55
votes
Accepted
Iteration can replace Recursion?
It's possible to replace recursion by iteration plus unbounded memory.
If you only have iteration (say, while loops) and a finite amount of memory, then all you have is a finite automaton. With a ...
49
votes
Accepted
Will this program terminate for every Integer?
The correct answer is that this function does not terminate for all integers (specifically, it does not terminate on -1). Your friend is correct in stating that this is pseudocode and pseudocode does ...
48
votes
Accepted
Why is tail recursion better than regular recursion?
if that is the only reason, why the compiler would not be able to optimize the regular recursive call?
You are focusing on the wrong thing here: the reason the optimization works is because of the ...
- 5,830
35
votes
Iteration can replace Recursion?
Every recursion can be converted to iteration, as witnessed by your CPU, which executes arbitrary programs using a fetch-execute infinite iteration. This is a form of the Böhm-Jacopini theorem. ...
- 275k
28
votes
Accepted
Difference between Tail-Recursion and structural recursion
Structural recursion: recursive calls are made on structurally smaller arguments.
Tail recursion: the recursive call is the last thing that happens.
There is no requirement that the tail recursion ...
- 30k
26
votes
Iteration can replace Recursion?
As an example to the answer from Gilles, here is an "iterative" algorithm for the Ackermann function (using the common Ackermann-Péter version mentioned by Wikipedia $a(n,m)$).
We need a ...
- 533
21
votes
Accepted
Is the Berkeley tutorial on Fibonacci trees using wrong figures?
I would like to say both you and that Berkeley tutorial are correct.
As commented by chepner, the trees in Berkeley tutorial and the trees you thought are the same semantically; only the labels of the ...
- 38.6k
19
votes
Accepted
Why are loops faster than recursion?
The reason that loops are faster than recursion is easy.
A loop looks like this in assembly.
...
- 1,040
17
votes
Why are loops faster than recursion?
These other answers are somewhat misleading. I agree that they state implementation details that can explain this disparity, but they overstate the case. As correctly suggested by jmite, they are ...
16
votes
Iteration can replace Recursion?
There are already some great answers (which I can't even hope to compete with), but I'd like to pitch this simple explanation.
Recursion is just the manipulation of the runtime stack. Recursing adds ...
- 516
16
votes
Accepted
What's the Big O runtime of a DFS word search through a matrix?
The complexity will be $O(m*n*4^{s})$ where m is the no. of rows and n is the no. of columns in the 2D matrix and s is the length of the input string.
When we start searching from a character we ...
- 1,195
15
votes
Accepted
What property of cons allows elimination of tail recursion modulo cons?
While GCC likely uses ad-hoc rules, you can derive them in the following way. I'll use pow to illustrate since you're foo is so ...
13
votes
Why is tail recursion better than regular recursion?
The way a standard function/procedure (from now on I'll just say "function") call works is that the caller needs to store onto the stack whatever state it needs for computations that occur ...
- 21.6k
12
votes
Accepted
Is this a generic way to convert any recursive procedure to tail-recursion?
Your description of your algorithm is really too vague to evaluate it at this point. But, here are some things to consider.
CPS
In fact, there is a way to transform any code into a form that uses ...
- 622
12
votes
Accepted
Count total number of k length paths in a tree
This can be solved in $\mathcal{O}(n \log n)$ by using the smaller-to-larger merging technique. Root the tree at an arbitrary vertex. We will calculate for every subtree an array where the $d$th ...
- 1,096
11
votes
Correct name for a recursive descent parser that uses loops to handle left recursion?
It is just an LL(1) parser implemented with recursive descent.
Starts with:
...
- 3,024
10
votes
Accepted
How can the class of tail recursive functions be compared to the classes of PR and R?
Every computable function can be expressed in continuation-passing-style, in which all calls are tail-calls.
The trick is to add a "continuation" parameter to every function. Instead of making a non-...
- 29.7k
9
votes
What property of cons allows elimination of tail recursion modulo cons?
I’m going to beat around the bush for a while, but there is a point.
Semigroups
The answer is, the associative property of the binary reduction operation.
That’s pretty abstract, but multiplication ...
- 1,241
8
votes
Can Breadth-First Search be Implemented Recursively without Data Structures?
You basically have two choices: "cheating" by embedding a queue in the nodes, and simulating BFS with higher complexity.
Embedded-Queue Cheating
If you look at virtually any description of BFS, e.g.,...
- 238
8
votes
Accepted
How to derive dependently typed eliminators?
The canonical reference for this is Peter Dybjer, Inductive Families, which gives a pretty comprehensive treatment of inductive families based on eliminators.
- 8,114
7
votes
How is the complexity of recursive algorithms calculated and do they admit better complexity than non-recursive algorithms?
Typically, by writing a recurrence relation for the running time and then solving the recurrence relation. See How to come up with the runtime of algorithms? and Solving or approximating recurrence ...
D.W.♦
- 156k
7
votes
Accepted
Algorithm to find maximum number of floors you can check with N eggs and D maximum drops
First we'll clarify the problem a bit. We have $n$ eggs, which we can drop from any floor we want. We also have the constraint that we are allowed a total of $d$ drops of these eggs (rather than $d$ ...
- 14.8k
7
votes
Accepted
Worst-case input for median-of-medians with groups of size 3
Whether or not the median-of-medians algorithm with groups of size 3 runs in linear time is an open problem as said in [1] (while they proposed a variant running in linear time). I checked some follow-...
- 7,395
7
votes
Is the Berkeley tutorial on Fibonacci trees using wrong figures?
Your trees are showing the same thing; you are just labeling each node by the call, and the Berkeley tutorial is labeling each node by the result of that call. Compare the two pictures of fibtree(3), ...
- 451
6
votes
Accepted
Efficient algorithm for getting from 1 to n with 3 specific operations
Find the shortest path from $1$ to $n$ on an appropriate graph on vertices $\{1, \dots, n\}$. This approach will work whenever it's guaranteed that intermediate values in the calculations will ...
- 81.4k
6
votes
How to derive dependently typed eliminators?
You might find some of our recent papers on this useful, as we derive eliminators for lambda-encoded datatypes. For example, see this one for generic derivation of eliminators, and this one for the ...
- 61
6
votes
Is there any recursive function f whose code is unique?
No. The Padding Lemma states that there is a primitive recursive function $\sf pad$ such that, if $n$ is a code for $f$, then ${\sf pad}(n)$ is another code for $f$ which is larger than $n$.
...
- 14.4k
6
votes
Accepted
How to solve recurrence. T(n). = T(n-1) + T(n/2) + n?
Let $S(n) = T(n) - 2n - 2$. You can check that $S(n) = S(n-1) + S(n/2)$ (ignoring the fact that $n/2$ need not be an integer). This shows that the additive $n$ term doesn't make a big difference.
For ...
- 275k
6
votes
Accepted
Why is there no "traditional"-mathy way to describe the general algorithm and give a more math-friendly definition of algorithm?
If you're looking for algebraic structure, then you should look at the field of denotational semantics. This is exactly what you describe: using algebra, and often Category Theory, to model ...
- 29.7k
5
votes
How to calculate the mergesort time complexity?
Imagine that we have an array $A$ of size $n$. Mergesort splits this array into two equal halves and sorts them individually. So in context of the paragraph you have provided, each node corresponds to ...
- 1,198
Only top scored, non community-wiki answers of a minimum length are eligible