1. Minimum Spanning Tree 吳光哲、江盈宏、陳彥宏、 鍾至衡、張碧娟、余家興

2. Outline History of MST Background Knowledge: Soft Heap Algorithm at a Glance The MST algorithm Notations Detail description Correctness

3. History of MST(1/2) 1926, Boruvka present the first model to solve MST. Textbook algorithms run in O(mlogn), Kruskal’s Greedy Algorithm, Prim’s Algorithm 1975, Yao, O(mloglogn) 1984, Freeman and Tarjan, O(mβ(m,n)) 1987, Gabow, O(m logβ(m,n)), involved a new data structure : Fibonacci Heap

4. β(m,n) β(m,n) is a very slowly growing function defined as follows: β(m,n) = min { i : log i (n)<= m/n} The worst case is O(m log*m) (m = O(n))

5. History of MST(2/2) 1997, Bernard Chazelle, O(mα(m, n)log α(m, n)) Problem: Does there exists a linear deterministic algorithm which solves MST problem? Hope : for a given weighted connected graph G with m edges the algorithm finds a MST of G in at most Km steps?

6. α(m,n) α(m,n) is the “inverse” of Ackermann's function (a very slowly growing function) α(m,n) = { k>=1: A(k,  m/n  )>lgn} Ackermann’s function is defined by A(1,m) = 2 m A(n,1) = A(n-1,2) A(n,m) = A(n-1,A(n,m-1))

7. TODAY A Minimum Spanning Tree Algorithm with Inverse-Ackermann Type Complexity -- Bernard ChazelleBernard Chazelle -- Princeton University, and NEC Research Institute -- Journal of the ACM, November 2000 -- involved a new data structure : Soft Heap -- Running time is O(mα(m,n)) An Optimal Minimum Spanning Tree Algorithm -- Seth Pettie And Vijaya RamachandranSeth PettieVijaya Ramachandran -- The University of Texas at Austin, Austin, Texas -- Journal of the ACM, January 2002 -- involved a new data structure : Soft Heap -- Runs in time O(T*(m,n)), T*(m,n) is beteween  (m) & O(mα(m,n))

8. Soft Heap A variant of a binomial heap -create(H): Create an empty soft heap -insert(H,x): Add new item x to H -meld(H,H’): Union in H and H’. -delete(H,x): Remove item x from H. -findmin(H): return the smallest key in H. The amortized complexity of each operation is constant, except for insert, which takes O(log 1/  )  is the error rate in soft Heap.

9. Binomial Heap In Worst-case Make-Heap  (1) Insert O(lgn) Meld O(lgn) Delete  (lgn) Findmin  (lgn)

10. Key Point The entropy of the data structure is reduced by Artificially raising the values of certain keys. (corrupted) Move items across the data structure not individually, but in groups.(car pooling)  is the error rate in soft Heap. Deletemin() -> sift()

11. Algorithm at a Glance (1/4) Input: an undirected graph G. –Each edge e is assigned a cost c(e) –Edge costs are distinct –MST (G) is unique –There are m edges –There are n vertices

12. Algorithm at a Glance (2/4) Contractibility (for a subgraph C of G) –C is contractible if C ∩ MST (G) is connected. –Computing MST (C) is easier. –How can we discover C without computing MST (G)? C is not contractible 1 2 3 4 5 C is contractible C C

13. Algorithm at a Glance (3/4) Decompose  Contract  Glue Step. 1 Step. 2 Step. 4Step. 3

14. Algorithm at a Glance (4/4) Iterating this process forms a hierarchy tree Each v of G is a leaf of The hierarchy tree

15. Finally, the MST algorithm. To computer MST of G, call msf (G, t) with msf (G, t) We will explain this later.

16. Any question? We are going into the details now….

17. And Let’s welcome 光哲兄 ~

18. Computed in O(m+d 3 n) Balance between height and node: Ackermann’s function Tree z CzCz height d z nznz

19. Ackermann’s function Choice n z as

20. By induction on t, we have Expansion of C z =S(t,d z ) 3 Compute tree in bt(m+d 3 n)=O(m+d 3 n) d=(m/n) 1/3 The choice of t and d implies that t=O(α(m,n)) MST is O(mα(m,n))

21. Active path z1z1 z2z2 z3z3 z 4 =z k

22. Border edges Exactly one vertex in C z1 C z2 C z3 CuCu z1z1 z2z2 z3z3 u

23. Corrupted edge Border edges are stored in soft heaps They may become corrupted due to soft heap operation. If it’s corrupted: corrupted CzCz CzCz Cost++ Soft heap

24. Bad edge It’s called bad if corrupted bad When C z contracted into one vertex CzCz CzCz Once bad always bad

25. Working cost CzCz corrupted CzCz Cost++ bad CzCz Working cost=original costWorking cost=current cost

26. Stack view z1z1 z2z2 z3z3 z 4 =z k

27. Detail description of each steps Totally 5 steps. Many details for Step [3]. Detail but not too detail (hopefully).

28. Step[1, 2] Case t=1 is special. Solve this in O(n 2 ) –Because What if G is not connected? –Apply msf to each connected component. What if G is not simple? –Keep only the cheapest edge. The aim of performing Boruvka is to reduce the number of edges (to n/2 c ).

29. Step[3] Building the hierarchy –With t > 1 specified, target size n z is specified. –We discuss for a general case that the active path is z 1, z 2, …z k –Keep two invariants: INV1 and INV2 (explain later) –Two possible actions: Retraction and Extension.

30. INV1 For all i<k chain link e for any two pair (j 1, j 2 )<i

31. Step[3]: INV2 Each border edge (u, v) where u C z j are stored into H(j) or H(i, j) –No edge appears in more than one heap –Membership in H(j) implies v is incident to at least one edge in some H(i, j) –Membership in H(i, j) implies v is also incident to C z i but not any C z l that (i<l<j) CzjCzj CziCzi C z i-1 u v

32. Step[3]: INV2 A practice V Z1Z1 a b c d e f g Z5Z5 Z2Z2 Z3Z3 Z4Z4 A possible assignment of edges to heaps: a H(4,5),b H(4), c H(2,4), d H(2), e H(1,2), f H(1,2), g H(0,1)

33. Step[3]: Retraction If a last subgraph C z k has attained its target size. –Contract C z k into C z k-1 (as well as its links) –C z k-1 gains one vertex. –Destroy H(k) and H(k-1, k) Discard bad edges. For each cluster, insert the minimum edge implied by INV2. –Meld H(i, k) into H(i, k-1) a cluster

34. Step[3]: Extension Do findmin on all heaps and retrieve the border edge (u,v) with minimum cost. –(u,v) is the extension edge (chain-link also) –v is the first vertex in C z k+1 –Older border edges incident to v are not border edges now. Delete them from heap, update min- links and insert new border edges.

35. Step[3]: Extension If any min-link (a, b) is less than (u, v) –We have to do a fusion –Explain by the figure. C Zi C Zk C Zi a b v u a Becomes v fusion into a

36. Step[3]: Conclusion Keep doing Extension until target size is reached or no vertices is left. Perform Retraction when target size is reached. Maintain INV1 and INV2 at the same time.

37. Step[4] Recursing in Subgraphs of. –For every C z, do msf (C z -1, t-1) –The edge cost are resetted to original value. –The main goal is to modify the target size. –For every fusion edge (a, b), this fusion does not occur anymore in C z.

38. Step[5] The final recursion –The candidate edge set becomes F ∪ B now. –Adding edges contracted in Step[2] produces MST of G.

39. Correctness Contractibility –Subgraph C of G is strong contractible if each edge of π < min(e, f ) –That’s why we do fusion. Lemma 3.1: If an edge e is not bad and lies outside F, e lies outside of MST (G). C f e π

40. MST(G) and T CzCz z CzCz z: tree node C z : subgraph(vertex)

41. a cluster CzjCzj CziCzi C z i-1 u v 圖

42. V Z1Z1 a b c d e f g Z5Z5 Z2Z2 Z3Z3 Z4Z4 圖

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