5.27. Alice wants to throw a party and is deciding whom to call. She has n peopl

Question

5.27. Alice wants to throw a party and is deciding whom to call. She has n people to choose from, and
has made up a list of which pairs of these people know each other. She wants to pick as many
people as possible, subject to two constraints: at the party, each person should have at least ?ve
other people whom they know and ?ve other people whom they don’t know.
Give an ef?cient algorithm that takes as input the list of n people and the list of pairs who know
each other and outputs the best choice of party invitees. Give the running time in terms of n.

Explanation / Answer

Greedy algorithms can be characterized as being 'short sighted', and as 'non-recoverable'. They are ideal only for problems which have 'optimal substructure'. Despite this, greedy algorithms are best suited for simple problems (e.g. giving change). It is important, however, to note that the greedy algorithm can be used as a selection algorithm to prioritize options within a search, or branch and bound algorithm. There are a few variations to the greedy algorithm: Pure greedy algorithms Orthogonal greedy algorithms Relaxed greedy algorithms [edit] Applications Greedy algorithms mostly (but not always) fail to find the globally optimal solution, because they usually do not operate exhaustively on all the data. They can make commitments to certain choices too early which prevent them from finding the best overall solution later. For example, all known greedy coloring algorithms for the graph coloring problem and all other NP-complete problems do not consistently find optimum solutions. Nevertheless, they are useful because they are quick to think up and often give good approximations to the optimum. If a greedy algorithm can be proven to yield the global optimum for a given problem class, it typically becomes the method of choice because it is faster than other optimisation methods like dynamic programming. Examples of such greedy algorithms are Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees, Dijkstra's algorithm for finding single-source shortest paths, and the algorithm for finding optimum Huffman trees. The theory of matroids, and the more general theory of greedoids, provide whole classes of such algorithms. Greedy algorithms appear in network routing as well. Using greedy routing, a message is forwarded to the neighboring node which is "closest" to the destination. The notion of a node's location (and hence "closeness") may be determined by its physical location, as in geographic routing used by ad-hoc networks. Location may also be an entirely artificial construct as in small world routing and distributed hash table. [edit] Examples The Activity selection problem is characteristic to this class of problems, where the goal is to pick the maximum number of activities that do not clash with each other. In the Macintosh computer game Crystal Quest the objective is to collect crystals, in a fashion similar to the travelling salesman problem. The game has a demo mode, where the game uses a greedy algorithm to go to every crystal. The artificial intelligence does not account for obstacles, so the demo mode often ends quickly. The Matching pursuit is an example of greedy algorithm applied on signal approximation. A greedy algorithm finds the optimal solution to Malfatti's problem of finding three disjoint circles within a given triangle that maximize the total area of the circles; it is conjectured that the same greedy algorithm is optimal for any number of circles.

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