Detecting Negative Cost Cycles in Zero Space

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Detecting Negative Cost Cycles in Zero Space

• K. Subramani
• West Virginia University

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Topics to be covered

• Statement of Problem
• Motivation
• Related Work
• The Difference Constraint Approach
• The Stressing Algorithm
• Analysis (Space and Time)
• Visualizing the algorithm
• Correctness
• Differences between Stressing and Relaxing
• Producing the cycle
• Open Issues

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Statement of Problem
NCCD Given a network GltV,E,cgt, is there a
directed, simple cycle with cumulative edge cost
negative?
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Motivation (for studying NCCD)

• Shortest Paths
• Max Flows
• Image Segmentation
• Real-Time Scheduling
• System Verification

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Motivation (for our work)

• New Design Paradigms
• ? NCCD and Shortest Paths
• Space Savings
• Connectivity issues

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Related Work

• Relaxation-based approaches
• Contraction-based approaches
• Algebraic Methods

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The Difference Constraint Approach

• Def A constraint of the form
• Def A conjunction of difference constraints is
  called a
• difference constraint system
• Farkas Lemma.
• Main observation

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DCA (contd.)
Adding the non-positivity constraints makes the
constraint polyhedron pointed indeed there is a
unique optimal solution to the system
System (1)
(Proof in paper)
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DCA (contd.)

• If the RHS is positive, suffices.
• Pick the constraint , with
• Replace xi by xicij,
• throughout the constraint system.
• 4. oioicij

Analysis At most steps and the
algorithm can terminate.
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DCA (contd.)
The above algorithm terminates with the unique
solution to System (1), if the constraint system
is feasible.
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The Stressing Algorithm

• Repeat (n-1) times
• (a) for each vertex v in G
• if (v has an incoming negative edge)
• Stress(v)
• 2. If there still exists a negative cost arc in
  the graph
• declare G has a negative cost cycle
• else
• declare G does not have a negative cost
  cycle.

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Stressing Algorithm (contd.)
The Stress Function

• 1. Find the most negative edge into v
• 2. Subtract this cost from all the edges into v.
• 3. Add this cost to all the edges out of v

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Analysis (Space)
Clearly, only register space is needed to
determine the minimum and for loop indexing.
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Analysis (Time)

• O(m.n) on a network with m arcs and n vertices.

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Visualization
B
1
H
-3
C
-2
G
A
-3
3
3
-1
5
-4
D
F
-7
I
-4
5
J
E
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Visualization (contd.)
G is stressed
A is stressed
B
1
H
0
C
-3
G
A
-3
3
3
0
5
-7
D
F
-7
I
-4
5
J
E
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Visualization (contd.)
B is stressed
J is stressed
B
H
-2
0
C
0
G
A
-6
3
3
0
5
0
D
F
-7
I
-4
-2
J
E
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Visualization (contd.)
C is stressed
I is stressed
B
H
0
0
C
0
G
A
-6
1
5
0
3
0
D
F
-7
I
-4
0
J
E
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Visualization (contd.)
D is stressed
B
H
0
0
C
0
G
A
1
1
-2
0
3
0
D
F
0
I
-4
0
J
E
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Visualization (contd.)
E is stressed, F cannot be stressed first round
is over
B
H
0
0
C
0
G
A
1
1
-2
0
3
0
D
F
-4
I
0
0
J
E
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Visualization (contd.)
Round 2 begin by stressing C
B
H
2
0
C
0
G
A
1
1
0
0
1
0
D
F
-4
I
0
0
J
E
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Visualization (contd.)
Finally the negative cycle is isolated as a
negative edge
B
H
2
0
C
0
G
A
1
1
0
0
1
0
D
F
-4
I
0
0
J
E
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Correctness

• Observation The Stressing Algorithm is precisely
  the
• Difference Constraint Approach with two
  differences
• The Stressing Algorithm terminates after exactly
• (n-1) rounds.
• (b) When a vertex is stressed, we use the most
  negative
• incoming edge as the pivot.

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Correctness (contd.)
Theorem Stressing preserves costs around a cycle.
H
-1
G
1
0
I
0
J
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Correctness (contd.)

• .

Corollary Stressing does not create negative
cost cycles, if none existed in the first place.
Theorem If the graph does have a negative cost
cycle, the Stressing Algorithm declares that the
input graph has a negative cost cycle.
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Correctness (contd.)

• Theorem If the graph does not have a negative
  cost cycle, there will not be any negative cost
  edges in the input graph after (n-1) rounds.

Proof (sketch) Focus on the unique BFS of the
Difference Constraint system, which is a spanning
tree. Key Observation There is at least one
vertex which is never stressed at all. After each
round of stressing, one additional vertex is
never stressed again. (By induction)
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Differences between Bellman-Ford and Stressing

• Sourced versus sourceless
• Dynamic Programming versus Greedy
• Zero labeling space

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Producing the cycle

• Maintain a variable at each vertex.
• When a vertex is stressed store the
• label of the pivot edge!
• Total space used is O(n), which is
• strongly polynomial, unlike Bellman-Ford.

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Open Issues

• Implementation study
• Planar digraphs
• Parallelization

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The Beginning