Minimum Spanning Tree

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Minimum Spanning Tree

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Outline

• History of MST
• Background Knowledge Soft Heap
• Algorithm at a Glance
• The MST algorithm
• Notations
• Detail description
• Correctness

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History of MST(1/2)

• 1926, Boruvka present the first model to solve
  MST.
• Textbook algorithms run in O(mlogn), Kruskals
  Greedy Algorithm, Prims Algorithm
• 1975, Yao, O(mloglogn)
• 1984, Freeman and Tarjan, O(mß(m,n))
• 1987, Gabow, O(m logß(m,n)), involved a new data
  structure Fibonacci Heap

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ß(m,n)

• ß(m,n) is a very slowly growing function defined
  as follows
• ß(m,n) min i logi(n)lt m/n
• The worst case is O(m logm) (m O(n))

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History of MST(2/2)

• 1997, Bernard Chazelle, O(ma(m, n)log a(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?

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a(m,n)

• a(m,n) is the inverse of Ackermann's function
  (a very slowly growing function)
• a(m,n) kgt1 A(k,?m/n?)gtlgn
• Ackermanns function is defined by
• A(1,m) 2m
• A(n,1) A(n-1,2)
• A(n,m) A(n-1,A(n,m-1))

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TODAY

• A Minimum Spanning Tree Algorithm with
  Inverse-Ackermann Type Complexity
• -- Bernard Chazelle
• -- Princeton University, and NEC Research
  Institute
• -- Journal of the ACM, November 2000
• -- involved a new data structure Soft Heap
• -- Running time is O(ma(m,n))
• An Optimal Minimum Spanning Tree Algorithm
• -- Seth Pettie And Vijaya 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(ma(m,n))

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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.

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Binomial Heap

• In Worst-case
• Make-Heap ?(1) Insert O(lgn)
• Meld O(lgn) Delete ?(lgn)
• Findmin ?(lgn)

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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() -gt sift()

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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

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Algorithm at a Glance (2/4)

• Contractibility (for a subgraph C of G)
• C is contractible if C n MST(G) is connected.
• Computing MST(C) is easier.
• How can we discover C without computing MST(G)?

C
1
C
5
2
3
4
C is contractible
C is not contractible
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Algorithm at a Glance (3/4)

• Decompose ? Contract ? Glue

Step. 1
Step. 2
Step. 4
Step. 3
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Algorithm at a Glance (4/4)

• Iterating this process forms a hierarchy tree
• Each v of G is a leaf of

The hierarchy tree
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Finally, the MST algorithm.

• To computer MST of G, call msf(G, t) with

We will explain this later.
msf(G, t)
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Any question?

• We are going into the details now.

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And

• Lets welcome ???

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Tree

• Computed in O(md3n)
• Balance between height and nodeAckermanns
  function

Cz
z
nz
height dz
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Ackermanns function

• Choice nz as

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• By induction on t, we have
• Expansion of CzS(t,dz)3
• Compute tree in bt(md3n)O(md3n)d(m/n)1/3
• The choice of t and d implies that
  tO(a(m,n))MST is O(ma(m,n))

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Active path
z1
z2
z3
z4zk
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Border edges

• Exactly one vertex in

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Corrupted edge

• Border edges are stored in soft heaps
• They may become corrupted due to soft heap
  operation.
• If its corrupted

Cz
Cz
Soft heap
Cost
corrupted
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Bad edge

• Its called bad if

When Cz contracted into one vertex
Cz
Cz
corrupted
bad
Once bad always bad
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Working cost
Cz
bad
Working costoriginal cost
Working costcurrent cost
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Stack view
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Detail description of each steps

• Totally 5 steps.
• Many details for Step 3.
• Detail but not too detail (hopefully).

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Step1, 2

• Case t1 is special. Solve this in O(n2)
• 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/2c).

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Step3

• Building the hierarchy
• With t gt 1 specified, target size nz is
  specified.
• We discuss for a general case that the active
  path is z1, z2, zk
• Keep two invariants INV1 and INV2 (explain
  later)
• Two possible actions Retraction and Extension.

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INV1
For all iltk
chain link e
for any two pair (j1, j2)lti
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Step3 INV2

• Each border edge (u, v) where u Czj 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 Czi but not
• any Czl that (iltlltj)

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Step3 INV2

• A practice

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)
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Step3 Retraction

• If a last subgraph Czkhas attained its target
  size.
• Contract Czkinto Czk-1 (as well as its links)
• Czk-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)

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Step3 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 Czk1
• Older border edges incident to v are not border
  edges now. Delete them from heap, update
  min-links and insert new border edges.

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Step3 Extension

• If any min-link (a, b) is less than (u, v)
• We have to do a fusion
• Explain by the figure.

fusion into a
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Step3 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.

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Step4

• Recursing in Subgraphs of .
• For every Cz, do msf(Cz -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 Cz.

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Step5

• The final recursion
• The candidate edge set becomes F?B now.
• Adding edges contracted in Step2 produces MST
  of G.

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Correctness

• Contractibility
• Subgraph C of G is strong contractible if
• each edge of p lt min(e, f )
• Thats why we do fusion.
• Lemma 3.1 If an edge e is not bad and lies
  outside F, e lies outside of MST(G).

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MST(G) and T
Cz
z tree node Cz subgraph(vertex)
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