Undirected Connectivity in Log Space - O. Reingold
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1 CS 640 Course Presentation Undirected Connectivity in Log Space - O. Reingold Keerti Choudhary
2 Contents Background Main Idea Expander Graphs Powering Replacement Product Zig-zag Product Main Transformation Implementation Bounding the degree Rotation Map
3 Background
4 Background SL is class of problems solvable by symmetric nondeterministic TM in log space. USTCON (Undirected s-t connectivity) was shown to be complete for class SL. L SL NL. This paper proves that USTCON can be solved in log space, thus proving L=SL.
5 Background SL is class of problems solvable by symmetric nondeterministic TM in log space. L SL NL.
6 Background SL is class of problems solvable by symmetric nondeterministic TM in log space. L SL NL. USTCON (Undirected s-t connectivity) was proved to be complete for class SL.
7 Background SL is class of problems solvable by symmetric nondeterministic TM in log space. L SL NL. USTCON (Undirected s-t connectivity) was proved to be complete for class SL. This paper proves that USTCON can be solved in log space, thus shows L=SL.
8 Main Idea
9 Main Idea Suppose a undirected graph G has max degree bounded by D and diameter bounded by.
10 Main Idea Suppose a undirected graph G has max degree bounded by D and diameter bounded by. Then s-t connectivity can be solved in space.
11 Main Idea Suppose a undirected graph G has max degree bounded by D and diameter bounded by. Then s-t connectivity can be solved in space. So, we will convert G to a new graph satisfying degree bounded by some constant c diameter is.
12 Expander Graphs
13 Expander Graphs A D-regular graph is -expander if any set S of size less than n/2 has closed neighbourhood of size at least (1+ ) S.
14 Expander Graphs A D-regular graph is -expander if any set S of size less than n/2 has closed neighbourhood of size at least (1+ ) S. Diameter of such graphs is bounded by.
15 Expander Graphs A D-regular graph is -expander if any set S of size less than n/2 has closed neighbourhood of size at least (1+ ) S. Diameter of such graphs is bounded by. Idea is to convert G to an expander graph whose degree is also bounded.
16 Expander Graphs
17 Expander Graphs Regular graphs with small second largest eigen value ( ) of normalised adjacency matrix.
18 Expander Graphs Regular graphs with small second largest eigen value ( ) of normalised adjacency matrix. For every D, ( <1), there exist >0 such that all D-regular graphs with second largest eigen value bounded by are -expander.
19 Expander Graphs Regular graphs with small second largest eigen value ( ) of normalised adjacency matrix. For every D, ( <1), there exist >0 such that all D-regular graphs with second largest eigen value bounded by are -expander. To convert G to an expander graph we need to bound its.
20 Powering
21 Powering Squaring of the graph reduces the second largest eigen value to. As, so squaring the graph times would bound. But the cost is that degree will also be squared. Note - squaring should not affect connectivity, so before powering we add a loop on on all the vertices.
22 Powering Squaring of the graph reduces the second largest eigen value to. As, so squaring the graph times would bound.
23 Powering Squaring of the graph reduces the second largest eigen value to. As, so squaring the graph times would bound. But the cost is that degree will also be squared.
24 Bounding the degree
25 Two methods - Bounding the degree
26 Bounding the degree Two methods - 1. Replacement Product
27 Bounding the degree Two methods - 1. Replacement Product 2. Zig-zag Product
28 Rotation Map In graph G, iff, u v vertex in adjacency list of u is v, and vertex in adjacency list of v is u.
29 Replacement Product
30 Replacement Product Given a D-regular graph G, and a d-regular (d<<d) connected graph H on D vertices. Replace each vertex of G by H.
31 Replacement Product u v
32 Replacement Product u v
33 Replacement Product Given a D-regular graph G, and a d-regular (d<<d) connected graph H on D vertices. Replace each vertex of G by H. New graph is a d+1 regular.
34 Zig-zag Product
35 Zig-zag Product (a,d) is an edge in replacement product satisfying - iff a b c d is a path in
36 Zig-zag Product (a,d) is an edge in replacement product satisfying - iff a b c d is a path in (a,b) and (c,d) are edges in different instances of H
37 Zig-zag Product (a,d) is an edge in replacement product satisfying - iff a b c d is a path in (a,b) and (c,d) are edges in different instances of H (b,c) corresponds to edge of G.
38 Zig-zag Product (a,d) is an edge in replacement product satisfying - iff a b c d is a path in (a,b) and (c,d) are edges in different instances of H (b,c) corresponds to edge of G. a b a c d d
39 Zig-zag Product
40 Zig-zag Product So, if G is D-regular and H is d-regular than, is -regular.
41 Zig-zag Product So, if G is D-regular and H is d-regular than, is -regular. THEOREM - If G is an -graph and H is a aaaa -graph, then
42 Zig-zag Product So, if G is D-regular and H is d-regular than, is -regular. THEOREM - If G is an -graph and H is a aaaa -graph, then! Here of the new graph is bounded above.
43 Main Transformation
44 Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,.
45 Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,. Let. For i=1 to, define
46 Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,. Let. For i=1 to, define Then,. Proof Idea : - -
47 Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,. Let. For i=1 to, define Then,. Proof Idea : - -
48 Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,. Let. For i=1 to, define Then,. Proof Idea : - -
49 Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,. Let. For i=1 to, define Then,. Proof Idea : - -
50 Main Transformation Converting G to regular graph of size N= : Each vertex ( ) is replaced by a n-cycle ( ). If, then is an edge in. Multiple loops are added to each vertex in until its degree reaches. H is of constant size and can be encoded in TM.
51 Main Transformation Converting G to regular graph of size N= : Each vertex ( ) is replaced by a n-cycle ( ). If, then is an edge in. Multiple loops are added to each vertex in until its degree reaches. H is of constant size and can be encoded in TM.
52 Main Transformation Converting G to regular graph of size N= : Each vertex ( ) is replaced by a n-cycle ( ). If, then is an edge in. Multiple loops are added to each vertex in until its degree reaches. H is of constant size and can be encoded in TM.
53 Main Transformation Converting G to regular graph of size N= : Each vertex ( ) is replaced by a n-cycle ( ). If, then is an edge in. Multiple loops are added to each vertex in until its degree reaches. H is of constant size and can be encoded in TM.
54 Main Transformation Converting G to regular graph of size N= : Each vertex ( ) is replaced by a n-cycle ( ). If, then is an edge in. Multiple loops are added to each vertex in until its degree reaches. H is of constant size and can be encoded in TM.
55 Implementation
56 Simpler Case : Implementation
57 Implementation Simpler Case : Another copy of
58 Implementation Simpler Case : Another copy of Replacement product
59 Another copy of Replacement product Now, we want to compute its neighbour and,
60 Another copy of Replacement product Now, we want to compute its neighbour and,
61 Another copy of Replacement product Now, we want to compute its neighbour and,
62 Another copy of Replacement product Now, we want to compute its neighbour and,
63 Another copy of Replacement product Now, we want to compute its neighbour and,
64 Another copy of Replacement product Now, we want to compute its neighbour and,
65 Another copy of Replacement product Now, we want to compute its neighbour and,
66 Implementation
67 Implementation will be of size, and any vertex
68 Implementation will be of size, and any vertex! We need to compute the rotation map in log space.
69 Main Algorithm Implementation
70 THANK YOU
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