Iterative reduced DP search

1
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Iterative reduced DP search
• for polygonal approximation
• of digital curves

2
Min-? problem

• Approximate the given open N-vertex polygonal
  curve
• by another one consisting of at most M line
  segments
• with minimum error E
• E ? min subject to M Const

3
Fidelity vs Complexity
Algorithms Fidelity
Complexity Optimal 100
O(N2) O(N3) Heuristic 50
O(N) O(N2) Example N5000, M300 Optimal
(Dynamic Programming) 30 min Heuristic
(Douglas-Peucker) 1 s
Fidelity F (Eopt/E) 100
4
Paradigm of bounding corridor
1. Find rough approximation with any fast
heuristic algorithm 2. Construct a bounding
corridor in the state space along the path 3.
Apply DP search to the bounding corridor
5
Example of bounding corridor
W10
M
M300, W10 RS F
99.9, T 2 s FS F100, T 2000 s
6
Time and space complexity of IRS

• Processing time T kN2W2/M
• Time complexity O(N) ? O(N2)
• Space complexity O(NW)
• Speed-up for one iteration TRSDP/TDP (W/M)2
• Example
• If (W/M)2 (10/300)2 then (30 min/900) 2
  s

7
Example of approximation
N 5000, M 300
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Iterative Reduced Search algorithm

• Find a reference solution by any fast algorithm
• REPEAT
• Constuct bounding corrridor along the reference
  path
• Apply dynamic programming inside the bounding
  corridor
• UNTIL good enough

9
Results

• N5000, M300,
• Algorithm Time Fidelity,
• Heuristic (Douglas-Peucker) 1.0 s 50
• Fast near-optimal
• W 6 10.8 s 98.9
• W 8 11.2 s 99.7
• W 6 8 12.0 s 100
• Fast optimal 14.4 s 100
• Optimal 30 min 100
• 
  

10
Conclusions

• A fast near-optimal algorithm is developed
• a) Fidelity F ?? 100
• b) Time complexity O(N) ? O(N2)
• Processing time T k W2 N2/ M
• c) Space complexity O(NW)
• The trade-off between the run time and optimality
  is regulated by the corridor width and the number
  of iterations.