# Difference between revisions of "Chapter 1"

Problems

1.1. Show that a + b can be less than min(a, b).

1.2. Show that a × b can be less than min(a, b).

1.3. Design/draw a road network with two points a and b such that the fastest route between a and b is not the shortest route.

1.4. Design/draw a road network with two points a and b such that the shortest route between a and b is not the route with the fewest turns.

1.5. The knapsack problem is as follows: given a set of integers S = {s1, s2. . . , sn}}, and a target number T, find a subset of S that adds up exactly to T. For example, there exists a subset within S = {1, 2, 5, 9, 10} that adds up to T = 22 but not T = 23.
Find counterexamples to each of the following algorithms for the knapsack problem. That is, give an S and T where the algorithm does not find a solution that leaves the knapsack completely full, even though a full-knapsack solution exists.
(a) Put the elements of S in the knapsack in left to right order if they fit, that is, the first-fit algorithm.
(b) Put the elements of S in the knapsack from smallest to largest, that is, the best-fit algorithm.
(c) Put the elements of S in the knapsack from largest to smallest.

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