I think this D entry is closer to the literature value 1. The fractal dimension is now lower than the C version so the resulting tree is less dense , because the attach probability is higher, this causes a faster dendrite growth. After implementing and experimenting a while, I just thought about a restriction I placed on the movement of the particles I called them dust motes. I was contemplating about going back and re-doing the particle movements, but that would probably make the randomization of the movements much more complex and that version of a Moore's neighborhood would be harder to compute I should think.
I'm not sure making the REXX program more complex and bigger would add anything to the this task, but it would make the movements more "lifelike". Seeing that the link for the AutoHotkey image is dead , is it necessary to add an examle image as example output? Only 15 of the 35 languages have images embedded.
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You may also like:. World Russian baby tiger fights for life after frostbite, surgery. AFP 17 hours ago. The number of times this program computes fib 1 when computing fib n is precisely F n. The mistake of recomputation is compounded, exponentially. No imaginable computer will ever be able to do that many calculations.
Incidentally, a systematic technique known as memoization allows us to avoid this pitfall while still taking advantage of the compact recursive description of a computation. In memoization, we maintain an array that keeps track of the values we have computed so that we can return those values and make recursive calls only for new values. This technique is a form of dynamic programming , a well-studied technique for organizing computations that you will learn if you take courses in algorithms or operations research.
Are there situations when iteration is the only option available to address a problem? No, any loop can be replaced by a recursive function, though the recursive version might require excessive memory. Are there situations when recursion is the only option available to address a problem?
No, any recursive function can be replaced by an iterative counterpart. In Section 4. Whichever leads to the simpler, more easily understood, or more efficient code. I get the concern about excessive space and excessive recomputation in recursive code. Anything else to be concerned about? Be extremely wary of creating arrays in recursive code. The amount of space used can pile up very quickly, as can the amount of time required for memory management. Exercises What happens if you run factorial with negative value of n?
With a large value, say 35? Solution : The base case will never be reached. A call to ex 3 will result in calls to ex 0 , ex -3 , ex -6 , and so forth until the "maximum depth exceeded" run-time error occurs. Given four positive integers a , b , c , and d , explain what value is computed by gcd gcd a, b , gcd c, d. What are the values of mystery 2, 25 and mystery 3, 11? Given positive integers a and b , describe what value mystery a, b computes.
Compose a recursive program ruler. Assume n is a power of two. Prove by induction that the minimum possible number of moves needed to solve the Towers of Hanoi puzzle satisfies the same recurrence as the number of moves used by our recursive solution.
Prove by induction that the recursive program given above makes exactly F n recursive calls to fib 1 when computing fib n. Prove that the second argument to gcd decreases by at least a factor of two for every second recursive call, then prove that gcd p, q uses at most log 2 n recursive calls, where n is the larger of p and q.
Next, rearrange the order of the recursive calls and the base case , view the resulting animation, and explain each outcome. Binary representation. Compose a program that takes a positive integer n in decimal from the command line and writes its binary representation.
Recall that in Section 1. Instead, use the following simpler method: repeatedly divide 2 into n and read the remainders backwards. First, compose a while loop to carry out this computation and write the bits in the wrong order. Then, use recursion to write the bits in the correct order. A4 paper. The width-to-height ratio of paper in the ISO format is the square root of 2 to 1. Format A0 has an area of 1 square meter. Format A1 is A0 cut with a vertical line into two equal halves, A2 is A1 cut with a horizontal line into in two halves, and so on.
Write a program that takes a command-line argument n and uses stddraw to show how to cut a sheet of A0 paper into 2 n pieces. Here's a nice illustration of A size formats. Compose a program that takes a command-line argument n and writes all n! A permutation of n elements is one of the n! Do not worry about the order in which you enumerate them.
Permutations of size k. You need not write them in any particular order. Combinations of size k. Hamming distance. The Hamming distance between two bit strings of length n is equal to the number of bits in which the two strings differ.
Compose a program that takes an integer k and a bit string s from the command line, and writes all bit strings that have Hamming distance at most k from s. For example if k is 2 and s is then your program should write:. Recursive squares.
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