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Chapter 9: Problem 13

Evaluate each vertex in each game tree. The values of the terminal vertices are given.

Short Answer

Expert verified
To determine the value of each vertex in a game tree, we use a decision-making algorithm such as minimax. Starting from the terminal vertices, we work our way upwards, deciding at each step whether to 'minimize' or 'maximize' based on the current level of the tree. The value of the root of the tree is the outcome of the game.

Step by step solution

01

Understand the minimax algorithm

Minimax is a decision-making algorithm, useful for game trees. Here, two players play the game, one tries to get the maximum score (\(MAX\)) and the other tries to get the minimum score (\(MIN\)). The algorithm moves through the game tree, starting from the terminal vertices. At each step, it determines whether to 'minimize' or 'maximize' based on the current level of the tree.
02

Move upwards from the terminal vertices

Starting at the bottom of the tree (i.e., at the terminal vertices), the algorithm works its way upwards. At each non-terminal vertex, it notes the value of each of its children and then decides (based on the current level of the tree) whether to take the 'minimum' or 'maximum' of these values as its own value.
03

Evaluating all vertices

Repeat the process described in Step 2 until all vertices in the tree have been evaluated. The value of the root of the tree is considered to be the outcome of the game.

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Most popular questions from this chapter

Draw all nonisomorphic binary trees having four vertices.

Refer to tournament sort. Tournament Sort. We are given a sequence \(s_{1}, \ldots, s_{2^{k}}\) to sort in nondecreasing order. We will build a binary tree with terminal vertices labeled \(s_{1}, \ldots, s_{2^{k}} .\) An example is shown. Working left to right, create a parent for each pair and label it with the maximum of the children. Continue in this way until you reach the root. At this point, the largest value, \(m\), has been found. To find the second-largest value, first pick a value vless than all the items in the sequence. Replace the terminal vertex w containing \(m\) with \(v\). Relabel the vertices by following the path from w to the root, as shown. At this point, the secondlargest value is found. Continue until the sequence is ordered. Show that any algorithm that finds the largest value among \(n\) items requires at least \(n-1\) comparisons.

Write an algorithm based on breadth-first search that finds the minimum length of each path in an unweighted graph from a fixed vertex \(y\) to all other vertices.

The subset-sum problem is: Given a set \(\left\\{c_{1}, \ldots, c_{n}\right\\}\) of positive integers and a positive integer \(M,\) find all subsets \(\left\\{c_{k_{1}}, \ldots, c_{k_{j}}\right\\}\) of \(\left\\{c_{1}, \ldots, c_{n}\right\\}\) satisfying $$\sum_{i=1}^{j} c_{k_{i}}=M$$ Write a backtracking algorithm to solve the subset-sum problem.

Write a breadth-first search algorithm to test whether a graph is connected.
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