The recursive nature of D&C leads to recurrences, or functions defined in terms of:
In the case of the Insertion Sort we saw Incremental Strategy for designing algorithms. Another strategy which is very powerfull is to Divide and Conquer:
Merge Sort (which you should have seen in ICS 211 (Complex Sorting Algorithms)) takes this strategy:
The strategy can be written simply and elegantly in recursive code ...
Here are examples when the input is a power of two, and another example when it is not a power of two:
Now let's look in detail at the merge procedure, implemented using ∞ as sentinels (what do lines 1-2 do? lines 3-9 ? lines 10-17?):
Here's an example of how the final pass of MERGE(9, 12, 16) happens in an array, starting at line 12. Entries with slashes have had their values copied to either L or R and have not had a value copied back in yet. Entries in L and R with slashes have been copied back into A.
We can also dance this one: http://youtu.be/XaqR3G_NVoo
A loop invariant is used in the book to establish correctness of the Merge procedure. Since the loop is rather straightforward, we will leave it to the above example. Once correctness of Merge is established, induction can be used to show that Merge-Sort is correct for any N.
Merge Sort provides us with our first example of using recurrence relations and recursion trees for analysis.
Analysis of the Merge procedure is straightforward. The first two for loops (lines 4 and 6) take Θ(n1+n2) = Θ(n) time, where n1+n2 = n. The last for loop (line 12) makes n iterations, each taking constant time, for Θ(n) time. Thus total time is Θ(n).
Recurrence equations are used to describe the run time of Divide & Conquer algorithms. Let T(n) be the running time on a problem of size n.
Then the total time to solve a problem of size n by dividing into a problems of size n/bcan be expressed as:
Merge-Sort is called with p=1 and r=n. For simplicity, assume that n is a power of 2. (We can always raise a given n to the next power of 2, which gives us an upper bound on a tighter Θ analysis.) When n≥2, the time required is:
Suppose you have an array of numbers and need to find the subarray with the maximum sum of elements in the subarray. (The problem is trival unless there are negative numbers involved.)
The following algorithm is not the fastest known (a linear solution exists), but it illustrates divide and conquer. The solution strategy, given an array A[low .. high], is:
The strategy works because any subarray must lie in one of these three positions:
Recursion will handle the lower and upper halves. The algorithm relies on a helper to find the crossing subarray. Any maximum subarray crossing the midpoint must include arrays ending at A[mid] and starting at A[mid+1]:
Therefore the pseudocode finds the maximum array on each side and adds them up:
It should be clear that the above is Θ(n). The recursive solution follows.
Check your understanding: Where is the work done? What adds up the values in the left and right subarrays?
The analysis relies on the simplifying assumption that the problem size is a power of 2 (the same assumption for merge sort). Let T(n) denote the running time of FIND-MAXIMUM-SUBARRAY on a subarray of n elements.
The resulting recurrence is the same as for merge sort:
So how do we solve these recurrences? We have three methods: Substitution, Recursion Trees, and the Master Method.
Don't you love it when a "solution method" starts with ...
Recursion trees (next section) are one way to guess solutions. Experience helps too. For example, if a problem is divided in half we may expect to see lg n behavior.
As an example, let's solve the recurrence for merge sort and maximum subarray. We'll start with an exact rather than asymptotic version:
Induction would require that we show our solution holds for the boundary conditions. This is discussed in the textbook.
Normally we use asymptotic notation rather than exact forms:
If we want Θ, sometimes we can prove big-O and Ω separately "squeezing" the Θ result.
But be careful when using asymptotic notation. For example, suppose you have the case where a=4 and b=4 and want to prove T(n) = O(n) by guessing that T(n) ≤ cn and writing:
One must prove the exact form of the inductive hypothesis, T(n) ≤ cn.
See the text for other strategies and pitfalls.
Problems 4.3-1 and 4.3-2 are good practice problems.
Recursion trees provide an intuitive understanding of the above result. Although recursion trees can be considered a proof format, for a formal analysis, they must be applied very carefully. Instead, they are used to generate guesses that are verified by substitution.To construct the recursion tree, do the following:
Let's choose a constant c that is the largest of all the constant costs in the algorithm (the base case and the divide steps). Then the recurrence can be written:
It costs cn to divide the original problem in half and then to merge the results. We then have to pay cost T(n/2) twice to solve the subproblems:
For each of the two subproblems, n/2 is playing the role of n in the recurrence. So, it costs cn/2 to divide and then merge the n/2 elements, and T(n/4) to solve the subproblems:
If we continue in this manner we eventually bottom out at problems of size 1:
Notice that if we sum across the rows each level has cost cn. So, all we have to do is multiply this by the number of levels. Cool, huh?
But how many levels are there? A little thought (or a more formal inductive proof you'll find in the book) shows that there are about (allowing for the fact that n may not be a power of 2) lg(n)+1 levels of the tree. This is because you can only divide a power of two in half as many times as that power before you reach 1, and n = 2lg(n). The 1 counts the root node before we start dividing: there is always at least one level.
Questions? Does it make sense, or is it totally mysterious?
A more complex example is developed in the textbook for
T(n) = 3T(n/4) + Θ(n2)
which is rewritten (making the implied constant explicit) as
T(n) = 3T(n/4)+ cn2node, T(n) = 3T(n/4) +cn2.
We can develop the recursion tree in steps, as follows. First, we begin the tree with its root:
Now let's branch the tree for the three recursive terms 3T(n/4). There are three children nodes with T(n/4) as their cost, and we leave the cost cn2 behind at the root node:
We repeat this for the subtrees rooted at each of the nodes for T(n/4): Since each of these costs 3T((n/4)/4) +c(n/4)2, we make three branches, each costing T((n/4)/4) = T(n/16), and leave the c(n/4)2 terms behind at their roots:
Continuing this way until we reach the leaf nodes where the recursion ends at trivial subproblems T(1), the tree looks like this:
Subproblem size for a node at depth i is n/4i, so
the subproblem size reaches n = 1 when (assuming n a power of 4)
n/4i = 1, or when i = log4n.
Including i = 0, there are log4n + 1 levels. Each level has 3i nodes.
Substituting i = log4n into 3i, there are 3log4n nodes in the bottom level.
Using alogbc = clogba, there are nlog43 in the bottom level (not n, as in the previous problem).
Adding up the levels, we get:
It is easier to solve this summation if we change the equation to an inequality and let the
summation go to infinity (the terms are decreasing geometrically), allowing us to apply equation
A.6 (∑k=0,∞xk = 1/1-x):
Additional observation: since the root contributes cn2, the root dominates the cost of the tree, and the recurrence must also be Ω(n2), so we have Θ(n2).
Please see the text for an example involving unequal subtrees. For practice, exercises 4.4-6 and 4.4-9 have solutions posted on the book's web site.
If we have a divide and conquer recurrence of the form
T(n) = aT(n/b) + f(n)
where a ≥ 1, b > 1, and f(n) > 0 is asymptotically positive,
then we can apply the master method, which is based on the master theorem. We compare f(n) to nlogba under asymptotic (in)equality:Case 1: f(n) = O(nlogba - ε) for some constant ε > 0.
Important: there are functions that fall between the cases!
T(n) = 5T(n/2) + Θ(n2)
T(n) = 27T(n/3) + Θ(n3 lg n)
T(n) = 5T(n/2) + Θ(n3)
T(n) = 27T(n/3) + Θ(n3 / lg n)
Recapitulating our conclusions, we have seen that Insertion sort is quick on already sorted data, so it works well when incrementally adding items to an existing list. Due to its simplicity it is a good choice when the sequence to sort will always be small. But for large inputs Merge Sort will be faster than Insertion Sort, as n2 grows much faster than nlg(n). Each sort algorithm has different strengths and weaknesses, and performance depends on the data. Some of these points are made in the following visualizations (also watch for patterns that help you understand the strategies):
http://www.sorting-algorithms.com/ (set to 50 elements)