How to represent a triangle of integers?

Question

This addresses "a specific programming problem" from On-Topic

I am working on an interview question from Amazon Software Interview
The question is " Given a triangle of integers, find the path of the largest sum without skipping. "

My question is how would you represent a triangle of integers?

I looked this up on Triangle of Integers and saw that a triangle of integers looked something like

1
2      3
4      5      6
7      8      9      10
11     12     13     14     15

What is the best way(data structure) to represent something like this? My idea was having something like

int[] r1 = {1};
int[] r2 = {2, 3};
int[] r3 = {4, 5, 6};
int[] r4 = {7, 8, 9, 10};
int[] r5 = {11, 12, 13, 14, 15};

Is this the best way to represent this triangle integer structure? I thought about using a 2 dimensional matrix structure but those have to have arrays of the same size.

If you want to get fancy and stupid at the same time, you could use a leaning tree. — Jon Egeland, Feb 21 '15 at 04:07
unless there are negatives, largest sum path is clearly traversing each row, summing all the integers in the triangle...:) — גלעד ברקן, Feb 21 '15 at 17:25
@גלעדברקן There's probably negatives, making it the largest sub-sequence problem? — committedandroider, Feb 21 '15 at 17:40

Christian Sarofeen · Accepted Answer · 2015-02-21T13:21:12.877

3

You should put them in linear memory and access them as:

int triangular(int row){
 return row * (row + 1) / 2 + 1;
}

int[] r = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15};
for(int i=0; i<n_rows; i++){
 for(int j=0; j<=i; j++){
  System.out.print(r[triangular(i)+j]+" ");
 }System.out.println("");
}

row, column
if row>column:
 index=triangular(row)+column

Since it's a predictable structure there's an expression for the offset of the beginning of each row. This will be the most efficient way.

edited Feb 21 '15 at 13:21

answered Feb 21 '15 at 04:07

Christian Sarofeen

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So linear memory meaning array? – committedandroider Feb 21 '15 at 04:26
Updated for a full example. – Christian Sarofeen Feb 21 '15 at 13:21

score 2 · Answer 2 · answered Feb 21 '15 at 04:51

2

I thought about using a 2 dimensional matrix structure but those have to have arrays of the same size.

Not correct.

In Java you can use arrays to represent non-rectangular data structures; e.g.

int[][] triangle = {{1}, 
                    {2, 3},
                    {4, 5, 6},
                    {7, 8, 9, 10},
                    {11, 12, 13, 14, 15}};

It is an option, though not necessarily the most convenient option.

answered Feb 21 '15 at 04:51

Stephen C

698,415
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811
1,216

What would be more convenient? – committedandroider Feb 21 '15 at 07:32
Possibly one of the other solutions proposed by the other answers. You can make up your own mind. – Stephen C Feb 21 '15 at 08:25
I cannot give you a specific answer on that because I have not considered the implications for the algorithms, and their ease of implementation. However, I'm pretty sure that any of the alternatives will be easier to deal with than N different named 1-D arrays ... as in your sample code. – Stephen C Feb 22 '15 at 09:08

score 1 · Answer 3 · answered Feb 21 '15 at 04:54

1

I'd use a 2D array with guard-bands. In the example below, the 0's represent invalid entries in the array. The top and bottom rows, as well as the leftmost and rightmost columns are the guard-bands. The advantage is that your pathfinding algorithm can wander around the array without having to constantly check for out-of-bounds array indices.

int[][] array =
{
    { 0, 0, 0, 0, 0, 0, 0 },
    { 0, 1, 0, 0, 0, 0, 0 },
    { 0, 2, 3, 0, 0, 0, 0 },
    { 0, 4, 5, 6, 0, 0, 0 },
    { 0, 7, 8, 9,10, 0, 0 },
    { 0,11,12,13,14,15, 0 },
    { 0, 0, 0, 0, 0, 0, 0 }
};

answered Feb 21 '15 at 04:54

user3386109

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but woudn't all those 0s take up too much space? – committedandroider Feb 21 '15 at 07:32
How big is the biggest triangle? – user3386109 Feb 21 '15 at 07:47
Question doesn't say. So no stated limit, I guess – committedandroider Feb 21 '15 at 07:52
@committedandroider Well, if you go beyond 10,000 rows, then maybe the extra space taken up by the 0s matters, otherwise no. – user3386109 Feb 21 '15 at 07:54
What I was thinking about was just using one array and using some formula like what duskwuff suggested to identify the different rows of the triangle. That way you save space? – committedandroider Feb 21 '15 at 08:00
@committedandroider Right, you would save space, but is that important? With algorithms you need to consider the space complexity and the time complexity. My intuition is that time complexity will be the limiting factor with this problem, so using more space to reduce the time is a good trade-off. – user3386109 Feb 21 '15 at 08:03
But if you have one array with that formula, why would you need the extra space? – committedandroider Feb 21 '15 at 09:08
@committedandroider I think you need to go through the process of writing all the code before I can explain that to you. – user3386109 Feb 21 '15 at 09:32
This method increases both space and time complexity. It's a wasteful approach. – Christian Sarofeen Feb 21 '15 at 13:15

score 0 · Answer 4 · answered Feb 21 '15 at 04:11

0

You don't need to represent it at all! The starting number of row r (starting at 0) is given by the expression:

r * (r + 1) / 2 + 1

answered Feb 21 '15 at 04:11

So you represent this with one array? – committedandroider Feb 21 '15 at 04:24
If your array is literally just these numbers, you don't need an array at all. You can just calculate what the number at a given position will be without storing it anywhere. – Feb 21 '15 at 04:40
@duskwuff his array is not just those numbers. – user253751 Feb 21 '15 at 05:14
@duskwuff that array was just an example from another thread. I suppose the array for this problem can take any value in – committedandroider Feb 21 '15 at 07:31

How to represent a triangle of integers?

4 Answers4