Minimum Spanning Tree Union-Find Data Structure. Feb 28, 2018 CSCI211 - Sprenkle. Comcast wants to lay cable in a neighborhood. Neighborhood Layout

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1 Objec&ves Minimum Spanning Tree Union-Find Data Structure Feb, 0 CSCI - Sprenkle Started teasing out some algorithms. Laying Cable Focus on commonality: what should our final solution look like? Comcast wants to lay cable in a neighborhood Ø Reach all houses Ø Least cost Neighborhood Layout Cost of laying cable btw houses depends on amount of cable, landscaping, obstacles, etc. 9 Feb, 0 CSCI - Sprenkle

2 Minimum Spanning Tree (MST) Spanning tree: spans all nodes in graph Given a connected graph G = (V, E) with posi&ve edge weights c e, an MST is a subset of the edges T E such that T is a spanning tree whose sum of edge weights is minimized 0 G = (V, E) 9 9 T, Σ e T c e = 0 Feb, 0 CSCI - Sprenkle Examples Identify spanning trees and which is the minimal spanning tree. Graph A B C D Feb, 0 CSCI - Sprenkle

3 Examples Identify spanning trees and which is the minimal spanning tree. Graph A B A B MST: C D C D Other Spanning Trees: A B A B C D C D Feb, 0 CSCI - Sprenkle MST Applica&ons Network design Ø telephone, electrical, hydraulic, TV cable, computer, road Approxima&on algorithms for NP-hard problems Ø traveling salesperson problem, Steiner tree Indirect applica&ons Ø max bozleneck paths Ø image registra&on with Renyi entropy Ø learning salient features for real-&me face verifica&on Ø reducing data storage in sequencing amino acids in a protein Ø model locality of par&cle interac&ons in turbulent fluid flows Cluster analysis ~eppstein/gina/mst.html Feb, 0 CSCI - Sprenkle

4 Ideas for Solu&ons? Cayley's Theorem. There are n n- spanning trees Towards a solu&on Ø Where to start? can't solve by brute force 0 9 G = (V, E) Feb, 0 CSCI - Sprenkle Minimum Spanning Tree Given a connected graph G = (V, E) with posi&ve edge weights c e, an MST is a subset of the edges T E such that T is a spanning tree whose sum of edge weights is minimized 0 G = (V, E) 9 Why must the solution be a tree? Feb, 0 CSCI - Sprenkle T, Σ e T c e = 0 9

5 Minimum Spanning Tree Assume have a minimal solu&on that is not a tree, i.e., it has a cycle What could we do? Ø What do we know about the edges? Ø How does that change the cost of the solu&on? Feb, 0 CSCI - Sprenkle 9 Minimum Spanning Tree Proof by Contradic&on. Assume have a minimal solu&on V that is not a tree, i.e., it has a cycle Contains edges to all nodes because solu&on must be connected (spanning) Remove an edge from the cycle Ø Can s&ll reach all nodes (could go long way around ) Ø But at lower total cost Ø Contradic&on to our minimal solu&on Feb, 0 CSCI - Sprenkle 0

6 Greedy Algorithms All three algorithms produce a MST Prim's algorithm. Ø Start with some root node s and greedily grow a tree T from s outward Ø At each step, add cheapest edge e to T that has exactly one endpoint in T Ø Similar to Dijkstra s (but simpler) Kruskal's algorithm. Ø Start with T = φ Ø Consider edges in ascending order of cost Ø Insert edge e in T unless doing so would create a cycle Reverse-Delete algorithm. Ø Start with T = E Ø Consider edges in descending order of cost Ø Delete edge e from T unless doing so would disconnect T What do these algorithms have/do/check in common? Feb, 0 CSCI - Sprenkle What Do These Algorithms Have in Common? When is it safe to include an edge in the minimum spanning tree? Cut Property When is it safe to eliminate an edge from the minimum spanning tree? Cycle Property Feb, 0 CSCI - Sprenkle

7 Cut and Cycle Proper&es Simplifying assump&on: All edge costs c e are dis&nct MST is unique Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then MST contains e. Cycle property. Let C be any cycle, and let f be the max cost edge belonging to C. Then MST does not contain f. f C S e Cut Property: e is in MST Cycle Property: f is not in MST Let s try to prove these Feb, 0 CSCI - Sprenkle Cycles and Cuts Cycle. Set of edges in the form a-b, b-c, c-d,, y-z, z-a Cycle C = -, -, -, -, -, - Feb, 0 CSCI - Sprenkle

8 Cycles and Cuts Cycle. Set of edges in the form a-b, b-c, c-d,, y-z, z-a Cycle C = -, -, -, -, -, - Cutset. A cut is a subset of nodes S. The corresponding cutset D is the subset of edges with exactly one endpoint in S. Feb, 0 CSCI - Sprenkle Cut S = {,, } Cutset D = -, -, -, -, - Cycle-Cut Intersec&on Claim. A cycle and a cutset intersect in an even number of edges Cycle C = -, -, -, -, -, - Cut S = {,, } Cutset D = -, -, -, -, - Intersection = -, - What are the possibilities for the cycle? Feb, 0 CSCI - Sprenkle

9 Cycle-Cut Intersec&on Claim. A cycle and a cutset intersect in an even number of edges Proof sketch Cycle C = -, -, -, -, -, - Cut S = {,, } Cutset D = -, -, -, -, - Intersection = -, -. Cycle all in S. Cycle not in S. Cycle has to go from SàV-S and back (Cut) S Edges link to not-cut V - S Feb, 0 CSCI - Sprenkle Proving Cut Property: OK to Include Edge Simplifying assump&on: All edge costs c e are dis&nct. Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then the MST T* contains e. Pf.? Feb, 0 CSCI - Sprenkle 9

10 Proving Cut Property: OK to Include Edge Simplifying assump&on: All edge costs c e are dis&nct. Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then the MST T* contains e. Pf. (exchange argument) Ø Suppose there is an MST T* that does not contain e What do we know about T, by defn? What do we know about the nodes e connects? Feb, 0 CSCI - Sprenkle 9 Proving Cut Property: OK to Include Edge Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then the MST T* contains e. Pf. (exchange argument) Ø Suppose there is an MST T* that does not contain e Ø Adding e to T* creates a cycle C in T* Ø Edge e is in cycle C and in cutset corresponding to S there exists another edge, say f, that is in both C and S s cutset Which means? e Feb, 0 CSCI - Sprenkle 0 S f 0

11 Proving Cut Property: OK to Include Edge Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then the MST T* contains e. Pf. (exchange argument) Ø Suppose there is an MST T* that does not contain e Ø Adding e to T* creates a cycle C in T* Ø Edge e is in cycle C and in cutset corresponding to S there exists another edge, say f, that is in both C and S s cutset Ø T' = T* { e } - { f } is also a spanning tree Ø Since c e < c f, cost(t') < cost(t*) f Ø This is a contradic&on. S e Feb, 0 CSCI - Sprenkle Proving Cycle Property: OK to Remove Edge Simplifying assump&on: All edge costs c e are dis&nct Cycle property. Let C be any cycle in G, and let f be the max cost edge belonging to C. Then the MST T* does not contain f. Ideas about approach? Feb, 0 CSCI - Sprenkle

12 Cycle Property: OK to Remove Edge Cycle property. Let C be any cycle in G, and let f be the max cost edge belonging to C. Then the MST T* does not contain f. Pf. (exchange argument) Ø Suppose f belongs to T* Ø Dele&ng f from T* creates a cut S in T* Ø Edge f is both in the cycle C and in the cutset S there exists another edge, say e, that is in both C and S Ø T' = T* { e } - { f } is also a spanning tree Ø Since c e < c f, cost(t') < cost(t*) f Ø This is a contradic&on. S Feb, 0 CSCI - Sprenkle e Summary of What Just Proved Simplifying assump&on: All edge costs c e are dis&nct MST is unique Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then MST contains e. Cycle property. Let C be any cycle, and let f be the max cost edge belonging to C. Then MST does not contain f. f C S e Cut Property: e is in MST Cycle Property: f is not in MST Feb, 0 CSCI - Sprenkle

13 Prim s Algorithm [Jarník 90, Dijkstra 9, Prim 99] Start with some root node s and greedily grow a tree T from s outward. At each step, add the cheapest edge e to T that has exactly one endpoint in T. How can we prove its correctness? Feb, 0 CSCI - Sprenkle Prim s Algorithm: Proof of Correctness Ini&alize S to be any node Apply cut property to S Ø Add min cost edge (v, u) in cutset corresponding to S, and add one new explored node u to S S Ideas about implementation? Feb, 0 CSCI - Sprenkle

14 Looking Ahead Problem Set due Friday Feb, 0 CSCI - Sprenkle