Data Structure & Algorithm 11 – Minimal Spanning Tree JJCAO Steal some from Prof. Yoram Moses &...
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Transcript of Data Structure & Algorithm 11 – Minimal Spanning Tree JJCAO Steal some from Prof. Yoram Moses &...
Data Structure & Algorithm
11 – Minimal Spanning Tree
JJCAO
Steal some from Prof. Yoram Moses & Princeton COS 226
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MST Origin
Otakar Boruvka (1926).• Electrical Power Company of Western Moravia in
Brno.• Most economical construction of electrical power
network.• Concrete engineering problem is now a
cornerstone problem in combinatorial optimization.
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MST describes arrangement of nuclei in the epithelium for cancer research
http://www.bccrc.ca/ci/ta01_archlevel.html
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Normal Consistency
[Hoppe et al. 1992] • Based on angles between unsigned normals• May produce errors on close-by surface sheets
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MST is fundamental problem with diverse applications
• Network design.– telephone, electrical, hydraulic, TV cable, computer, road
• Approximation algorithms for NP-hard problems.– traveling salesperson problem, Steiner tree
• Indirect applications.– max bottleneck paths– LDPC codes for error correction– image registration with Renyi entropy– learning salient features for real-time face verification– reducing data storage in sequencing amino acids in a protein– model locality of particle interactions in turbulent fluid flows– autoconfig protocol for Ethernet bridging to avoid cycles in a network
• Cluster analysis.
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Minimum Spanning Tree
Input: a connected, undirected graph - G, with a weight function on the edges – wt
Goal: find a Minimum-weight Spanning Tree for G
Fact:If all edge weights are distinct, the MST is unique
Brute force: Try all possible spanning trees• problem 1: not so easy to implement• problem 2: far too many of them
Ex: [Cayley, 1889]: V^{V-2} spanning trees on the complete graph on V vertices.
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Main algorithms of MST
1. Kruskal’s algorithm 2. Prim’s algorithm
Both O(ElgV) using ordinary binary heapsBoth greedy algorithms => Global solution
3. …
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Two Greedy Algorithms
• Kruskal's algorithm. Consider edges in ascending order of cost. Add the next edge to T unless doing so would create a cycle.
• Prim's algorithm. Start with any vertex s and greedily grow a tree T from s. At each step, add the cheapest edge to T that has exactly one endpoint in T.
Greed is good. Greed is right. Greed works. Greedclarifies, cuts through, and captures the essence of theevolutionary spirit."
- Gordon Gecko
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Cycle Property• Let T be a minimum spanning tree
of a weighted graph G• Let e be an edge of G that is not in
T and C be the cycle formed by e with T
• For every edge f of C, weight(f) ≤ weight(e)
Proof:• By contradiction• If weight(f) > weight(e) we can
get a spanning tree of smaller weight by replacing e with f
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Cut (/Partition) PropertyLemma:Let G =(V,E) and X ⊂ V.If e = a lightest edge connecting X and V-Xthen e appears in some MST of G.
Proof:• Let T be an MST of G• If T does not contain e, consider the cycle C
formed by e with T and let f be an edge of C across the partition
• By the cycle property, weight(f) ≤ weight(e)
• Thus, weight(f) = weight(e)• We obtain another MST by replacing f with e
locally optimal choice(of lightest edges)
globally optimal solution(MST)
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Worst-Case Example• n: the number of MAKE-SET operations,• m: the total number of MAKE-SET, UNION, and FIND-
SET operations• we can easily construct a sequence of m operations
on n objects that requires (n^2) time
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Weighted Union Heuristic
• Each set id includes the length of the list• In Union - append shorter list at end of
longer
Theorem: Performing m > n operations takes O(m + nlgn) time
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Simple Forest Implementation
Find-Set(x) -follow pointersfrom x up to root
Union(c,f) - make c a child of f and return f
∪
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Weighted Union Heuristic
• Each node includes a weight fieldweight = # elements in sub-tree rooted at node
• Find-Set(x) - as before O(depth(x))
• Union(x,y) - always attach smaller tree below the root of larger tree O(1)
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Weighted Union
Theorem:Any k-node tree created using theweighted-union heuristic, has height ≤ lg(k)
Proof: By induction on k
Find-Set Running Time: O(lg n)
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The function lg n
lg n = the number of times we have to take the log2 n repeatedly to reach root node
Lg 2 = 1Lg 2^2 = 2Lg 2^16 = lg 65536 = 16
=> Lg n < 16 for all practical values of n
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Theorem(Tarjan): IfS = a sequence of O(n) Unions and Find-SetsThe worst-case time for S with
– Weighted Unions, and– Path Compressions
is O(nlgn)
The average time is O(lgn) per operation
in Linked List Implementation
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Theorem(Tarjan): LetS = a sequence of O(n) Unions and Find-SetsThe worst-case time for S with
– Weighted Unions, and– Path Compressions
is O(nα(n))
The average time is O(α(n)) per operation, α(n) < 5 in practice
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Connected Components usingUnion-Find
Reminder:
• Every node v is connected to itself• if u and v are in the same connected
component then v is connected to u and u is connected to v
• Connected components form a partition of the nodes and so are disjoint:
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MST-Kruskal
Kruskal's algorithm for minimum spanning tree works by inserting edges in order of increasing cost, adding as edges to the tree those which connect two previously disjoint components.
Kruskal's algorithm on a graph of distances between 128 North American cities
The minimum spanning tree describes the cheapest network to connect all of a given set of vertices
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Decrease_key(v,x)
We use a min-Heap to hold the edges in G-THow can we implement Decrease key(v,x)?
Simple solution:• Change value for v• Follow strategy for Heap_insert from v
upwards
• Cost: O(lgV)
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Euclidean MSTGiven N points in the plane, find MST connecting them, where the distances between point pairs are their Euclidean distances.
Brute force. Compute ~ /2 distances and run Prim's algorithm.Ingenuity. Exploit geometry and do it in ~ c N lg N.
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Scientific application: clusteringk-clustering. Divide a set of objects classify into k coherent groups.Distance function. Numeric value specifying "closeness" of two objects.
Goal. Divide into clusters so that objects in different clusters are far apart.
Applications.• Routing in mobile ad hoc networks.• Document categorization for web search.• Similarity searching in medical image databases.• Skycat: cluster 109 sky objects into stars, quasars, galaxies.
outbreak of cholera deaths in London in 1850s (Nina Mishra)
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Single-link clustering
k-clustering. Divide a set of objects classify into k coherent groups.Distance function. Numeric value specifying "closeness" of two objects.
Goal. Divide into clusters so that objects in different clusters are far apart.
Single link. Distance between two clusters equals the distance between the two closest objects (one in each cluster).
Single-link clustering. Given an integer k, find a k-clustering that maximizes the distance between two closest clusters.
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Single-link clustering algorithm
“Well-known” algorithm for single-link clustering:• Form V clusters of one object each.• Find the closest pair of objects such that each object is
in a different cluster, and merge the two clusters.• Repeat until there are exactly k clusters.
Observation. This is Kruskal's algorithm (stop when k connected components).
Alternate solution. Run Prim's algorithm and delete k-1 max weight edges.