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Data Structure & Algorithm 11 – Minimal Spanning Tree JJCAO Steal some from Prof. Yoram Moses & Princeton COS 226 Weighted Graphs G =(V,E),wt wt: E → R wt(G) = (,)∈[] (, ) 2 Sub-Graphs Note: G' is not a spanning sub-graph of G 3 Minimum Spanning Tree • • • • A Subgraph A tree Spans G Of minimal weight 4 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. 5 MST describes arrangement of nuclei in the epithelium for cancer research http://www.bccrc.ca/ci/ta01_archlevel.html 6 Normal Consistency [Hoppe et al. 1992] • Based on angles between unsigned normals • May produce errors on close-by surface sheets 7 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. 8 Minimum Spanning Tree on Surface of Sphere 5000 Vertices 9 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. 10 Main algorithms of MST 1. Kruskal’s algorithm 2. Prim’s algorithm Both O(ElgV) using ordinary binary heaps Both greedy algorithms => Global solution 3. … 11 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. Greed clarifies, cuts through, and captures the essence of the evolutionary spirit." - Gordon Gecko 12 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 13 Edges cross the cut 14 Cut (/Partition) Property Lemma: Let G =(V,E) and X ⊂ V. If e = a lightest edge connecting X and V-X then 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) 15 Disjoint Set ADT 16 An application of disjoint-set data structures 17 Linked List Implementation 18 Union in Linked List Implementation 19 20 Worst-Case Example • n: the number of MAKE-SET operations, • m: the total number of MAKE-SET, UNION, and FINDSET operations • we can easily construct a sequence of m operations on n objects that requires Θ(n^2) time 21 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 22 Simple Forest Implementation Find-Set(x) follow pointers from x up to root Union(c,f) - make c a child of f and return f ∪ 23 Worst-Case Example n … 3 2 1 24 Weighted Union Heuristic • Each node includes a weight field weight = # 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) 25 Weighted Union Theorem: Any k-node tree created using the weighted-union heuristic, has height ≤ lg(k) Proof: By induction on k Find-Set Running Time: O(lg n) 26 2nd heuristic: Path Compression 27 The function lg n lg n = the number of times we have to take the log2 n repeatedly to reach root node Lg 2 = 1 Lg 2^2 = 2 Lg 2^16 = lg 65536 = 16 => Lg n < 16 for all practical values of n 28 Theorem(Tarjan): If S = a sequence of O(n) Unions and Find-Sets The 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 29 Theorem(Tarjan): Let S = a sequence of O(n) Unions and Find-Sets The 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 30 Connected Components using Union-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: 31 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. The minimum spanning tree describes the cheapest network to connect all of a given set of vertices Kruskal's algorithm on a graph of distances between 128 North American cities 32 Example 33 34 MST-Kruskal 35 MST-Kruskal 36 MST-Kruskal Running Time: 37 MST-Prim-Jarnik 38 Example 39 MST-Prim-Jarnik 40 MST-Prim 41 MST-Prim 42 MST-Prim 43 Decrease_key(v,x) We use a min-Heap to hold the edges in G-T How can we implement Decrease key(v,x)? Simple solution: • Change value for v • Follow strategy for Heap_insert from v upwards • Cost: O(lgV) 44 MST-Prim Running Time: 45 Does a linear-time MST algorithm exist? 46 Euclidean MST Given N points in the plane, find MST connecting them, where the distances between point pairs are their Euclidean distances. Brute force. Compute ~ 2 /2 distances and run Prim's algorithm. Ingenuity. Exploit geometry and do it in ~ c N lg N. 47 Scientific application: 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. outbreak of cholera deaths in London in 1850s (Nina Mishra) 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. 48 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. 49 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. 50 Dendrogram Tree diagram that illustrates arrangement of clusters. 51 Dendrogram of cancers in human Tumors in similar tissues cluster together 52