Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

Analyzing Recursive Algorithms using Recursion Traces
We can analyze a recursive algorithm by using a visual tool known as a recursion
trace. We used recursion traces, for example, to analyze and visualize the
recursive Fibonacci function of Section 3.5, and we will similarly use recursion
traces for the recursive sorting algorithms of Sections 11.1 and 11.2.
To draw a recursion trace, we create a box for each instance of the method and
label it with the parameters of the method. Also, we visualize a recursive call by
drawing an arrow from the box of the calling method to the box of the called
method. For example, we illustrate the recursion trace of the LinearSum
algorithm of Code Fragment 3.31 in Figure 3.24. We label each box in this trace
with the parameters used to make this call. Each time we make a recursive call,
we draw a line to the box representing the recursive call. We can also use this
diagram to visualize stepping through the algorithm, since it proceeds by going
from the call for n to the call for n − 1, to the call for n − 2, and so on, all the way
down to the call for 1. When the final call finishes, it returns its value back to the
call for 2, which adds in its value, and returns this partial sum to the call for 3, and
so on, until the call for n − 1 returns its partial sum to the call for n.
Figure 3.24: Recursion trace for an execution of
LinearSum(A,n) with input parameters A = {4,3,6,2,5}
and n = 5.
From Figure 3.24, it should be clear that for an input array of size n, Algorithm
LinearSum makes n calls. Hence, it will take an amount of time that is roughly
proportional to n, since it spends a constant amount of time performing the
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