Data Structures Graph Algorithms Virendra Singh Associate Professor Computer Architecture and Dependable Systems Lab Department of Electrical Engineering Indian Institute of Technology Bombay http://www.ee.iitb.ac.in/~viren/ E-mail: viren@ee.iitb.ac.in EE-717/453:Advance Computing for Electrical Engineers Lecture 11 (29 Aug 2013)
Algorithm Design Methods Greedy method. Divide and conquer. Dynamic Programming. Backtracking. Branch and bound.
Other Methods Linear Programming. Integer Programming. Simulated Annealing. Neural Networks. Genetic Algorithms. Tabu Search.
Optimization Problem A problem in which some function (called the optimization or objective function) is to be optimized (usually minimized or maximized) subject to some constraints.
Feasible And Optimal Solutions A feasible solution is a solution that satisfies the constraints. An optimal solution is a feasible solution that optimizes the objective/optimization function.
Greedy Algorithm
Greedy Method Solve problem by making a sequence of decisions. Decisions are made one by one in some order. Each decision is made using a greedy criterion. A decision, once made, is (usually) not changed later.
Problems: Greedy Method Single Source all destination shortest path Dijkstra’s Algorithm Minimum Spanning Tree (MST) Prim’s Algorithm Bin Packing Job Scheduling
Application: Shortest Path Problems Directed weighted graph. Path length is sum of weights of edges on path. The vertex at which the path begins is the source vertex. The vertex at which the path ends is the destination vertex.
Shortest Path Problems Single source single destination. Single source all destinations. All pairs (every vertex is a source and destination).
Single Source Single Destination Possible greedy algorithm: Leave source vertex using cheapest/shortest edge. Leave new vertex using cheapest edge subject to the constraint that a new vertex is reached. Continue until destination is reached.
Greedy Shortest 1 To 7 Path 8 6 2 1 3 3 1 16 7 5 6 4 10 4 2 4 7 5 3 14 Path length is 12. Not shortest path. Algorithm doesn’t work!
Shortest Path Algorithms 1 10 100 • Dijkstra’s algorithm 30 Greedy algorithm 2 5 Make local decision greedily 10 60 Gives shortest path from a 4 3 source node 20
Data Structures For Dijkstra’s Algorithm • The greedy single source all destinations algorithm is known as Dijkstra’s algorithm. • Implement d() and p() as 1D arrays. • Keep a linear list L of reachable vertices to which shortest path is yet to be generated. • Select and remove vertex v in L that has smallest d() value. • Update d() and p() values of vertices adjacent to v.
Dijkstra’s Algorithms Iter. S V-S w D[2] D[3] D[4] D[5] 1 10 100 Init {1} {2,3,4,5} - 10 ∞ 30 100 30 2 5 1 {1,2} {3,4,5} 2 10 60 30 100 10 50 2 {1,2,4} {3,5} 4 10 50 30 90 60 4 3 3 {1,2,4,3} {5} 3 10 50 30 60 20 4 {1,2,4,3,5} Φ 5 10 50 30 60
Dijkstra’s Algorithms • Begin • S = {1} 1 • For I = 2 to n do 10 100 – D[i] = C [1,i] – initialize 30 2 5 • For i = 1 to n-1 do begin 10 – Choose a vertex w in V-S s.t. D[w] is minimum 60 – Add w to S 4 3 20 – For each vertex v in V-S do – D[v] = Min{D[v], D[w]+C[w,v]} • end
Complexity • O(n) to select next destination vertex. • O(out-degree) to update d() and p() values when adjacency lists are used. • O(n) to update d() and p() values when adjacency matrix is used. • Selection and update done once for each vertex to which a shortest path is found. • Total time is O(n2 + e) = O(n2).
Minimum-Cost Spanning Tree weighted connected undirected graph Free tree that connects all the vertices spanning tree cost of spanning tree is sum of edge costs find spanning tree that has minimum cost
Edge Selection Greedy Strategies • Start with a 1-vertex tree and grow it into an n-vertex tree by repeatedly adding a vertex and an edge. When there is a choice, add a least cost edge. Prim’s method. • Start with an n-vertex 0-edge forest. Consider edges in ascending order of cost. Select edge if it does not form a cycle together with already selected edges. Kruskal’s method.
MST: Prims Algorithms • Prim’s Algorithm Greedy algoritthm Start from an intial node U = {1} Grows ST, one edge at a time At each step, it finds a shortest edge (u,v) that connects U and V-U and adds v to V-U from U
Prim’s Algorithm 1 1 6 5 1 1 5 5 5 2 4 2 4 3 3 4 4 6 3 3 2 2 6 6 5 5 6
Prim’s Algorithm 1 1 1 6 5 1 1 1 5 2 4 5 3 2 4 2 4 3 3 4 4 6 3 2 6 5 6 5 6 5 6 1 1 1 1 1 5 1 5 2 4 2 4 3 3 2 4 3 4 4 2 2 4 3 2 6 6 6 5 5 5
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