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Penn CIT 594 - Greedy Algorithms

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Greedy AlgorithmsA short list of categoriesOptimization problemsExample: Counting moneyA failure of the greedy algorithmA scheduling problemAnother approachAn optimum solutionHuffman encodingMinimum spanning treeTraveling salesmanAnalysisOther greedy algorithmsDijkstra’s shortest-path algorithmAnalysis of Dijkstra’s algorithm IAnalysis of Dijkstra’s algorithm IIConnecting wiresCollecting coinsThe End1Greedy Algorithms22A 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 algorithms33Optimization 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 optimum44Example: 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 solution55A failure of the greedy algorithmIn some (fictional) monetary system, “krons” come in 1 kron, 7 kron, and 10 kron coinsUsing a greedy algorithm to count out 15 krons, you would getA 10 kron pieceFive 1 kron pieces, for a total of 15 kronsThis requires six coinsA better solution would be to use two 7 kron pieces and one 1 kron pieceThis only requires three coinsThe greedy algorithm results in a solution, but not in an optimal solution66A 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 available201815 141110653P1P2P3Time to completion: 18 + 11 + 6 = 35 minutesThis solution isn’t bad, but we might be able to do better77Another 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 now6 + 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 maximum201815141110653P1P2P388An optimum solution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Better solutions do exist:201815141110 653P1P2P399Huffman 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 solution22 12 24 6 27 9 A B C D E F15274654100A=00B=100C=01D=1010E=11F=10111010Minimum 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 added12345633332224441111Traveling 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 14EA B CD23 344 41212AnalysisA greedy algorithm typically makes (approximately) n choices for a problem of size n(The first or last choice may be forced)Hence the expected running time is:O(n * O(choice(n))), where choice(n) is making a choice among n objectsCounting: Must find largest useable coin from among k sizes of coin (k is a constant), an O(k)=O(1) operation;Therefore, coin counting is (n)Huffman: Must sort n values before making n choicesTherefore, Huffman is O(n log n) + O(n) = O(n log n)Minimum spanning tree: At each new node, must include new edges and keep them sorted, which is O(n log n) overallTherefore, MST is O(n log n) + O(n) = O(n log n)1313Other 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 tree1414Dijkstra’s shortest-path algorithmDijkstra’s algorithm finds the shortest paths from a given node to all other nodes in a graphInitially, Mark the given node as known (path length is zero)For each out-edge, set the distance in each neighboring node equal to the cost (length) of the out-edge, and set its predecessor to the initially given nodeRepeatedly (until all nodes are known),Find an unknown node containing the smallest distanceMark the new node as knownFor each node adjacent to the new node, examine its neighbors to see whether their estimated distance can be reduced (distance to known node plus cost of


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Penn CIT 594 - Greedy Algorithms

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