Greedy Algorithms Reading Material: Chapter 8 (Except Section 8.5)

Greedy Algorithms Like dynamic programming algorithms, greedy algorithms are usually designed to solve optimization problems Unlike dynamic programming algorithms, –greedy algorithms are iterative in nature. –An optimal solution is reached from local optimal solutions. –This approach does not work all the time. –A proof that the algorithm does what it claims is needed, and usually not easy to get.

Fractional Knapsack Problem Given n items of sizes s 1, s 2, …, s n and values v 1, v 2, …, v n and size C, the problem is to find x 1, x 2, …, x n  that maximize subject to

Solution to Fractional Knapsack Problem Consider y i = v i / s i What is y i ? What is the solution?

Shortest Paths Problems Input: A graph with non-negative weights or costs associated with each edge. Output: The list of edges forming the shortest path. Sample problems: –Find shortest path between two named vertices –Find shortest path from S to all other vertices –Find shortest path between all pairs of vertices Will actually calculate only distances, not paths.

Shortest Paths Definitions  (A, B) is the shortest distance from vertex A to B. length(A, B) is the weight of the edge connecting A to B. –If there is no such edge, then length(A, B) = . A D C B

Single-Source Shortest Paths Problem: Given G=(V,E), start vertex s, find the shortest path from s to all other vertices. –Assume V={1, 2, …, n} and s=1 Solution: A greedy algorithm called Dijkstra’s Algorithm

Dijkstra’s Algorithm Outline Partition V into two sets: X={1} and Y= {2, 3, …, n} Initialize [i] for 1  i  n as follows: … Select y  Y such that [y] is minimum – [y] is the length of the shortest path from 1 to y that uses only vertices in set X. Remove y from Y, add it to X, and update [w] for each w  Y where (y,w)  E if the path through y is shorter.

Example ABCDE

Dijkstra’s Algorithm ABCDE Initial0  Process A Process

Dijkstra’s Algorithm Input: A weighted directed graph G = (V,E), where V = {1,2,…,n} Output: The distance from vertex 1 to every other vertex in G. 1. X = {1}; Y = {2,3,…,n}; [1]=0; 2. for y = 2 to n do 3. if y is adjacent to 1 then [y]=length[1,y]; 4. else [y]=  end if 5. end for 6. for j = 1 to n – 1 do 7. Let y  Y be such that [y] is minimum; 8. X = X  {y} // add vertex y to X 9. Y = Y  - {y} // delete vertex y from Y 10. for each edge (y,w) do 11. if w  Y and [y] + length[y,w] < [w] then 12. [w] = [y] + length[y,w] 13. end for 14. end for

Correctness of Dijkstra’s Algorithm Lemma: In Dijkstra’s algorithm, when a vertex y is chosen in Step 7, if its label [y] is finite then [y] =  [y]. Proof:

Time Complexity Mainly depends on how we implement step 7, i.e., finding y s.t. [y] is minimum. Approach 1: Scan through the vector representing current distances of vertices in Y: Approach 2: Use a min-heap to maintain vertices in the set Y:

Minimum Cost Spanning Trees Minimum Cost Spanning Tree (MST) Problem: Input: An undirected weighted connected graph G. Output: The subgraph of G that 1) has minimum total cost as measured by summing the weights of all the edges in the subset, and 2) keeps the vertices connected. –What does such subgraph look like?

MST Example ABCDE

Kruskal’s Algorithm Initially, each vertex is in its own MST. Merge two MST’s that have the shortest edge between them. –Use a priority queue to order the unprocessed edges. Grab next one at each step. How to tell if an edge connects two vertices that are already in the same MST? –Use the UNION/FIND algorithm with parent- pointer representation.

Example ABCDE

Kruskal’s MST Algorithm Sort the edges of E(G) by weight in non-decreasing order; For each vertex v  V(G) do New_Tree(v); // creating tree with one root node v T=  ; // MST initialized to empty While |T| < n - 1 do Let (u,v) be the next edge in E(G) if FIND(u)  FIND(v) then T = T  (u,v); UNION(u,v); Return T;

Asymptotic Analysis of Kruskal’s Algorithm

Correctness of Kruskal’s Algorithm Lemma: Algorithm Kruskal correctly finds a minimum cost spanning tree in a weighted undirected graph. Proof: Theorem: Algorithm Kruskal’s finds a minimum cost spanning tree in a weighted undirected graph in O(m log m) time, where m is the number of edges in G.

Prim’s Algorithm Very similar to Dijkstra’s algorithm Grows a minimum spanning tree from an arbitrary vertex u  V

Idea of Prim’s Algorithm X Y

Idea of Prim’s Algorithm (Cont.) X Y

Prim’s Algorithm 1. T =  ; X={1}; Y = V(G) – {1}; 2. while Y   do 3. Let (x,y) be of minimum weight such that x  X and y  Y. 4. X = X  {y}; 5. Y = Y – {y}; 6. T = T  {(x,y)}; 7. end while;

Example ABCDE

Prim’s MST Implementation As with Dijkstra’s algorithm, the key issue is determining which vertex is next closest. As with Dijkstra’s algorithm, the alternative is to use a priority queue (min-heap). Running times for the two implementations are identical to the corresponding Dijkstra’s algorithm implementations. For implementation details of the corresponding algorithms, check the book pages 245 and 247.

Correctness of Prim’s Algorithm Lemma: Algorithm Prim correctly finds a minimum cost spanning tree in a connected undirected graph. Proof: