CS 5321: Advanced Algorithms Minimum Spanning Trees. Acknowledgement. Minimum Spanning Trees

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1 CS : Advanced Algorithms Minimum Spanning Trees Ali Ebnenasir Department of Computer Science Michigan Technological University Acknowledgement Eric Torng Moon Jung Chung Charles Ofria Minimum Spanning Trees Definition Algorithms Prim Kruskal Boruvka

2 Problem Definition Input Weighted, connected undirected graph =(V,E) Weight (length) function w on each edge e in E W: E R Task Compute a spanning tree of of minimum total weight Spanning tree If there are n nodes in, a spanning tree consists of n- edges such that no cycles are formed A eneric Solution Kruskal s Algorithm reedy approach to edge selection Select the minimum weight edge that does not form a cycle Implementation Sort edges by increasing lengths Alternatively use heap to store edges Either way: O(E log E) time Include minimum weight edge if it does not form a cycle. How do we test if edge forms a cycle?

3 Kruskal s Algorithm Disjoint-Set Data Structure Use disjoint-set data structure (Chapter ) Each vertex belongs to a set containing the vertices in its current tree (initially only itself) 4 6 Find-set(A) = Find-set(B) = Find-set(E) = A Find-set(C) = Find-set(D) = C Find-set(F) = F Find-set() = Cycle Detection Take minimum weight edge and test both endpoints to see if they are in the same set 4 6 Min weight edge: (A,E). Same Set. Cycle. Min weight edge: (B,C): Different Sets. No cycle.

4 Update Disjoint-set Data Structure Merge sets corresponding to node B and node C We need the result to now be: 4 6 Find-set(A) = Find-set(B) = Find-set(C) = Find-set(D) = Find-set(E) = A (or C) Find-set(F) = F Find-set() = Time for Disjoint-Set Data Structure Operations Initialization: Makeset(v) for all vertices v: O(V) time Unions and Find-Set operations O(V) Unions. Why? O(E) Find-sets. Why? Each can be implemented in amortized α(v) time where α is a very slow growing function O(E α(v)) time overall which is essentially O(E) for all practical purposes Overall Running Time Initialization: Makeset(v) for all vertices v: O(V) time Unions and Find-set operations O(E) of them Each can be implemented in amortized α(v) time where α is a very slow growing function O(E α(v)) time overall for cycle detection O(E log E) time for sorting edges by weight dominates Since log E = O(log V), we have O(E log V) 4

5 Prim s Algorithm Different greedy approach to edge selection Initialize connected component N to be any node v Select the minimum weight edge connecting N to V-N Implementation Maintain a priority queue for the nodes in V-N based on how close they are to any node in N When a new node v is added to N, we need to update the weight of the neighbors of v in V-N Prim s Algorithm // r is the starting vertex // key[u] is min weight of any edge connecting u to component N // π[u] is parent of u Priority Queue Maintain priority queue of nodes in V-N 4 6 If we started with node D, N is now {C,D} Priority Queue values of other nodes: A, E, F: infinity B: 4 : 6

6 Updating Priority Queue Node B is added to N; edge (B,C) is added to T 4 6 Need to update priority queue values of A, E, F Decrease-key operation Priority Queue values of other nodes: A: E: F: : 6 Updating Priority Queue Node A is added to N; edge (A,B) is added to T 4 6 Need to update priority queue values of E Decrease-key operation Priority Queue values of other nodes: E: (unchanged because is smaller than ) F: : 6 Running time Analysis Assume binary heap implementation Build initial heap takes O(V) time V extract-min operations for O(V log V) time For each edge, potentially decrease-key operation, so O(E log V) time Overall: O(E log V) time which is asymptotically equivalent to our implementation of Kruskal s algorithm Use of Fibonacci heap can improve running time to O(E + V log V) time Decrease-key drops to O() amortized time 6

7 Proof of Correctness - Kruskal Let T O be a minimum spanning tree for Let T K be the graph formed by Kruskal s algorithm T K must be a tree. Why? T K utilizes edges in T O whenever a tie needs to be broken Proof by contradiction: T K has more weight than T O Otherwise Kruskal s algorithm has produced an optimal tree Proof of Correctness - Kruskal Let e be the smallest weight edge in T K that is not in T O Add e to T O and consider the cycle C that is formed Why exactly one cycle? There must be some edge e on C that has weight greater than e. Why? There must be at least one edge e in C that is not in T K Why? w(e ) > w(e) because T K utilizes edges in T O whenever a tie needs to be broken Proof of Correctness - Kruskal Replace e by e in T O We have a new spanning tree T O such that the weight of T O is smaller than the weight of T O This contradicts the fact that T O was optimal This implies no such edge e can be found, and thus T K must be optimal 7

8 Proof of Correctness - Prim Let T O be a minimum spanning tree for Let T P be the tree formed by Prim s algorithm that utilizes edges in T O whenever a tie needs to be broken Assumption: T P has more weight than T O Otherwise Prim s algorithm has produced an optimal tree Let e be the first edge added to T P that is not in T O Finish this argument Boruvka s Algorithm. Initialize a spanning forest F. Update F For each tree T in F, join a vertex of T to a vertex of some other tree by a minimum weight edge. Denote the resulting tree as T. If the number of edges in F is V - then stop; Otherwise goto Step 4 6 Example of Boruvka s algorithm

9 Spanning Tree in Action A distributed system of K processes P.,, P.k A binary, irreflexive and symmetric adjacency relation connects all machines Only adjacent processes can communicate Processes/links may fail, thereby partitioning the network The state of the system maybe in a variety of cases Single nodes Cycles A forest of trees Continually maintain a spanning tree even when processes fail or their adjacency list changes due to link failures Program Representation The code of a process may consists of a set of actions executed in parallel act [] act []. [] act n where [] denotes parallel execution of actions, and An action is of the form <guard> <assignment statement> A guard is a Boolean expression over the variables of a process and its neighbors An assignment statement may update the variables of the process executing the action Program Variables N.i : a variable that keeps the indices of the non-faulty neighbors of P.i root.i : index of the current root process f.i : current father of P.i Spanning tree is maintained by the father relationship Thus the initial values of f.i in each process represents the initial graph, which may be a forest, include cycles, etc. d.i : the length of shortest path from P.i to P.(root.i) 9

10 Actions of Process i