UMass Lowell Computer Science Analysis of Algorithms Prof. Karen Daniels Fall, 2008 Lecture 2 Tuesday, 9/16/08 Design Patterns for Optimization Problems Greedy Algorithms

Algorithmic Paradigm Context Subproblem solution order Make choice, then solve subproblem(s) Solve subproblem(s), then make choice

Greedy Algorithms

What is a Greedy Algorithm? ä Solves an optimization problem ä Optimal Substructure: ä optimal solution contains in it optimal solutions to subproblems ä Greedy Strategy: ä At each decision point, do what looks best “locally” ä Choice does not depend on evaluating potential future choices or presolving overlapping subproblems ä Top-down algorithmic structure ä With each step, reduce problem to a smaller problem ä Greedy Choice Property: ä “locally best” = globally best

Greedy Strategy Approach 1. Determine the optimal substructure of the problem. 2. Develop a recursive solution. 3. Prove that, at any stage of the recursion, one of the optimal choices is the greedy choice. 4. Show that all but one of the subproblems caused by making the greedy choice are empty. 5. Develop a recursive greedy algorithm. 6. Convert it to an iterative algorithm. source: textbook Cormen, et al.

Examples ä Activity Selection ä Minimum Spanning Tree ä Dijkstra Shortest Path ä Huffman Codes ä Fractional Knapsack

Activity Selection

Activity Selection Optimization Problem ä Problem Instance: ä Set S = {1,2,...,n} of n activities ä Each activity i has: ä start time: s i ä finish time: f i ä Activities i, j are compatible iff non-overlapping: ä Objective: ä select a maximum-sized set of mutually compatible activities source: textbook Cormen, et al.

Activity Selection Activity Time Duration Activity Number

Algorithmic Progression ä “Brute-Force” ä (board work) ä Dynamic Programming #1 ä Exponential number of subproblems ä (board work) ä Dynamic Programming #2 ä Quadratic number of subproblems ä Greedy Algorithm

Activity Selection Solution to S ij including a k produces 2 subproblems: 1) S ik (start after a i finishes; finish before a k starts) 2) S kj (start after a k finishes; finish before a j starts) source: textbook Cormen, et al. c[i,j]=size of maximum-size subset of mutually compatible activities in S ij.

source: web site accompanying textbook Cormen, et al. Errors from earlier printing are corrected in red. High-level call: RECURSIVE-ACTIVITY-SELECTOR(s,f,0,n) Returns an optimal solution for Recursive Activity Selection

source: web site accompanying textbook Cormen, et al.

Running time? Greedy Algorithm ä Algorithm: ä S’ = presort activities in S by nondecreasing finish time ä and renumber ä GREEDY-ACTIVITY-SELECTOR(S’) ä n length[S’] ä A {1} ä j1 ä for i 2 to n ä do if ä then ä j i ä return A source: textbook Cormen, et al.

Streamlined Greedy Strategy Approach 1. View optimization problem as one in which making choice leaves one subproblem to solve. 2. Prove there always exists an optimal solution that makes the greedy choice. 3. Show that greedy choice + optimal solution to subproblem optimal solution to problem. source: textbook Cormen, et al. Greedy Choice Property: “locally best” = globally best

Minimum Spanning Tree

A B C D E F G source: textbook Cormen et al. for Undirected, Connected, Weighted Graph G=(V,E) Produces minimum weight tree of edges that includes every vertex. Invariant: Minimum weight spanning forest Becomes single tree at end Invariant: Minimum weight tree Spans all vertices at end Time: O(|E|lg|E|) given fast FIND-SET, UNION Time: O(|E|lg|V|) = O(|E|lg|E|) slightly faster with fast priority queue

Dijkstra Shortest Path

Single Source Shortest Paths: Dijkstra’s Algorithm source: textbook Cormen et al. for (nonnegative) weighted, directed graph G=(V,E) A B C D E F G

Huffman Codes

source: textbook Cormen, et al.

source: web site accompanying textbook Cormen, et al.

Fractional Knapsack

Knapsack item1item2item3 “knapsack” Value: $60 $100$120 Each item has value and weight. Goal: maximize total value of items chosen, subject to weight limit. 0-1: take all or none of an item fractional: can take part of an item

source: web site accompanying textbook Cormen, et al.

Additional Examples On course web site under Miscellaneous Docs ä Patriotic Tree ä 404 review handout ä Tree Vertex Cover ä midterm

Greedy Heuristic ä If optimization problem does not have “greedy choice property”, greedy approach may still be useful as a heuristic in bounding the optimal solution ä Example: minimization problem Optimal (unknown value) Upper Bound (heuristic) Lower Bound Solution Values

Homework 1 T 9/9 T 9/ review & Chapter 15 2 T 9/16 T 9/23 Chapter 16 HW# Assigned Due Content