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Geometric Graphs
Sathish Govindarajan
Department of Computer Science and AutomationIndian Institute of Science, Bangalore
CSA Undergraduate Summer School, 2013
Sathish Govindarajan (CSA, IISc) Geometric Graphs 1 / 27
Geometric Graph
v
u
x
y
z
w
Sathish Govindarajan (CSA, IISc) Geometric Graphs 2 / 27
Geometric Graph
v
u
x
y
z
w
V = set of geometric objects
Sathish Govindarajan (CSA, IISc) Geometric Graphs 2 / 27
Geometric Graph
v
u
x
y
z
w
V = set of geometric objects
E = {(u, v)} based on some geometric condition (ex. intersection)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 2 / 27
Questions on Geometric Graphs
Problems on graphs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 3 / 27
Questions on Geometric Graphs
Problems on graphsIndependent set, Coloring, Clique, etc.
Sathish Govindarajan (CSA, IISc) Geometric Graphs 3 / 27
Questions on Geometric Graphs
Problems on graphsIndependent set, Coloring, Clique, etc.
Combinatorial/Structural questions
Sathish Govindarajan (CSA, IISc) Geometric Graphs 3 / 27
Questions on Geometric Graphs
Problems on graphsIndependent set, Coloring, Clique, etc.
Combinatorial/Structural questionsObtain BoundsCharacterization
Sathish Govindarajan (CSA, IISc) Geometric Graphs 3 / 27
Questions on Geometric Graphs
Problems on graphsIndependent set, Coloring, Clique, etc.
Combinatorial/Structural questionsObtain BoundsCharacterization
Computational questions
Sathish Govindarajan (CSA, IISc) Geometric Graphs 3 / 27
Questions on Geometric Graphs
Problems on graphsIndependent set, Coloring, Clique, etc.
Combinatorial/Structural questionsObtain BoundsCharacterization
Computational questionsEfficient AlgorithmApproximation
Sathish Govindarajan (CSA, IISc) Geometric Graphs 3 / 27
Geometric graphs
V - set of geometric objects
E - object i and j satisfy certain geometric condition
Broad classes of geometric graphs (based on edge condition)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 4 / 27
Geometric graphs
V - set of geometric objects
E - object i and j satisfy certain geometric condition
Broad classes of geometric graphs (based on edge condition)
Intersection graphs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 4 / 27
Geometric graphs
V - set of geometric objects
E - object i and j satisfy certain geometric condition
Broad classes of geometric graphs (based on edge condition)
Intersection graphs
Proximity graphs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 4 / 27
Geometric graphs
V - set of geometric objects
E - object i and j satisfy certain geometric condition
Broad classes of geometric graphs (based on edge condition)
Intersection graphs
Proximity graphs
Distance based graphs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 4 / 27
Intersection Graphs
Interval Graph - Classic example
S - set of geometric objects si (intervals on the real line)
a b c
ed
f
Sathish Govindarajan (CSA, IISc) Geometric Graphs 5 / 27
Intersection Graphs
Interval Graph - Classic example
S - set of geometric objects si (intervals on the real line)
a b c
ed
f
V - set of objects si
Sathish Govindarajan (CSA, IISc) Geometric Graphs 5 / 27
Intersection Graphs
Interval Graph - Classic example
S - set of geometric objects si (intervals on the real line)
a b c
ed
f
V - set of objects si
(si , sj ) ∈ E if objects si and sj intersect
Sathish Govindarajan (CSA, IISc) Geometric Graphs 5 / 27
Interval Graphs
S - set of intervals on the line
a b c
ed
f
a b c
ed
f
Sathish Govindarajan (CSA, IISc) Geometric Graphs 6 / 27
Interval Graphs
S - set of intervals on the line
a b c
ed
f
a b c
ed
f
V - set of intervals si
Sathish Govindarajan (CSA, IISc) Geometric Graphs 6 / 27
Interval Graphs
S - set of intervals on the line
a b c
ed
f
a b c
ed
f
V - set of intervals si
(si , sj ) ∈ E if intervals si and sj intersect
Sathish Govindarajan (CSA, IISc) Geometric Graphs 6 / 27
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Sathish Govindarajan (CSA, IISc) Geometric Graphs 7 / 27
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:
Sathish Govindarajan (CSA, IISc) Geometric Graphs 7 / 27
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 7 / 27
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 7 / 27
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 7 / 27
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)Connection between this problem and interval graphs?
Sathish Govindarajan (CSA, IISc) Geometric Graphs 7 / 27
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)Connection between this problem and interval graphs?Maximum independent set in Interval graph
Sathish Govindarajan (CSA, IISc) Geometric Graphs 7 / 27
Applications of Interval Graphs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 8 / 27
Applications of Interval Graphs
Classroom allocation problem:
Sathish Govindarajan (CSA, IISc) Geometric Graphs 8 / 27
Applications of Interval Graphs
Classroom allocation problem:Classes in a university:(6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 8 / 27
Applications of Interval Graphs
Classroom allocation problem:Classes in a university:(6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Decide how many classrooms are needed to holds all these classes
Sathish Govindarajan (CSA, IISc) Geometric Graphs 8 / 27
Applications of Interval Graphs
Classroom allocation problem:Classes in a university:(6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Decide how many classrooms are needed to holds all these classesConnection between this problem and interval graphs?
Sathish Govindarajan (CSA, IISc) Geometric Graphs 8 / 27
Applications of Interval Graphs
Classroom allocation problem:Classes in a university:(6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Decide how many classrooms are needed to holds all these classesConnection between this problem and interval graphs?Minimum Coloring in Interval graph
Sathish Govindarajan (CSA, IISc) Geometric Graphs 8 / 27
Intervals
S - set of intervals on the real line
Suppose every 2 intervals in S intersect
Sathish Govindarajan (CSA, IISc) Geometric Graphs 9 / 27
Intervals
S - set of intervals on the real line
Suppose every 2 intervals in S intersect
Claim: All the intervals have a common intersection
Sathish Govindarajan (CSA, IISc) Geometric Graphs 9 / 27
Intervals
S - set of intervals on the real line
Suppose every 2 intervals in S intersect
Claim: All the intervals have a common intersection
Sathish Govindarajan (CSA, IISc) Geometric Graphs 10 / 27
Intervals
S - set of intervals on the real line
Suppose every 2 intervals in S intersect
Claim: All the intervals have a common intersection
Constructive/Extremal proofConstruct a point p that is contained in all the intervals
Sathish Govindarajan (CSA, IISc) Geometric Graphs 10 / 27
Intervals
S - set of intervals on the real line
Every 2 intervals intersect
Extremal proofConstruct a point p that is contained in all the intervals
Sathish Govindarajan (CSA, IISc) Geometric Graphs 11 / 27
Intervals
S - set of intervals on the real line
Every 2 intervals intersect
Extremal proofConstruct a point p that is contained in all the intervals
p : Leftmost right endpoint
Sathish Govindarajan (CSA, IISc) Geometric Graphs 11 / 27
Intervals
S - set of intervals on the real line
Every 2 intervals intersect
Extremal proofConstruct a point p that is contained in all the intervals
p : Leftmost right endpoint
Claim: All the intervals contain p
Sathish Govindarajan (CSA, IISc) Geometric Graphs 11 / 27
Intervals
Construct a point p that is contained in all the intervals
p : Leftmost right endpoint
Claim: All the intervals contain p
Sathish Govindarajan (CSA, IISc) Geometric Graphs 12 / 27
Intervals
Construct a point p that is contained in all the intervals
p : Leftmost right endpoint
Claim: All the intervals contain p
Proof by contradiction
Sathish Govindarajan (CSA, IISc) Geometric Graphs 12 / 27
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Sathish Govindarajan (CSA, IISc) Geometric Graphs 13 / 27
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
We proved this theorem for d = 1
Sathish Govindarajan (CSA, IISc) Geometric Graphs 13 / 27
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
We proved this theorem for d = 1
Proof using different approachesRadon’s theoremInductionShrinking ball techniqueBrouwer’s theoremConstructive/Extremal proof
Sathish Govindarajan (CSA, IISc) Geometric Graphs 13 / 27
Interval Graph Problems
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graph
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctness
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctnessExtension: What if jobs have different profits?
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctnessExtension: What if jobs have different profits?
Classroom allocation problem:
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctnessExtension: What if jobs have different profits?
Classroom allocation problem:Given a set of n classes in a university ci = (si , fi)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctnessExtension: What if jobs have different profits?
Classroom allocation problem:Given a set of n classes in a university ci = (si , fi)Decide how many classrooms are needed to holds all these classes
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctnessExtension: What if jobs have different profits?
Classroom allocation problem:Given a set of n classes in a university ci = (si , fi)Decide how many classrooms are needed to holds all these classesMinimum Coloring in Interval graph
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Interval Graph Problems
Consultant problem:Given a set of n jobs ji = (si , fi)Choose the maximum number of (non-conflicting) jobsMaximum independent set in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctnessExtension: What if jobs have different profits?
Classroom allocation problem:Given a set of n classes in a university ci = (si , fi)Decide how many classrooms are needed to holds all these classesMinimum Coloring in Interval graphAlgorithm to solve the problem (Exercise)with Proof of correctness
Sathish Govindarajan (CSA, IISc) Geometric Graphs 14 / 27
Proximity Graphs
P - point set in plane
Ri ,j - proximity region defined by i and j
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V - point set P
(i , j) ∈ E if Ri ,j is empty
Examples - Delaunay, Gabriel, Relative Neighborhood Graph
Applications - Graphics, wireless networks, GIS, computer vision
Sathish Govindarajan (CSA, IISc) Geometric Graphs 15 / 27
Delaunay Graph - Classic Example
P - point set in plane
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V - point set P
(i , j) ∈ E if ∃ some empty circle thro’ i and j
Triangle (i , j , k) if circumcircle(i , j , k) is empty(Equivalent condition)
Applications - Graphics, mesh generation, computer vision, etc.
Sathish Govindarajan (CSA, IISc) Geometric Graphs 16 / 27
Questions on Delaunay Graph
Combinatorial - Bounds on
Maximum size of edge set?Chromatic number?Maximum independent set?
(Over all possible point sets P)
ComputationalEfficient Algorithm
Sathish Govindarajan (CSA, IISc) Geometric Graphs 17 / 27
Delaunay Graph - Classic Example
P - point set in plane
Observations:
Sathish Govindarajan (CSA, IISc) Geometric Graphs 18 / 27
Delaunay Graph - Classic Example
P - point set in plane
Observations: Planar?
Sathish Govindarajan (CSA, IISc) Geometric Graphs 19 / 27
Delaunay Graph - Planar
Let, if possible, 2 edges cross
Sathish Govindarajan (CSA, IISc) Geometric Graphs 20 / 27
Delaunay Graph - Planar
Let, if possible, 2 edges cross
Sathish Govindarajan (CSA, IISc) Geometric Graphs 21 / 27
Delaunay Graph - Planar
Let, if possible, 2 edges cross
Sathish Govindarajan (CSA, IISc) Geometric Graphs 22 / 27
Delaunay Graph - Planar
Let, if possible, 2 edges cross
Circles c’ant intersect like this (why?)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 23 / 27
Delaunay Graph - Planar
Let, if possible, 2 edges cross
Circles c’ant intersect like this (why?)One endpoint of an edge lies within the other circle
Contradiction
Alternate proof using angles
Sathish Govindarajan (CSA, IISc) Geometric Graphs 24 / 27
Questions on Delaunay Graph
Given any n-point set P in the planeDelaunay graph is planar
Maximum size of edge set
Sathish Govindarajan (CSA, IISc) Geometric Graphs 25 / 27
Questions on Delaunay Graph
Given any n-point set P in the planeDelaunay graph is planar
Maximum size of edge set≤ 3n − 6 (Euler’s formula)
Chromatic number
Sathish Govindarajan (CSA, IISc) Geometric Graphs 25 / 27
Questions on Delaunay Graph
Given any n-point set P in the planeDelaunay graph is planar
Maximum size of edge set≤ 3n − 6 (Euler’s formula)
Chromatic number≤ 4 (Four color theorem)
Maximum independent set
Sathish Govindarajan (CSA, IISc) Geometric Graphs 25 / 27
Questions on Delaunay Graph
Given any n-point set P in the planeDelaunay graph is planar
Maximum size of edge set≤ 3n − 6 (Euler’s formula)
Chromatic number≤ 4 (Four color theorem)
Maximum independent set≥ n/4 (Chromatic number)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 25 / 27
Open Problem
Maximum independent set on Delaunay graph?
Sathish Govindarajan (CSA, IISc) Geometric Graphs 26 / 27
Open Problem
Maximum independent set on Delaunay graph?Is there a polynomial time algorithm (or is it NP-complete)
Sathish Govindarajan (CSA, IISc) Geometric Graphs 26 / 27
Open Problem
Maximum independent set on Delaunay graph?Is there a polynomial time algorithm (or is it NP-complete)
Related results
Sathish Govindarajan (CSA, IISc) Geometric Graphs 26 / 27
Open Problem
Maximum independent set on Delaunay graph?Is there a polynomial time algorithm (or is it NP-complete)
Related resultsSolvable for the special case when all points lie on boundary
Sathish Govindarajan (CSA, IISc) Geometric Graphs 26 / 27
Open Problem
Maximum independent set on Delaunay graph?Is there a polynomial time algorithm (or is it NP-complete)
Related resultsSolvable for the special case when all points lie on boundaryNP-complete for planar graphs
Sathish Govindarajan (CSA, IISc) Geometric Graphs 26 / 27
Questions?
Sathish Govindarajan (CSA, IISc) Geometric Graphs 27 / 27
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