1 / 36

Undirected ST-Connectivity in Log-Space

Undirected ST-Connectivity in Log-Space. By Omer Reingold (Weizmann Institute) Year 2004 Presented by Maor Mishkin. s. t. ST-Connectivity Problem. Given an Undirected Graph and given two Vertices in the Graph, s and t ,

minjonet-roussel
Download Presentation

Undirected ST-Connectivity in Log-Space

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Undirected ST-Connectivity in Log-Space By Omer Reingold (Weizmann Institute) Year 2004 Presented by Maor Mishkin

  2. s t ST-Connectivity Problem • Given an Undirected Graph and given two Vertices in the Graph,s and t, We need to return “true” if s and t are connected and to return “false” if they are not connected. TRUE FALSE

  3. ST-Connectivity Problem • Is called USTCONor MazeProblem. • The STCON problem is on a Directed Graph. Exit t Entrance s

  4. ST-Connectivity Problem • Related problem (Cook late 70’s)- Universal TraversalProblem, where we need to also return the path, which connects s and t– Not the focus of this lecture. s t

  5. Basic Search Algorithms • Breadth First Search (BFS) and Depth First Search (DFS) solve the problem in directed graphs (denoted STCON), therefore will solve in Undirected also. • Let N be the input Graph size. • Time O(N) -> this is also the lower Time bound for USTCON problem. • Space O(N).

  6. Space vs. Time • Algorithms have central resources of computation, Space & Time. • Space is the work memory –not including Input memory. • Trade-Off example: Given a bit array, calculate the parity bit (XOR over all the bits) in Time O(1). • Algorithm: Pre-Calculation - Create an array in the size of 2^N and put in place j the parity bit of the bit array representation of j. Input:101101 Output:0 Input:100101 Output:1

  7. Log-Space Algorithms • Definition of a Log-Space algorithm: Given an input of size N, algorithm’s Spaceis O(logN). • Claim. If the algorithm is Log-Space & Deterministic => Time is polynomial. • Proof. Lets assume that it is not polynomial Time and is Log-Space, therefore will have differentSpace states. By looking on Space at time i (Si), we know that we will have Si & Sj that Si = Sj and i^=j (since#Time is bigger than #Space states)-> a contradiction to algorithm finishing the calculation. • That is, class LOG-SPACE is contained in P.

  8. Previous Related Work • STCON– J. Savitch (70), Log^2 Space & asuper polynomial Time. • USTCON– R. Aleliunas (79), RandomizedLog-Space. Use a pointer to current vertex and a counter. Randomly start finding a path from s to t, stop when counter hits a limit. * the Random walk has a one side error.

  9. Previous Related Work • USTCON– Massive work to solve the problem without Randomization, but still pseudo-randomized algorithms [AKS87, BNS89, Nis92b, INW94] – We will fully remove the Randomization factor. • Universal Traversal Sequence– Noam Nisan 92b, quasi-polynomialTime in Space Log^2.

  10. Previous Related Work • USTCON– [NSW89] improved Savitch(70) to Space Log^1.5, and not polynomial Time. • USTCON– [ATSWZ00] improved the previous one to Space Log^1.333, but still not polynomial Time –most Space efficient until this work.

  11. s t Approach • To solve theconnectivity problem between s & t, improvethe connectivity of every connected component in the Graph, that is: Transform the input Graph into a Graph, which has Logarithmic Diameter (with the same connected components). We also make it a Constant Degree one.

  12. Approach • Since the Graph will have Logarithmic Diameter, we can build all Logarithmiclengthpaths, starting from s, and to see if one visits t . • Since the Degree is Constant and does not depend on N, the number of such paths is (polynomial). We later explain how to execute this in Log-Space.

  13. Open issues • How can we enumerate paths on a Graph that is Constant Degree & Logarithmic Diameter Graph in Log-Space (relatively easy). • How can we Transform the input Graph into a Constant Degree & Logarithmic Diameter Graph (the mainissue).

  14. Powering • Definition. the k’th Power of G contains an edge between two vertices v & w, iff there exists a path of length k from v to w in G. • Repeatedly squaring the graph logarithmicnumber oftimes will turn G into a Logarithmic Diameter Graph. • Powering increases the Degree of the Graph and will not maintain the Graph as a Constant Degree one.

  15. Decreasing Degrees • Replacement Product - an operation with two Graphs, a D-regularGraphG with N vertices and a d-regular Graph H on Dvertices (with d<<D). • Each vertex v of G is replaced with a “copy”Hvof H. • Each of the d vertices of Hvis connected to its neighbors in Hvand also to one vertex in Hw, where (v,w) is one of the D edges going out of v in G. • The Degree of the Product Graph is d+1.

  16. ( ) ( ) È ³ + a N S S 1 S Expander Graphs • Definition. For a d-regular GraphG(V,E), |V|=N. • , , • VertexExpansionProperties give us that the Expander Graph will have Logarithmic Diameter.

  17. Expander Graphs • We can turn a Graph into an Expander by Squaring it Logarithmic Times. • Algebraic ExpansionProperties will give us a way to measure the Graph ExpansionProperties. • Introduced in the 1970’s. • Widely used for De-Randomization, Error Correction, CS theory.

  18. Not Damaging Logarithmic Diameter • It turns out that if H is a “good enough”Expander,the expansion properties of the Replacement Product are not worse by much than those of the original Graph. • Formal statements to this effect were proved by Reingold, Vadhan & Wigderson [RVW01] for the Replacement Product and for the Zig-Zag Product (to be described later).

  19. Informal USTCON algorithm 1. First turn the input Graph into a constant-degree, regular Graph with each connected component being non-bipartite (notReplacement Product). 2. The main transformation turns each connected component of the Graph, in a logarithmic number of phases, into an Expander (a logarithmic diameter) 2.1. Each phase starts by raising the current graph to some constant power and then reducing the degree back via a Replacement or Zig-Zag product, using a constant size Expander.

  20. Informal USTCON algorithm 3. Now solve USTCON on the resulting Graph that has Logarithmic Diameter & Constant Degree.

  21. Graph Representation • Adjacency Matrix - A way to represent a Graph G(V,E) –not the input graph –will be used for theoretic discussion only. • At entry (v,u) will have a non-negative integer that equals to the number of edges that go from vertex v to vertex u. • A Graph is undirected iff it’s adjacency matrix is symmetric. • A Graph is D-regular if the sum of entries in a row (and column) is D.

  22. Graph Representation • Given a D-regular Graph, we can assume that for vertex v, the edges are labeled 1…D, and we can talk about the i’th neighbor of v. • When taking a step from v to w, it may be useful to keep track of the edges traversed to get to w (rather then just remembering that we are now at w).

  23. Graph Representation • For a D-regular undirected graph G, let us define Rotation Map: RotG : [N]*[D] --> [N]*[D] is defined as RotG(v,i) = (w,j) if the i’th edge incident to v leads to w, and this is the j’th edge incident to w. • Rotation map will be the input Graph representation at this work.

  24. Measure of Graph Expansion • We would like to make sure that our iterations will give us a “good”Expander, therefore we would like to measure our Graph Expansion. • ExpansionProperties can be calculated. we will look at the Normalized Adjacency MatrixMG of a D-regular Graph G, that is the AdjacencyMatrix of G divided by D. • Formally ->

  25. Eigenvalues and Eigenvectors • Eigenvalue - The factor by which alinear transformationmultiplies one of its Eigenvectors. • Eigenvector– For matrix M, vector x is an Eigenvector with Eigenvalueλ iff Mx= λ*x. • All one Eignvalue

  26. Graph Expansion Measurement • It turns out that the Eigenvalues of MG are at most 1. • We denote by λ(G) the second largest Eigenvalue of MG(in the absolute value) • It is known that λ(G)is a good measure of the Expansion property of G.[alpha vs. lambda] • We refer to a D-regular undirected Graph G with Nvertices such that λ(G)< λ as an (N,D,λ)-graph.

  27. Solving USTCON given a Constant Degree Expander. • USTCON in Constant Degree Expanders can be solved in Log-Space: • Let λ <1 be some constant, then there exists an O(logD*logN) Space algorithm A, such that when a D-regular undirected Graph G with N vertices is given to A as an input the following holds: • If s & t are in the same connected component & this component is an (N’,D, λ)-Graph then A outputs ‘connected’. • If A outputs connected then s & t are indeed in the same connected component.

  28. Solving USTCON given a Constant Degree Expander (cont.) • The algorithm A simply enumerates all D^lpaths of length l=O(logN) froms, where the leading constant in the big-O notation depends on λ. The algorithm A outputs ‘connected’ iff at least one of these paths encounter t. • Following any path from s with length l requires O(logN) Space. • Enumerating all D^l paths requires O(logD*logN) Space. When D is a constant we get O(logN) Space.

  29. s t logd logN Solving USTCON given a Constant Degree Expander (cont.) 0 2 1 3 1 3 2 0 0 0 2 3 1 1 1 3 2 0 2 0 2 3 1 1 3 2 0 0 1 2 0 1 0

  30. Powering & Expansion Measure • Our main transformation will take a graph and transform each one of its connected components into a Constant Degree Expander. If we ignore the constant degree requirement, a simple way of amplifying the Graph is by Powering. • Let G be a D-regular multi-graph with Nvertexes, given by rotation map RotG. The t’th power of G is the D^t-regular GraphG^t whose rotation map is given by Where these values are computed via the rule:

  31. Powering & Expansion Measure Lemma - If G is an (N,D,λ)-graph then G^t is an (N,D^t, λ^t)-graph. Proof– The normalized adjacency matrixMG^tof G^t is the t’th power of the Normalized Matrix MG of G, so all the Eigenvalues also get raised to the t’th power. MGtx = MGt-1MGx = MGt-1λx= λ MGt-1x = λtx

  32. Two Graph Products • Reminder - Replacement Product • Zig-Zag Product of G & H correspond to a subset of the paths of length three in the Replacement Product of these Graphs. • The degree of the output graph is d^2 (d^2<<D)

  33. Zig-Zag Product • Definition. If G is a D-regular Graph with N vertexes with rotation map RotG & H is a d-regular Graph with D vertexes with rotation map RotH, then their Zig-Zag ProductG H is defined to be the d^2-regular graph with N*D vertexes whose rotation map RotG H is as follows: • RotG H((v,a),(i,j)): 1. Let(a’,i’)=RotH (a,i) 2. Let(w,b’)=RotG(v,a’) 3. Let(b,j’)=RotH(b’,j) 4. Output((w,b),(j’,i’)) z z z

  34. Expansion Properties of G H z • Theorem. ([RVW01]) If G is an (N,D,λG)-graph & H is a (D,d, λH)-graph, then G H is a (N*D,d^2,f(λG, λH))-graph, where: • We get, that if is a “good”Expander (λH ->0) then f(λG, λH)-> λG z

  35. Universal Traversal & exploration sequence • The mentioned Algorithm also solves Universal Traversal (i.e. finding the path from s to t if such a path exist). • Every edge in the logarithmic long path of the final G H Graph is a sequence in G (original input) & can be followed by the “Rotation GraphLabeling” in the Zig-Zag Product. z

  36. Issues Summary • USTCON • Log-Space & Polynomial Time • Powering • Replacement Product • Expander Graphs • Zig-Zag Product

More Related