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If there is some way for all processes to have their pending requests satisfied and run to completion, then deadlock does not exist. Otherwise, deadlock does exist, and some action must be taken to break the deadlock. The deadlock detection algorithm works by maintaining several data structures. First, the algorithm maintains a vector with the count of available resources of each type. This vector is called Available. If a system has seven printers, two CD-ROM drives, and six plotters, and all devices are allocated to one or another process (none is available), then the vector Available = [0, 0, 0] A matrix called Allocated records the counts of resources of each type already allocated to each active process. In a particular case of five active processes, where all of the resources listed in the previous paragraph have been allocated, the matrix Allocated might look like this: Allocated CD 1 0 0 1 0

BASIS AND RECURSIVE PARTS To work correctly, every recursive function must have a basis and a recursive part. The basis is what stops the recursion. The recursive part is where the function calls itself. EXAMPLE 9.4 The Basis and Recursive Parts of the Factorial Function

In the Java method that implements the factorial function in Example 9.2, the basis and the recursive parts are labeled with comments. The recursive part invokes the method, passing a smaller value of n. So starting with a positive value like 5, the values on the successive invocations will be 4, 3, 2, 1, and 0. When 0 is passed, the basis executes, thereby stopping the recursion and beginning the chain of returns, returning 1, 1, 2, 6, 24, and finally 120.

P 0 2 3 2 0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

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0 1 1 2 3 5 8 13 21 34 55 89 144 233 377

A matrix called Requests maintains the counts of all pending resource requests, at this particular moment, by each process. Suppose this is the matrix Requests for the same set of five processes at a particular instant: Requests CD 0 0 0 0 0

The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . Each number after the second is the sum of the two preceding numbers. This is a naturally recursive definition: Fn = 0, if n = 0 1, if n = 1 F n 1 + F n 2 , if n > 1

The first 15 values of the Fibonacci sequence are shown in Table 9.2. The first two values, F0 and F1, are defined by the first two parts of the definition: F0 = 0 (for n = 0) and F1 = 1 (for n = 1). These two parts form the basis of the recursion. All the other values are defined by the recursive part of the definition:

P 0 2 0 1 0

CHAP. 9]

For n = 2, F2 = Fn = Fn 1 + Fn 2 = F(2) 1 + F(2) 2 = F1 + F0 = 1 + 0 = 1. For n = 3, F3 = Fn = Fn 1 + Fn 2 = F(3) 1 + F(3) 2 = F2 + F1 = 1 + 1 = 2. For n = 4, F4 = Fn = Fn 1 + Fn 2 = F(4) 1 + F(4) 2 = F3 + F2 = 2 + 1 = 3. For n = 5, F5 = Fn = Fn 1 + Fn 2 = F(5) 1 + F(5) 2 = F4 + F3 = 3 + 2 = 5. For n = 6, F6 = Fn = Fn 1 + Fn 2 = F(6) 1 + F(6) 2 = F5 + F4 = 5 + 3 = 8. For n = 7, F7 = Fn = Fn 1 + Fn 2 = F(7) 1 + F(7) 2 = F6 + F5 = 8 + 5 = 13.

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