Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
An O(n ) Space and Polynomial Time Algorithm
for Reachability in Directed Layered Planar Graphs
Diptarka Chakraborty
Raghunath Tewari
Department of Computer Science & Engineering,
Indian Institute of Technology Kanpur
11th December, 2015
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Outline
1
Introduction
Problem Definition & Motivation
Previous Work
Main Result
2
Preliminaries
3
Proof Sketch of Main Result
Overall Idea
Solving LGGR
4
Conclusion
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Problem Definition & Motivation
Previous Work
Main Result
1
Introduction
Problem Definition & Motivation
Previous Work
Main Result
2
Preliminaries
3
Proof Sketch of Main Result
Overall Idea
Solving LGGR
4
Conclusion
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Problem Definition & Motivation
Previous Work
Main Result
Reachability Problem in Graphs
Computational Problem
Given a directed graph G and two fixed vertices (say s and t) in G ,
checking if there is a path from s to t in G .
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
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Problem Definition & Motivation
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Main Result
Reachability Problem in Graphs
Computational Problem
Given a directed graph G and two fixed vertices (say s and t) in G ,
checking if there is a path from s to t in G .
Different variations of the graph reachability problem characterize
various space bounded computational models.
Over directed (layered) graphs = NL(Nondeterministic logarithmic space)
Over undirected graphs
Certain other variation
= L (Deterministic logarithmic space)
[Reingold (2005)]
= RL (Randomized logarithmic space)
[Reingold, Trevisan, Vadhan (2005)]
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Introduction
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Problem Definition & Motivation
Previous Work
Main Result
Motivation
Definition (TISP(t(n), s(n)))
TISP(t(n), s(n)) is the class of problems decidable by a
deterministic Turing machine in O(t(n)) time and O(s(n)) space.
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
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Problem Definition & Motivation
Previous Work
Main Result
Motivation
Definition (TISP(t(n), s(n)))
TISP(t(n), s(n)) is the class of problems decidable by a
deterministic Turing machine in O(t(n)) time and O(s(n)) space.
?
Question: NL ⊆ SC = TISP(poly, polylog).
Diptarka Chakraborty, [email protected]
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Introduction
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Problem Definition & Motivation
Previous Work
Main Result
Motivation
Definition (TISP(t(n), s(n)))
TISP(t(n), s(n)) is the class of problems decidable by a
deterministic Turing machine in O(t(n)) time and O(s(n)) space.
?
Question: NL ⊆ SC = TISP(poly, polylog).
RL ⊆ SC. [Nisan (1995)]
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Time-Space Bound for LPGR
Introduction
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Problem Definition & Motivation
Previous Work
Main Result
Motivation
Definition (TISP(t(n), s(n)))
TISP(t(n), s(n)) is the class of problems decidable by a
deterministic Turing machine in O(t(n)) time and O(s(n)) space.
?
Question: NL ⊆ SC = TISP(poly, polylog).
RL ⊆ SC. [Nisan (1995)]
?
Question: NL ⊆ TISP(poly, n ), for any > 0.
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Problem Definition & Motivation
Previous Work
Main Result
Time-Space Complexity of Reachability
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Problem Definition & Motivation
Previous Work
Main Result
Time-Space Complexity of Reachability
- DFS/BFS solves reachability in linear time but they require
linear space as well.
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Introduction
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Proof Sketch of Main Result
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Problem Definition & Motivation
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Main Result
Time-Space Complexity of Reachability
- DFS/BFS solves reachability in linear time but they require
linear space as well.
- Savitch showed that reachability is in O(log2 n) space but his
algorithm requires nΩ(log n) time [Savitch (1970)].
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Problem Definition & Motivation
Previous Work
Main Result
Time-Space Complexity of Reachability
- DFS/BFS solves reachability in linear time but they require
linear space as well.
- Savitch showed that reachability is in O(log2 n) space but his
algorithm requires nΩ(log n) time [Savitch (1970)].
Question: Is reachability in TISP(poly, n1− )?
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
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Problem Definition & Motivation
Previous Work
Main Result
Time-Space Complexity of Reachability
Class of Directed
Graphs
Space Bound (simultaneously with polynomial time)
General Graphs
n/2θ( log n) [Barnes, Buss, Ruzzo, Scheiber
(1992)]
√
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Main Result
Time-Space Complexity of Reachability
Class of Directed
Graphs
General Graphs
Grid Graphs
Space Bound (simultaneously with polynomial time)
√
n/2θ( log n) [Barnes, Buss, Ruzzo, Scheiber
(1992)]
n1/2+ [Asano, Doerr (2011)]
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Problem Definition & Motivation
Previous Work
Main Result
Time-Space Complexity of Reachability
Class of Directed
Graphs
General Graphs
Grid Graphs
Planar Graphs
Space Bound (simultaneously with polynomial time)
√
n/2θ( log n) [Barnes, Buss, Ruzzo, Scheiber
(1992)]
n1/2+ [Asano, Doerr (2011)]
n1/2+ [Imai, Nakagawa, Pavan, Vinodchandran, Watanabe (2013)]
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Problem Definition & Motivation
Previous Work
Main Result
Time-Space Complexity of Reachability
Class of Directed
Graphs
General Graphs
Grid Graphs
Planar Graphs
H-minor-free
Graphs
Space Bound (simultaneously with polynomial time)
√
n/2θ( log n) [Barnes, Buss, Ruzzo, Scheiber
(1992)]
n1/2+ [Asano, Doerr (2011)]
n1/2+ [Imai, Nakagawa, Pavan, Vinodchandran, Watanabe (2013)]
n2/3+ [Chakraborty, Pavan, Vinodchandran, Tewari, Yang (2014)]
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Introduction
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Problem Definition & Motivation
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Main Result
Main Result
Theorem
For any > 0, the reachability problem over directed layered planar
graphs is in TISP(poly, n ).
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
1
Introduction
Problem Definition & Motivation
Previous Work
Main Result
2
Preliminaries
3
Proof Sketch of Main Result
Overall Idea
Solving LGGR
4
Conclusion
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
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Layered Planar Graph
Definition (Layered Planar Graph)
A graph G = (V , E ) is referred as layered planar if
it is possible to represent V as a union of disjoint partitions,
V = V1 ∪ V2 ∪ · · · ∪ Vk , for some k > 0,
there is no edge between two vertices of non-consecutive
partitions, and
there is a planar embedding of edges between the vertices of
any two consecutive partitions Vi and Vi+1 .
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Layered Grid Graph
Definition (Layered Grid Graph)
A directed graph G is said to be a n × n layered grid graph if
it can be drawn on a square grid of size n × n,
two vertices are neighbors if their L1 -distance is one, and
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Time-Space Bound for LPGR
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Layered Grid Graph
Definition (Layered Grid Graph)
A directed graph G is said to be a n × n layered grid graph if
it can be drawn on a square grid of size n × n,
two vertices are neighbors if their L1 -distance is one, and
we are allowed to have edges with only two directions, say
north and east.
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Layered Grid Graph
Figure : An example of layered grid graph G
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Complexity Class nSC
Definition (Complexity Class near-SC or nSC)
For a fixed > 0, we define nSC := TISP(nO(1) , n ). The
complexity class nSC is defined as
\
nSC :=
nSC .
>0
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
1
Introduction
Problem Definition & Motivation
Previous Work
Main Result
2
Preliminaries
3
Proof Sketch of Main Result
Overall Idea
Solving LGGR
4
Conclusion
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Proof Sketch
Theorem
If A ≤l B and B ∈ nSC, then A ∈ nSC.
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Proof Sketch
Theorem
If A ≤l B and B ∈ nSC, then A ∈ nSC.
Proposition ([Allender, Barrington, Chakraborty, Datta, Roy
(2009)])
Reachability problem in directed layered planar graphs is log-space
reducible to the reachability problem in layered grid graphs
(LGGR).
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Proof Sketch
Theorem
If A ≤l B and B ∈ nSC, then A ∈ nSC.
Proposition ([Allender, Barrington, Chakraborty, Datta, Roy
(2009)])
Reachability problem in directed layered planar graphs is log-space
reducible to the reachability problem in layered grid graphs
(LGGR).
Theorem
LGGR ∈ nSC.
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Time-Space Bound for LPGR
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Overall Idea
Solving LGGR
The Auxiliary Graph H
Divide G into k 2 many blocks (will be defined shortly) of
dimension n/k × n/k.
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
The Auxiliary Graph H
Divide G into k 2 many blocks (will be defined shortly) of
dimension n/k × n/k.
Include vertices on the border to V (H).
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
The Auxiliary Graph H
Divide G into k 2 many blocks (will be defined shortly) of
dimension n/k × n/k.
Include vertices on the border to V (H).
Add the edge (u, v ) to E (H) if and only if there is a path from
u to v within a block, unless they are on the same gridline.
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
The Auxiliary Graph H
Divide G into k 2 many blocks (will be defined shortly) of
dimension n/k × n/k.
Include vertices on the border to V (H).
Add the edge (u, v ) to E (H) if and only if there is a path from
u to v within a block, unless they are on the same gridline.
For any two corner vertices u and v , we add edge if and only
if there is a path from u to v within a block.
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
The Auxiliary Graph H
t
G7
G8
t
G9
Lh (3)
Lh (3)
G4
G5
Lv (2, 2)
G6
Lh (2)
D1 Lh (2, 2)
G1
s
G2
Lv (2)
G3
Lv (3)
(a)
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s
Lv (2)
Lv (3)
(b)
Time-Space Bound for LPGR
Lh (2)
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
The Auxiliary Graph H
Lemma
There is a path from s to t in G if and only if there is path from s
to t in the auxiliary graph H.
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
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Solving LGGR
The Auxiliary Graph H
Lemma
There is a path from s to t in G if and only if there is path from s
to t in the auxiliary graph H.
Lemma
Any path between s and t in H is of length 2k.
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Time-Space Bound for LPGR
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Overall Idea
Solving LGGR
Description of the Algorithm
Run a slightly modified DFS algorithm on H.
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Description of the Algorithm
Run a slightly modified DFS algorithm on H.
Why do we need linear space in normal DFS?
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Description of the Algorithm
Run a slightly modified DFS algorithm on H.
Why do we need linear space in normal DFS?
1
2
to remember the current path
to mark already visited vertices
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Description of the Algorithm
Run a slightly modified DFS algorithm on H.
Why do we need linear space in normal DFS?
1
2
to remember the current path
to mark already visited vertices
To remember the current path path takes O(k log n) space.
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Description of the Algorithm
Run a slightly modified DFS algorithm on H.
Why do we need linear space in normal DFS?
1
2
to remember the current path
to mark already visited vertices
To remember the current path path takes O(k log n) space.
Maintain two arrays of size k + 1 each, say Av and Ah , one for
vertical and the other for horizontal gridlines respectively.
For every i ∈ [k + 1], Av (i) is the topmost visited vertex in
Lv (i) and analogously Ah (i) is the leftmost visited vertex in
Lh (i).
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Proof of Correctness
z
w 00
Lh (j + 1)
x
Gl
Lv (i − 1)
w
w0
Lh (j)
Lv (i)
Figure : Crossing between two paths
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Time-Space Bound for LPGR
Introduction
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Complexity Analysis
Let T (n) and S(n) be the time and space complexity functions
respectively for layered grid graph of size n × n.
(
8n2 (T (n/k) + O(1)) if n > k
T (n) =
O(k 2 )
otherwise.
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Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Complexity Analysis
Let T (n) and S(n) be the time and space complexity functions
respectively for layered grid graph of size n × n.
(
8n2 (T (n/k) + O(1)) if n > k
T (n) =
O(k 2 )
otherwise.
log n 3
T (n) = O n log k .
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Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Complexity Analysis
Let T (n) and S(n) be the time and space complexity functions
respectively for layered grid graph of size n × n.
(
8n2 (T (n/k) + O(1)) if n > k
T (n) =
O(k 2 )
otherwise.
log n 3
T (n) = O n log k .
(
S(n/k) + O(k log n) if n > k
S(n) =
O(k 2 )
otherwise.
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Complexity Analysis
Let T (n) and S(n) be the time and space complexity functions
respectively for layered grid graph of size n × n.
(
8n2 (T (n/k) + O(1)) if n > k
T (n) =
O(k 2 )
otherwise.
log n 3
T (n) = O n log k .
(
S(n/k) + O(k log n) if n > k
S(n) =
O(k 2 )
otherwise.
S(n) = O
k
log k
log2 n + k 2 .
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Overall Idea
Solving LGGR
Complexity Analysis
Let T (n) and S(n) be the time and space complexity functions
respectively for layered grid graph of size n × n.
(
8n2 (T (n/k) + O(1)) if n > k
T (n) =
O(k 2 )
otherwise.
log n 3
T (n) = O n log k .
(
S(n/k) + O(k log n) if n > k
S(n) =
O(k 2 )
otherwise.
S(n) = O
k
log k
log2 n + k 2 . Set k = n/2 .
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
1
Introduction
Problem Definition & Motivation
Previous Work
Main Result
2
Preliminaries
3
Proof Sketch of Main Result
Overall Idea
Solving LGGR
4
Conclusion
Diptarka Chakraborty, [email protected]
Time-Space Bound for LPGR
Introduction
Preliminaries
Proof Sketch of Main Result
Conclusion
Questions
Can we extend the same bound for the class of planar graphs?
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
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Questions
Can we extend the same bound for the class of planar graphs?
Obtaining n1− space bound for general graphs.
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Time-Space Bound for LPGR
Introduction
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Proof Sketch of Main Result
Conclusion
THANK YOU!!!
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Time-Space Bound for LPGR