Draw a picture of the decision tree for an optimal algorithm to find the maximum number in the l

Draw a picture of the decision tree for an optimal algorithm...

Key Concepts

Decision Tree

A decision tree is a graphical representation of the sequence of comparisons and conditional decisions an algorithm makes while processing input data. In the context of comparison-based algorithms, each node represents a comparison between elements, and each branch represents the outcome of that comparison. The tree structure helps visualize the flow of decision-making that ultimately leads to identifying the desired output, such as the maximum element in a list.

Comparison-Based Algorithm

Comparison-based algorithms determine outcomes by comparing items pairwise rather than using any structural properties of the items themselves. When finding the maximum element in a list, the algorithm compares elements to eliminate candidates until the largest element is identified. The decision tree in this context maps out every possible sequence of comparisons based on varying outcomes of each comparison.

Optimal Algorithm for Finding the Maximum

An optimal algorithm for finding the maximum element minimizes the number of comparisons needed to guarantee the correct result. The decision tree framework helps in understanding that the minimum number of comparisons required to find the maximum in a list of n elements is n - 1. This lower bound is achieved in the simplest form of the algorithm, often visualized as a series of consecutive comparisons in the decision tree.

Tournament Method

The tournament method is an approach to finding the maximum element that frames the problem similarly to a knockout tournament. Each comparison is like a match in which the larger element 'wins' and advances to the next round. The structure of the tournament can be naturally represented as a decision tree, where the final champion is the maximum element. This visualization assists in demonstrating how the total number of comparisons aligns with the optimal bound.

Transcript

00:01 We're given a weighted graph and we're asked to find a maximum spanning tree for this graph.

00:08 The weighted graph is the one in exercise three.

00:13 If you look in the book, you see that this is a graph on vertices a, b, c, d, e, f, g, h, i with weighted edges.

00:24 And to find a maximum spanning tree for this graph, we're going to use a modified version of cruikull's algorithm, which we derived in earlier exercise.

00:42 So the first step of cruikull's algorithm is to start with the graph that contains only the vertices and no edges of our giving graph.

00:51 So first we just pointed to all the vertices, a, b, c, d, e, f, j, g, h, and i.

01:08 In the next step, i'm going to find the edge with the largest weight in the given graph this is 8 and it occurs between d and h so i'll add the edge dh in the next step find in the remaining graph the edge with the largest weight this is 7 and it occurs between d and e so i'll add this edge to our graph so it does not create a circuit in the next step i'll find the edge with the largest weight in the remaining graph.

02:02 This is now six, and this occurs between d and g and f.

02:08 So first, consider dg.

02:12 If we are to add edge dg, this will not create a circuit, so this is perfectly valid.

02:20 And if we are to add the edge ef, we also won't create a circuit there, so that's also valid.

02:30 I'm sorry, d and f, my mistake...

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