Most uniform path partitioning and its use in image processing

1. Most uniform path partitioning and its use in image processing Mario Lucertini, Yehoshua Perl, Bruno Simeone, Discrete Applied Mathematics 42(1993)227-256 報告人 王弘倫

2. Introduction • Most uniform path partitioning • (L, U)-partitioning • Minimum (L, U)-partitioning • Maximum (L, U)-partitioning • (L, U)-partitioning into p components Each can be solved in O(n), except MUP.

3. A greedy method Minimum (9, 13)-partitioning 11 2 4 4 5 6 11 11 2 4 4 5 6 11 illegal edge 11 2 4 4 5 6 11

4. Preprocessing 4-5-3-2-6-4-3-5-2-1-1-1-1-5-6-11 U=13, L=9 1.總重=11, Q=(11) 2.總重=17>U, Q=(11,6) 總重=17-11=6<L 6需與前一點合併 3.總重=11, Q=(11(6+5)) 4.總重=12, Q=(11,1) 5.總重=13, Q=(11,1,1) 11

5. Preprocessing(2) The result of preprocessing: 4-8-2-10-3-9-1-1-11-11 4-5-3-2-6-4-3-5-2-1-1-1-1-5-6-11 After the preprocessing, the greedy method works. stable

6. Lemma Assume that edges ei =(i,i+1) and ej =(j,j+1), i <j, are both illegal in the path Q. Then: • ej remains illegal in the path Q’ obtained by compressing the vertices i and i+1. • ei remains illegal in the path Q’’ obtained by compressing the vertices j and j+1.

7. Lemma There exists one and only one stable compression of the given path. Ex: P= 11-6-5-4-7-8-5, U=13, L=9 illegal edge

8. Lemma (continue) P= 11---6-5---4-7---8-5 P P65 P47 P85 P65,47 P65,85 P47,85 P65,47,85

9. Theorem The greedy procedure outputs an (L,U )-partition with the smallest number of components.

10. Partition into p components Given L=3, U=5 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 Minimum (3,5)-partition into 6 components 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 Maximum (3,5)-partition into 9 components 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 (3,5)-partition into 7 components 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 (3,5)-partition into 8 components

11. Theorem For an arbitrary p such that r<p<s, there exists always an (L,U)-partition of the path Q into p components. Furthermore, there exists an (L,U)-partition  of Q into p classes, which has the following properties: (1)  is an “hybrid” of a max-partition and of a min-partition, in the sense that each cut of  is either a max-cut or a min-cut(or both). (2) There is a vertex m such that all cuts of  on the left of m are max-cuts, while all cuts of  on the right of m are min-cuts.

12. Theorem 給定一介於min-partition和max-partition間的數p,一定可找到一合法的(L,U)-partition  將此path分成p份, 且具有以下性質： •  由min-cut及max-cut混合而成. • 在path中存在一點m, 其左邊均為max-cut, 右邊為min-cut.

13. How to find m ? • m = max{i : i S } • S = {i : i =p-r and edge (i -1,i ) bears a max-cut}.

14. Most uniform partitioning U W W/p L O W/p W

15. Example Path 10 49 2 7 50 3 50 10 10 10 , p=7 Max-min partition (L=10, U=57) 10 49 2 7 50 3 50 10 10 10 Min-max partition (L=3, U=50) 10 49 2 7 50 3 50 10 10 10 Most uniform partition (L=9,U=53) 10 49 2 7 50 3 50 10 10 10

16. Example (L, U)-partitioning where L=3 , U=6 1 2 5 3 3 3

17. Example Minimum (3, 6)-partitioning Maximum (3, 6)-partitioning 1 2 5 3 3 3 1 2 5 3 3 3