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Theory of Algorithms: Greedy Techniques

Explore the concept of greedy techniques through examples like Change Making and Huffman Coding. Discuss their strengths, weaknesses, and applications in optimization problems. Learn about algorithms such as Prim’s, Kruskal’s, and Dijkstra’s for MST and shortest paths. 8 Relevant

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Theory of Algorithms: Greedy Techniques

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  1. Theory of Algorithms:Greedy Techniques James Gain and Edwin Blake {jgain | edwin} @cs.uct.ac.za Department of Computer Science University of Cape Town August - October 2004

  2. Objectives • To introduce the mind set of greedy techniques • To show some examples of greedy algorithms: • Change Making • Huffman Coding • To discuss the strengths and weaknesses of greedy techniques • “Greed, for lack of a better word, is good! Greed is right! Greed works!” Gordon Grecko (Michael Douglas) in Wall Street

  3. Greedy Algorithms • Optimization problems solved through a sequence of choices that are: • Feasible - satisfy problem constraints • Locally Optimal - best choice among all feasible options for that step • Irrevocable- no backing out • A Greedy grab of the best alternative, hoping that a sequence of locally optimal steps will lead to a globally optimal solution • Even if not optimal, sometimes an approximation is acceptable • Not all optimization problems can be approached in this manner!

  4. Applications of the Greedy Strategy • Optimal solutions: • Change making • Minimum Spanning Trees (MST) - Prim’s and Kruskal’s Algorithms • Single-source shortest paths - Dijkstra’s Algorithm • Simple scheduling problems • Huffman codes • Approximations: • Traveling Salesman Problem (TSP) • Knapsack problem • Other combinatorial optimization problems

  5. Change Making • Problem: give change for a specific amount n, with the least number of coins of the denominations d1 > d2 > … > dm • Example: • Smallest change for R2.54 using R5, R2, R1, 50c, 20c, 10c, 5c, 2c, 1c • R2 + 50c + 2c + 2c = R2.54 • Algorithm: • At any step choose the coin of the largest denomination that doesn’t exceed the remaining total; Repeat • Note: Optimal for reasonable sets of coins

  6. Minimum Spanning Tree Problem • Problem: Given n points, connect them in the cheapest way so that there is a path between any pair of points • Spanning Tree (of a connected graph G ): • A connected acyclic subgraph of G that includes all of G’s vertices. • Minimum Spanning Tree (of a weighted graph G ): • A spanning tree of G with minimum total weight • Algorithms: Prim’s and Kruskal’s Algorithms • Example: 4 3 3 1 1 1 1  6 2 2 4 4 2 3 2 3

  7. Shortest Paths Problem • Single Source Shortest Paths Problem: • Given a weighted graph G, find the shortest paths from a source vertex sto each of the other vertices • Solution: • Dijkstra’s Algorithm (a relative of Prim’s MST) for positive weights 1 4 3 2 1 2 1 4 3 6 1 2 1 4 2 3 2 4 2 3

  8. Text Compression • Variable length encoding: • Compresses text by assigning codes of different length to characters based on their probability of occurrence • Used by Samuel Morse in constructing telegraph codes • Prefix-free Codes: • Codes are unique in that no code is a prefix for a code of another character • Tree for Binary Codes: • Leaves are characters • Left edge codes 0, right edge codes 1 • The code of a character is a simple walk from root to leaf

  9. Algorithm for Huffman Coding • Invented by David Huffman as part of a class assignment while he was an undergraduate • Algorithm: • Construct a Huffman Tree that assigns shorter strings to higher frequencies. This defines a Huffman Code • Initialize n one-node trees labeled with the characters of the alphabet. Record the frequency (weight) of each character in the root • REPEAT until a single tree is obtained: Find two trees with the smallest weight Make them the left and right sub-tree of a new tree and record the sum of their weights in the root

  10. Example: Huffman Coding 0.55 B 0.45 C 0.15 A 0.4 B 0.45 C 0.15 A 0.4 1.0 0.55 B 0.45 C 0.15 A 0.4

  11. Notes on Huffman Coding • Compression Ratio: • Standard measure of compression • CR = (x - y) / y * 100%, where x is compressed and y is uncompressed • Typically, 20-80% in the case of Huffman Coding • Yields optimal compression provided: • The probabilities of character occurrences are independent • Probabilities are known in advance

  12. Strengths and Weaknesses of Greedy Techniques • Strengths: • Intuitively simple and appealing • Weaknesses: • Only applicable to optimization problems • Doesn’t always produce an optimal solution

  13. Summary To-Date

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