Why lookup in a Binary Search Tree is O(log(n))?

Question

I can see how, when looking up a value in a BST we leave half the tree everytime we compare a node with the value we are looking for.

However I fail to see why the time complexity is O(log(n)). So, my question is:

If we have a tree of N elements, why the time complexity of looking up the tree and check if a particular value exists is O(log(n)), how do we get that?

Solution 1

Your question seems to be well answered here but to summarise in relation to your specific question it might be better to think of it in reverse; "what happens to the BST solution time as the number of nodes goes up"?

Essentially, in a BST every time you double the number of nodes you only increase the number of steps to solution by one. To extend this, four times the nodes gives two extra steps. Eight times the nodes gives three extra steps. Sixteen times the nodes gives four extra steps. And so on.

The base 2 log of the first number in these pairs is the second number in these pairs. It's base 2 log because this is a binary search (you halve the problem space each step).

Solution 2

For me the easiest way was to look at a graph of log2(n), where n is the number of nodes in the binary tree. As a table this looks like:

          log2(n) = d

          log2(1) = 0
          log2(2) = 1 
          log2(4) = 2
          log2(8) = 3
          log2(16)= 4 
          log2(32)= 5
          log2(64)= 6             

and then I draw a little binary tree, this one goes from depth d=0 to d=3:

            d=0          O
                        / \
            d=1        R   B
                      /\   /\
            d=2      R  B R  B
                    /\ /\ /\ /\
            d=3    R B RB RB R B

So as the number of nodes, n, in the tree effectively doubles (e.g. n increases by 8 as it goes from 7 to 15 (which is almost a doubling) when the depth d goes from d=2 to d=3, increasing by 1.) So the additional amount of processing required (or time required) increases by only 1 additional computation (or iteration), because the amount of processing is related to d.

We can see that we go down only 1 additional level of depth d, from d=2 to d=3, to find the node we want out of all the nodes n, after doubling the number of nodes. This is true because we've now searched the whole tree, well, the half of it that we needed to search to find the node we wanted.

We can write this as d = log2(n), where d tells us how much computation (how many iterations) we need to do (on average) to reach any node in the tree, when there are n nodes in the tree.

Solution 3

This can be shown mathematically very easily.

Before I present that, let me clarify something. The complexity of lookup or find in a balanced binary search tree is O(log(n)). For a binary search tree in general, it is O(n). I'll show both below.

In a balanced binary search tree, in the worst case, the value I am looking for is in the leaf of the tree. I'll basically traverse from root to the leaf, by looking at each layer of the tree only once -due to the ordered structure of BSTs. Therefore, the number of searches I need to do is number of layers of the tree. Hence the problem boils down to finding a closed-form expression for the number of layers of a tree with n nodes.

This is where we'll do a simple induction. A tree with only 1 layer has only 1 node. A tree of 2 layers has 1+2 nodes. 3 layers 1+2+4 nodes etc. The pattern is clear: A tree with k layers has exactly

n=2^0+2^1+...+2^{k-1}

nodes. This is a geometric series, which implies

n=2^k-1,

equivalently:

k = log(n+1)

We know that big-oh is interested in large values of n, hence constants are irrelevant. Hence the O(log(n)) complexity.

I'll give another -much shorter- way to show the same result. Since while looking for a value we constantly split the tree into two halves, and we have to do this k times, where k is number of layers, the following is true:

(n+1)/2^k = 1,

which implies the exact same result. You have to convince yourself about where that +1 in n+1 is coming from, but it is okay even if you don't pay attention to it, since we are talking about large values of n.

Now let's discuss the general binary search tree. In the worst case, it is perfectly unbalanced, meaning all of its nodes has only one child (and it becomes a linked list) See e.g. https://www.cs.auckland.ac.nz/~jmor159/PLDS210/niemann/s_fig33.gif

In this case, to find the value in the leaf, I need to iterate on all nodes, hence O(n).

A final note is that these complexities hold true for not only find, but also insert and delete operations.

(I'll edit my equations with better-looking Latex math styling when I reach 10 rep points. SO won't let me right now.)

Solution 4

Whenever you see a runtime that has an O(log n) factor in it, there's a very good chance that you're looking at something of the form "keep dividing the size of some object by a constant." So probably the best way to think about this question is - as you're doing lookups in a binary search tree, what exactly is it that's getting cut down by a constant factor, and what exactly is that constant?

For starters, let's imagine that you have a perfectly balanced binary tree, something that looks like this:

                     *
              /             \
             *               *
          /     \         /     \
         *       *       *       *
        / \     / \     / \     / \
       *   *   *   *   *   *   *   *

At each point in doing the search, you look at the current node. If it's the one you're looking for, great! You're totally done. On the other hand, if it isn't, then you either descend into the left subtree or the right subtree and then repeat this process.

If you walk into one of the two subtrees, you're essentially saying "I don't care at all about what's in that other subtree." You're throwing all the nodes in it away. And how many nodes are in there? Well, with a quick visual inspection - ideally one followed up with some nice math - you'll see that you're tossing out about half the nodes in the tree.

This means that at each step in a lookup, you either (1) find the node that you're looking for, or (2) toss out half the nodes in the tree. Since you're doing a constant amount of work at each step, you're looking at the hallmark behavior of O(log n) behavior - the work drops by a constant factor at each step, and so it can only do so logarithmically many times.

Now of course, not all trees look like this. AVL trees have the fun property that each time you descend down into a subtree, you throw away roughly a golden ratio fraction of the total nodes. This therefore guarantees you can only take logarithmically many steps before you run out of nodes - hence the O(log n) height. In a red/black tree, each step throws away (roughly) a quarter of the total nodes, and since you're shrinking by a constant factor you again get the O(log n) lookup time you'd like. The very fun scapegoat tree has a tuneable parameter that's used to determine how tightly balanced it is, but again you can show that every step you take throws away some constant factor based on this tuneable parameter, giving O(log n) lookups.

However, this analysis breaks down for imbalanced trees. If you have a purely degenerate tree - one where every node has exactly one child - then every step down the tree that you take only tosses away a single node, not a constant fraction. That means that the lookup time gets up to O(n) in the worst case, since the number of times you can subtract a constant from n is O(n).

Solution 5

If we have a tree of N elements, why the time complexity of looking up the tree and check if a particular value exists is O(log(n)), how do we get that?

That's not true. By default, a lookup in a Binary Search Tree is not O(log(n)), where n is a number of nodes. In the worst case, it can become O(n). For instance, if we insert values of the following sequence n, n - 1, ..., 1 (in the same order), then the tree will be represented as below:

                  n
                 /
              n - 1
               /
            n - 2
             /
           ...
           1

A lookup for a node with value 1 has O(n) time complexity.

To make a lookup more efficient, the tree must be balanced so that its maximum height is proportional to log(n). In such case, the time complexity of lookup is O(log(n)) because finding any leaf is bounded by log(n) operations.

But again, not every Binary Search Tree is a Balanced Binary Search Tree. You must balance it to guarantee the O(log(n)) time complexity.

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Comments

[Himel Das]

Holy moly, the way you explained it I understood the reason in a few seconds. Thanks!