3) (1,3) (3,5) (1,2) (2,4) (1,5) (3,4)1

2

3

4

5


Results on Graph Problems (1)

Many problems require (n) bits of space.

Graph distances (even approximation), Connectivity testing, Planarity testing …

Consider streaming algorithms that use O(n·polylogn) space and O(1) passes. In such a model, we can compute or approximate: Spanning trees Graph distances

[ Feigenbaum-S. Kannan-McGregor-Suri-Zhang ]



(1+,)-approximation: Our algorithm outputs {(u,v)} s.t. (u,v) (1+ ) distG(u,v) + , where distG(u,v) is the true distance between vertices u and v.

The algorithm uses O(n1+1/k) space. Processing time per edge is O(n1/k). Needs multiple passes. 1/k and are arbitrarily small parameters. and the

number of passes are functions of k and 1/.

[ Elkin-Zhang ]

We give a randomized streaming algorithm that approximates graph distances:



We give a one-pass, streaming algorithm for approximating graph distances. (2t+1)-approximation: (u,v) (2t+1)·distG(u,v) O(t·n1+1/t ·logn) space Processing time per edge: O(t

2·n1/t ·logn) Needs one pass.

Lower bound: The space complexity of one-pass, t-approximation is (n1+1/t).

[ Feigenbaum-S. Kannan-McGregor-Suri-Zhang ]

For t = log n, this gives a one-pass, O(logn)-approximation algorithm using n·polylog space and polylog time per edge.


Publications

J. Feigenbaum, S. Kannan, and J. Zhang, “Computing Diameter in the Streaming and Sliding-Window Models,” Algorithmica 41 (2005), pp. 25-41

J. Feigenbaum, S. Kannan, A. McGregor, S. Suri, and J. Zhang, “On Graph Problems in a Semi-Streaming Model,” ICALP 2004, pp. 531-543. Journal version to appear in Theoretical Computer Science.

M. Elkin and J. Zhang, “Efficient Algorithms for Constructing (1+ε,β)-Spanners in the Distributed and Streaming Models,” PODC 2004, pp. 160-168

J. Feigenbaum, S. Kannan, A. McGregor, S. Suri, and J. Zhang, “Graph Distances in the Streaming Model: The Value of Space,” SODA 2005, pp 745-754


Other Results in Thesis

Streaming-space requirement can be reduced by annotating the stream.

J. Feigenbaum, S. Kannan, and J. Zhang, “Annotation and Computational Geometry in the Streaming Model,” Yale University Technical Report YALEU/DCS/TR-1249, 2003

Using streaming algorithms to detect BGP-update anomalies.

J. Zhang, J. Rexford, and J. Feigenbaum, “Learning-Based Anomaly Detection in BGP Updates,” to appear in SIGCOMM Workshop on Mining Network Data 2005


Talk Outline


Overview of results




Shortest-Path Distances

Distance is the length of the shortest path.

Fundamental problem in graph theory

Many algorithms and approximations

Most of them use BFS-like subroutines, which are hard to adapt to the streaming model.


The “Sketch” Approach

A two-stage approach First stage: While going through the stream,

construct a small sketch of the input graph. Second stage: Compute the distance using the

sketch, without further access to the stream. Perform BFS-like computations in the second

stage.


Graph Spanners as Sketches

Edge subgraph H of a graph G, s.t., for any pair of vertices u and v, their distance in H, distH(u,v), is not far from their distance in G, distG(u,v).

Multiplicative spanner [t-Spanner]: distH(u,v) t·distG(u,v). Spanners are sparse. A t-Spanner has O(n1+1/t) edges. Reduce streaming graph distance to streaming spanner

construction. BFS-like subroutines are used in most existing spanner

constructions.


Streaming Spanner Construction

For each incoming edge, decide whether it should be in the spanner.

If the edge causes a cycle of length t, do not put the edge in the spanner.

This gives a t-spanner, because there is a path P of length < t connecting the two endpoints of any discarded edge.

This spanner is sparse. Thm [Bollobás78] : A graph whose girth is larger than k can only

have O(n1+2/(k-1)) edges. Need to know: For an incoming edge, does the path P

exist?


Partial Solution: Clusters (1)

A cluster is a subset of vertices and a small diameter spanning tree built on these vertices.

Intra-cluster edge


Partial Solution: Clusters (2)

Inter-cluster edges

Bollobás’s result no longer applies. Need to control the number of clusters (i.e., make it ).


Summary of the One-Pass Algorithm

Use a vertex-labeling scheme to construct the clusters. Structure of the algorithm:

In the pre-processing phase, generate a multi-level set of labels. Go through the stream; for each edge:

According to the current assignment of labels to vertices, decide whether to put this edge in the spanner.

Depending on the type of edge, possibly assign more labels to one of its endpoints.

Next, an example with t = log n


Labels

logn/2 levels w.h.p., there are top-level labels. Semantics of labels:

The set of vertices assigned the same top-level label forms a cluster.

The set of vertices assigned the same lower-level label forms a “pre-cluster.”

(0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9)(0,10) (0,11) (0,12)

(1,2) (1,4) (1,7) (1,11)

(2,2) (2,7)

Level 0

Level 1

Level 2

(0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9) (0,10) (0,11) (0,12)

(1,2) (1,4) (1,7) (1,11)


Initial Label Assignment

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12

(0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (0,8) (0,9) (0,10)(0,11) (0,12)

(1,2) (1,4) (1,7) (1,11)

(2,2) (2,7)

Level 0

Level 1

Level 2


On arrival of an edge

Already know what to do with: Intra-cluster/pre-cluster edges Inter-cluster edges

Edges connecting pre-clusters: the sticky edges They are added to the spanner. They may lead to new label assignment and cluster

growth.


“Good” Neighbor (1)

(3,2)

(2,2)

(1,2)

(0,2)

(1,6)

(0,6)

(2,2)

(3,2)

v u

Has marked labels


Good Neighbor (2)

v uC(1,2)

C(2,2)

C(3,2)

C(1,6)


“Bad” Neighbor

(3,2)(1,6)

v u

No marked labels


Properties of the Clusters

Small diameter

Number of clusters bounded by .

Do not need to cover the whole graph with clusters, but the uncovered subgraph is sparse.

The uncovered subgraph consists of sticky edges, and there are not too many of them.


Sticky Edges are Rare

u1

u2

u3

u4

v u1, u2, u3, u4 …

A neighbor is good with probability at least ½. After seeing at most logn/2 good neighbors, v will be assigned a top-

level label and be included in a cluster. No more sticky edges for v. The number of sticky edges can be bounded by the length of the

shortest prefix in the above sequence that contains logn/2 good neighbors.


Talk Outline


Overview of results




Summary

We investigated two important problem domains. Exact computation is hard; approximation may

be feasible. For some problems, particularly graph problems,

considering a more general model is important, because polylog space is too restrictive.

Constructing a sketch of non-numerical input is an important tool in streaming-algorithm design.


Future Research Directions

Geometric problems: High-dimensional geometric problems Sliding-window with flexible size

Graph problems: Dynamic graph problems

massive data streams in graph theory and computational geometry ph.d. dissertation defense jian...

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