CS 640 Course Presentation Undirected Connectivity in Log Space - O. Reingold Keerti Choudhary
Contents Background Main Idea Expander Graphs Powering Replacement Product Zig-zag Product Main Transformation Implementation Bounding the degree Rotation Map
Background
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.
Background SL is class of problems solvable by symmetric nondeterministic TM in log space. L SL NL.
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.
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.
Main Idea
Main Idea Suppose a undirected graph G has max degree bounded by D and diameter bounded by.
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.
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.
Expander Graphs
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.
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.
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.
Expander Graphs
Expander Graphs Regular graphs with small second largest eigen value ( ) of normalised adjacency matrix.
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.
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.
Powering
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.
Powering Squaring of the graph reduces the second largest eigen value to. As, so squaring the graph times would bound.
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.
Bounding the degree
Two methods - Bounding the degree
Bounding the degree Two methods - 1. Replacement Product
Bounding the degree Two methods - 1. Replacement Product 2. Zig-zag Product
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.
Replacement Product
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.
Replacement Product u v
Replacement Product u v
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.
Zig-zag Product
Zig-zag Product (a,d) is an edge in replacement product satisfying - iff a b c d is a path in
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
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.
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
Zig-zag Product
Zig-zag Product So, if G is D-regular and H is d-regular than, is -regular.
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
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.
Main Transformation
Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,.
Main Transformation is non-bipartite -regular graph on N vertices, H is D-regular graph on vertices. Also,. Let. For i=1 to, define
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 : - -
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 : - -
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 : - -
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 : - -
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.
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.
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.
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.
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.
Implementation
Simpler Case : Implementation
Implementation Simpler Case : Another copy of
Implementation Simpler Case : Another copy of Replacement product
Another copy of Replacement product Now, we want to compute its neighbour and,
Another copy of Replacement product Now, we want to compute its neighbour and,
Another copy of Replacement product Now, we want to compute its neighbour and,
Another copy of Replacement product Now, we want to compute its neighbour and,
Another copy of Replacement product Now, we want to compute its neighbour and,
Another copy of Replacement product Now, we want to compute its neighbour and,
Another copy of Replacement product Now, we want to compute its neighbour and,
Implementation
Implementation will be of size, and any vertex
Implementation will be of size, and any vertex! We need to compute the rotation map in log space.
Main Algorithm Implementation
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