Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter C.1, Problem 7E
Program Plan Intro
To prove that:
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Suppose that you have n objects that arecompletely identical except that one is slightly heavier. The problem P is to find theheavier object. You have a scale. A single operation consists of placing any two disjointsets of the objects the two sides of the scale. If one side is heavier, then the scale tips over.Give matching upper and lower bounds for this problem.
This question is about the procedure linearSearch, which was discussed in the lectures.
1 procedure linearSearch(x, a1, a2 ⋯, an) 2 for k := 1 to n 3 if x = ak 4 then return k 5 return 0
The procedure searches a sequence of objects a1, a2 ⋯, an in order. If it finds an object that is equal to x at line 3, then it returns the object’s index k at line 4. If it does not find such an object, then it returns 0 at line 5 instead. Suppose that linearSearch searches five objects, and always finds one of them. The probability that an object ai will be found is p(ai), where 1 ≤ i ≤ 5. The probabilities for all objects are shown below.
p(a1)
=
0.5
p(a2)
=
0.3
p(a3)
=
0.1
p(a4)
=
0.05
p(a5)
=
0.05
What is the expected number of comparisions (=) executed by linearSearch at line 3? Your answer must be a number. Show how you got the answer.
Suppose a set of n objects is given, such that the size of the ith object satisfies that 0 < < 1. It is required to pack all the objects in a set of boxes of unit size. Each box can contain any subset of objects whose total size does not exceed 1. The heuristic first-fit takes each object and fits it into the first box that can hold it. Finally, let S be sum of the sizes of all objects: a. Show that the total number of boxes used by the first-fit heuristic is never greater than ⌈2S⌉. b. Show that the approximation factor of the first-fit heuristic is 2.
Chapter C Solutions
Introduction to Algorithms
Ch. C.1 - Prob. 1ECh. C.1 - Prob. 2ECh. C.1 - Prob. 3ECh. C.1 - Prob. 4ECh. C.1 - Prob. 5ECh. C.1 - Prob. 6ECh. C.1 - Prob. 7ECh. C.1 - Prob. 8ECh. C.1 - Prob. 9ECh. C.1 - Prob. 10E
Ch. C.1 - Prob. 11ECh. C.1 - Prob. 12ECh. C.1 - Prob. 13ECh. C.1 - Prob. 14ECh. C.1 - Prob. 15ECh. C.2 - Prob. 1ECh. C.2 - Prob. 2ECh. C.2 - Prob. 3ECh. C.2 - Prob. 4ECh. C.2 - Prob. 5ECh. C.2 - Prob. 6ECh. C.2 - Prob. 7ECh. C.2 - Prob. 8ECh. C.2 - Prob. 9ECh. C.2 - Prob. 10ECh. C.3 - Prob. 1ECh. C.3 - Prob. 2ECh. C.3 - Prob. 3ECh. C.3 - Prob. 4ECh. C.3 - Prob. 5ECh. C.3 - Prob. 6ECh. C.3 - Prob. 7ECh. C.3 - Prob. 8ECh. C.3 - Prob. 9ECh. C.3 - Prob. 10ECh. C.4 - Prob. 1ECh. C.4 - Prob. 2ECh. C.4 - Prob. 3ECh. C.4 - Prob. 4ECh. C.4 - Prob. 5ECh. C.4 - Prob. 6ECh. C.4 - Prob. 7ECh. C.4 - Prob. 8ECh. C.4 - Prob. 9ECh. C.5 - Prob. 1ECh. C.5 - Prob. 2ECh. C.5 - Prob. 3ECh. C.5 - Prob. 4ECh. C.5 - Prob. 5ECh. C.5 - Prob. 6ECh. C.5 - Prob. 7ECh. C - Prob. 1P
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