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Uploaded byWaqar Akram
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Greedy Algorithm

This document provides an introduction to greedy algorithms. It defines greedy algorithms as algorithms that make locally optimal choices at each step in the hope of finding a global optimum. The document then provides examples of problems that can be solved using greedy algorithms, including counting money, scheduling jobs, finding minimum spanning trees, and the traveling salesman problem. It also provides pseudocode for a general greedy algorithm and discusses some properties of greedy algorithms.

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3 INTRODUCTION GroupGroup 22 Name Roll No Rubina Mustafa 3 Hafiz Jawad Mansoor 25 Sajad Ul Haq 5 Umair 16 Rao Waqar Akram 17
4 Greedy AlgorithmGreedy Algorithm
5 A SHORT LIST OF CATEGORIES  Algorithm types we will consider include:  Simple recursive algorithms  Backtracking algorithms  Divide and conquer algorithms  Dynamic programming algorithms  Greedy algorithms  Branch and bound algorithms  Brute force algorithms  Randomized algorithms
6 OPTIMIZATION PROBLEMS  An optimization problem is one in which you want to find, not just a solution, but the best solution  A “greedy algorithm” sometimes works well for optimization problems  A greedy algorithm works in phases. At each phase:  You take the best you can get right now, without regard for future consequences  You hope that by choosing a local optimum at each step, you will end up at a global optimum
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8 EXAMPLE: COUNTING MONEY  Suppose you want to count out a certain amount of money, using the fewest possible bills and coins  A greedy algorithm would do this would be: At each step, take the largest possible bill or coin that does not overshoot  Example: To make $6.39, you can choose:  a $5 bill  a $1 bill, to make $6  a 25¢ coin, to make $6.25  A 10¢ coin, to make $6.35  four 1¢ coins, to make $6.39  For US money, the greedy algorithm always gives the optimum solution
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10 EXAMPLE
11 A SCHEDULING PROBLEM  You have to run nine jobs, with running times of 3, 5, 6, 10, 11, 14, 15, 18, and 20 minutes  You have three processors on which you can run these jobs  You decide to do the longest-running jobs first, on whatever processor is available  Time to completion: 18 + 11 + 6 = 35 minutes  This solution isn’t bad, but we might be able to do better 20 18 15 14 11 10 6 5 3P1 P2 P3
12 ANOTHER APPROACH  What would be the result if you ran the shortest job first?  Again, the running times are 3, 5, 6, 10, 11, 14, 15, 18, and 20 minutes  That wasn’t such a good idea; time to completion is now 6 + 14 + 20 = 40 minutes  Note, however, that the greedy algorithm itself is fast  All we had to do at each stage was pick the minimum or maximum 20 18 15 14 11 10 6 5 3P1 P2 P3
13 AN OPTIMUM SOLUTION  Better solutions do exist:  This solution is clearly optimal (why?)  Clearly, there are other optimal solutions (why?)  How do we find such a solution?  One way: Try all possible assignments of jobs to processors  Unfortunately, this approach can take exponential time 20 18 15 14 11 10 6 5 3 P1 P2 P3
14 HUFFMAN ENCODING  The Huffman encoding algorithm is a greedy algorithm  You always pick the two smallest numbers to combine  Average bits/char: 0.22*2 + 0.12*3 + 0.24*2 + 0.06*4 + 0.27*2 + 0.09*4 = 2.42  The Huffman algorithm finds an optimal solution 22 12 24 6 27 9 A B C D E F 15 27 46 54 100 A=00 B=100 C=01 D=1010 E=11 F=1011
15 MINIMUM SPANNING TREE  A minimum spanning tree is a least-cost subset of the edges of a graph that connects all the nodes  Start by picking any node and adding it to the tree  Repeatedly: Pick any least-cost edge from a node in the tree to a node not in the tree, and add the edge and new node to the tree  Stop when all nodes have been added to the tree  The result is a least-cost (3+3+2+2+2=12) spanning tree  If you think some other edge should be in the spanning tree:  Try adding that edge  Note that the edge is part of a cycle  To break the cycle, you must remove the edge with the greatest cost  This will be the edge you just added 1 2 3 4 5 6 3 3 3 3 2 2 2 4 4 4
16 TRAVELING SALESMAN  A salesman must visit every city (starting from city A), and wants to cover the least possible distance  He can revisit a city (and reuse a road) if necessary  He does this by using a greedy algorithm: He goes to the next nearest city from wherever he is  From A he goes to B  From B he goes to D  This is not going to result in a shortest path!  The best result he can get now will be ABDBCE, at a cost of 16  An actual least-cost path from A is ADBCE, at a cost of 14 E A B C D 2 3 3 4 4 4
17 OTHER GREEDY ALGORITHMS  Dijkstra’s algorithm for finding the shortest path in a graph  Always takes the shortest edge connecting a known node to an unknown node  Kruskal’s algorithm for finding a minimum-cost spanning tree  Always tries the lowest-cost remaining edge  Prim’s algorithm for finding a minimum-cost spanning tree  Always takes the lowest-cost edge between nodes in the spanning tree and nodes not yet in the spanning tree
18 PSEOUDOCODE  Begin  Greedy(input I)  while (solution is not complete) do  Select the best element x in the  remaining input I;  Put x next in the output;  Remove x from the remaining input;  Endwhile  End
19 ALGORITHM  MAKE-CHANGE (n)         C ← {100, 25, 10, 5, 1}     // constant.         Sol ← {};                         // set that will hold the solution set.         Sum ← 0 sum of item in solution set         WHILE sum != n             x = largest item in set C such that sum + x ≤ n             IF no such item THEN                 RETURN    "No Solution"             S ← S {value of x}             sum ← sum + x         RETURN S 19
20 CONNECTING WIRES  There are n white dots and n black dots, equally spaced, in a line  You want to connect each white dot with some one black dot, with a minimum total length of “wire”  Example:  Total wire length above is 1 + 1 + 1 + 5 = 8  Do you see a greedy algorithm for doing this?  Does the algorithm guarantee an optimal solution?  Can you prove it?  Can you find a counterexample?
21 COLLECTING COINS  A checkerboard has a certain number of coins on it  A robot starts in the upper-left corner, and walks to the bottom left-hand corner  The robot can only move in two directions: right and down  The robot collects coins as it goes  You want to collect all the coins using the minimum number of robots  Example:  Do you see a greedy algorithm for doing this?  Does the algorithm guarantee an optimal solution?  Can you prove it?  Can you find a counterexample?
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