Sahami Mehran CS103B
Handout #10 January 23, 2009
Analysis of Algorithms: The Recursive Case
We again thank Maggie Johnson for portions of this handout.
Up until now, we have been analyzing non-recursive algorithms, looking at how big-Oh notation may be used to characterize the growth rate of running times for various algorithms. Such algorithm analysis becomes a bit more complicated when we turn our attention to analyzing recursive algorithms. As a result, we augment the analytic tools in our repertoire to help us perform such analyses. One such tool is an understanding of recurrence relations, which we discuss presently.
Recurrence Relations
Recall that a recursive or inductive definition has the following parts: 1. Base Case: the initial condition or basis which defines the first (or first few) elements of the sequence 2. Inductive (Recursive) Case: an inductive step in which later terms in the sequence are defined in terms of earlier terms. The inductive step in a recursive definition can be expressed as a recurrence relation, noting how earlier terms in a sequence relate to late terms. We more formally define this notion below. A recurrence relation for a sequence a1, a2, a3, ... is a formula that relates each term ak to certain of its predecessors ak-1, ak-2, ..., ak-i, where i is a fixed integer and k is any integer greater than or equal to i. The initial conditions for such a recurrence relation specify the values of a1, a2, a3, ..., ai-1. A recursive definition is one way of defining a sequence in mathematics (other ways to define sequences include enumerating the sequence or coming up with a formula to express the sequence). Suppose you have an enumerated sequence that satisfies a given recursive definition (or recurrence relation). It is frequently very useful to have the formula for the elements of this sequence in addition to the recurrence relation, especially if you need to determine a very large member of the sequence. Such an explicit formula is called a solution to the recurrence relation. If a member of the sequence can be calculated using a fixed number of elementary operations, we say it is a closed form formula. Solving recurrence relations is the key to analyzing recursive subprograms.
Example 1
A single pair of rabbits (one male and one female, of course) is born at the end of January 200 9. Assume the following interesting conditions: 1. Rabbit pairs give birth to one new male/female pair at the end of every month after the month in which they are born. 2. No deaths ever occur (not even from exhaustion...)
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Problem: Find a recurrence relation that can be used to tell us how many rabbits we will have at the end of one year (i.e., the end of January 2010).
Solution: We will keep track of time by using the variable x to denote the number of months that have passed since January 2009, and the function Tx to denote how many rabbits are currently in the population as a function of x. Thus, the end of January 2009 is denoted by x = 0. Since we know we have two rabbits at the end of January 2009, we can write: T0 = 2. Note that this function determines the base case in our recursive definition. Now, we need to determine how many rabbits there are in the population as a function of the number of rabbits that were in the population in the previous month. Since every month, each male/female pair produces a new male/female pair, the number of rabbits in the population at the end of a month is equal to the number of rabbits that were already alive, plus one new rabbit pair for each rabbit pair that was previously alive. More formally, we can write this as a recurrence relation: Tx = Tx-1 + Tx-1 Given this recurrence relation, and the value of the base case, T0, we can compute the number of rabbits in the population after the passage of any number of months.
In Death, as in Life Bring on the Recurrence Relation
Many problems in the biological, management, and social sciences lead to sequences that satisfy recurrence relations. Here are the famous Lancaster Equations of Combat: Two armies engage in combat. Each army counts the number of soldiers still in combat at the end of the day. Let a0 and b0 denote the number of soldiers in the first and second armies, respectively, before combat begins, and let an and bn denote the number of men in the two armies at the end of the nth day. Thus, an-1 - an represents the number of soldiers lost by the first army on the nth day. Similarly, bn-1 - bn represents the number of soldiers lost by the second army on the nth day. Suppose it is known that the decrease in the number of soldiers in each army is proportional to the number of soldiers in the other army at the beginning of each day. Thus, we have constants A and B such that an-1 - an = A * bn-1 and bn-1 - bn = B * an-1. These constants measure the effectiveness of the weapons of the different armies, and can be computed using the recurrence relations for a and b.
Solving Recurrence Relations Using Repeated Substitution
One method of solving recurrence relations is called repeated substitution. It works like magic. No, seriously, it works like this: you take the recurrence relation and use it to enumerate elements of the sequence back from k. To find the next element back from k, we substitute k-1 in for k in the recurrence relation.
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Example 2
Consider the following recurrence relation: a0 = 1 ak = ak-1 + 2 Problem: What is the closed form formula for this recurrence relation? We start by substituting k-1 into the recursive definition above to obtain: ak-1 = ak-2 + 2 Then, we substitute this value for ak-1 back in the original recurrence relation equation. We do this because we want to find an equation for ak, so if we enumerate back from k, but substitute those equations in the formula for ak, we will hopefully find a pattern. What we are looking for is a lot of different representations of the formula for ak based on previous elements in the sequence. So, ak = ak-1 + 2 = (ak-2 + 2) + 2 = ak-2 + 4 Then, we substitute k-2 for k in the original equation to get ak-2 = ak-3 + 2 and then substitute this value for ak-2 in the previous expression: ak = ak-2 + 4 = (ak-3 + 2) + 4 = ak-3 + 6 ak = ak-3 + 6 = (ak-4 + 2) + 6 = ak-4 + 8 ak = ak-4 + 8.... The pattern that emerges is obvious: ak = ak-i + 2i. We need to get rid of some of these variables though. Specifically, we need to get rid of the ak-i on the right-hand side of the equation to get a closed form formula. In other words, we do not have to evaluate previous members of the sequence to get the one we want. So, we set i = k, yielding: ak = a0 + 2k ak = 1 + 2k To substantiate this substitution (setting i = k), we would have to do a proof by induction to show that the recurrence relation represents the same sequence as the closed form formula. Since, theres no better time than the present, we present this proof here presently. Proof (by induction): Let P(n) denote the following: if a1, a2, a3, ..., an is the sequence defined by: 1. a0 = 1 2. ak = ak-1 + 2 then, an = 1 + 2n for all n >= 1 Base Case: prove that P(1) is true. Using the definition of the recurrence relation, we have: a1 = a0 + 2 = 1 + 2 = 3. Now, we show that we get the same answer using the formula: an = 1 + 2n for n = 1. an = 1 + 2n = 1 + 2 = 3. Therefore, the base case holds.
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Inductive Case: The induction hypothesis is to assume P(k): ak = 1 + 2k. We then need to show that P(k+1) = ak+1 = 1 + 2(k + 1). ak+1 = a((k+1)1) + 2 ak+1 = ak + 2 ak+1 = 1 + 2k + 2 ak+1 = 1 + 2(k + 1) from the recurrence relation (this is a "given") algebra substitution from inductive hypothesis (ak = 1 + 2k) algebra
Thus, we have shown the desired result. Since P(k+1) is true when P(k) is true, and we have shown P(1) holds, we have shown inductively that P(n) is true for all n >= 1.
Example 3
Recurrence Relations and Graphs: Two Great Tastes that Taste Great Together Let Kn be a complete graph on n vertices (i.e., between any two of the n vertices lies an edge).
Define a recurrence relation for the number of edges in Kn, and then solve this by enumerating and guessing a formula. Solution: By inspection, we can see that K1 = 0, since that graph has no edges. Now, we conjecture that adding another vertex to the graph will require us to add edges from each of the existing vertices in the graph to the new vertex we added. Thus, if there were already n vertices in the graph, when we add the (n + 1)st vertex, we must add n new edges to the graph to make it a complete graph. This gives us the recurrence relation: Kn+1 = Kn + n or, alternatively Kn = Kn-1 + (n 1) Enumerating the first few values of this sequences yields: K2 = K1 + 1 = 0 + 1 = 1 K3 = K2 + 2 = 1 + 2 = 3 K4 = K3 + 3 = 3 + 3 = 6 K5 = K4 + 4 = 6 + 4 = 10 These numbers seem to match the graphs given. So far, so good. Now, we just need to generalize a formula from this sequence. From the recurrence relation, we can see that this sequence may be given by the formula (for all n >= 1):
n-1
Kn
0 + 1 + 2 + ... + (n 1) =
i=0
i
= n(n 1)/2
5 We dont give a full inductive proof here that this formula is, in fact, correct, but the curious reader can quickly prove this themselves to get a little more comfortable with using induction.
Example 4
Consider the Tower of Hanoi puzzle (which you probably saw already in CS106B/X), which is defined by three pegs and a series of disks of various sizes that may be placed on the pegs. The disks all start by being placed in ascending order by size on the first peg. This configuration is depicted below.
The object of the Towers of Hanoi is to move all the disks from the first peg to the third peg while always following the following rules: 1. Only one disk may be moved at a time (specifically, the top disk on any peg). 2. At no time can a larger disk be placed on a smaller one. Thus, the goal configuration of the puzzle is:
The pseudocode for a recursive algorithm that can solve this problem is given below, where k is the number of disks to move, Beg is the peg to move the disks from, Aux is the peg that may be used an "auxiliary storage" during the move, and End is the peg that the disks must ultimately be moved to.
Tower(k, Beg, Aux, End) { if (k == 1) move disk from Beg to End; else { Tower(k-1, Beg, End, Aux); move disk from Beg to End; Tower(k-1, Aux, Beg, End); } }
6 Let Hn denote the number of moves to solve the Tower of Hanoi puzzle with n disks. Moving the n 1 disks from the beginning peg to the auxiliary peg requires Hn-1 moves. Then, use one move to transfer the bottom disk from the beginning peg to the ending peg. Finally, the n 1 disks on the auxiliary peg are moved to the ending peg, requiring an additional Hn-1 moves. This give us the recurrence relation: Hn = 2Hn-1 + 1 The base case for this relations is H0 = 0, since if there are no disks, there are no moves required to have all the disks on the third peg. To determine a closed form formula where there is no Hn on the right side of the equation, we will try repeated substitution: Hn = 2Hn-1 + 1 = 2(2Hn-2 + 1) + 1 = 22 * Hn-2 + 2 + 1 = 22 (2Hn-3 + 1) + 2 + 1 = 23 * Hn-3 + 22 + 2 + 1
The pattern to this recurrence relation is: Hn = (2i * Hn-i) + 2i-1 + 2i-2 + ... + 22 + 21 + 20 We can set n = i (this will require an inductive proof): Hn = (2n * H0) + 2n-1 + 2n-2 + ... + 22 + 21 + 1 Substituting H0 = 0, yields the formula: Hn = 2n-1 + 2n-2 + ... + 22 + 21 + 1 This is a well known summation formula that can easily be proven by induction to be equal to 2n 1. Therefore, the number of moves required for n disks is 2n 1. There is an ancient myth of this puzzle that says there is a tower in Hanoi where monks are transferring 64 gold disks from one peg to another according to the rules of the puzzle. They take one second to move one disk. The myth says that the world will end when they finish the puzzle. A quick calculation to see how much time this will take reveals: 264 1 = 1,446,744,073,709,551,615 seconds So, the monks should be busy for about 500 billion years to solve the puzzle. Lucky for us. Not so lucky for the monks who probably arent having much fun moving disks from peg to peg.
Example 5 (From Rosen)
Problem: Find a recurrence relation and give initial conditions for the number of bit strings (strings of 0s and 1s) of length n that do not have two consecutive 0s. Solution: Let an denote the number of bits strings of length n that do not have two consecutive 0s. To obtain a recurrence relation for an, note that the number of bit strings of length n that do not contain two consecutive 0s equals the number of such bit strings
7 ending with 0 plus the number of such bit strings ending with a 1. We assume n >= 3, so that the bit string has at least three bits. The bit strings of length n ending in 1 that do not contain two consecutive 0s are precisely the bit strings of length n 1 without two consecutives 0s with a 1 added at the end. Consequently, there are an-1 such strings. Bit strings of length n ending with a 0 that do not have two consecutive 0s must have 1 as their (n 1)st bit; otherwise they would end with a pair of 0s since the nth bit is a 0. It follows that the bit strings of length n ending with 0 that do not have two consecutive 0s are precisely the bit strings of length n 2 without two consecutive 0s with 10 added at the end. Consequently, there are an-2 such bit strings. Putting this all together, we conclude that an = an-1 + an-2 for n >= 3. The initial conditions are a1 = 2, since both strings of length 1, namely the strings 0 and 1, do not contain two consecutive 0s. Similarly, a2 = 3, since 01, 10, and 11 do not contain two consecutive 0s.
Example 6
Find a closed form solution for the following recurrence relations. For brevity, we omit the inductive proofs. a) Recurrence relation: a0 = 2 an = 3an-1 Solution: an = 3an-1 2 an = 3 (3an-2) = 3 an-2 2 3 an = 3 (3an-3) = 3 an-3 an = 3
i -1
(3an-i) = 3 an-i
i
Set i = n, yielding: n an = 3 a0 n an = 2 * 3
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b)
Recurrence relation: a0 = 1 an = 2an-1 1 Solution: 2 an = 2(2an-2 1) 1 = 2 an-2 2 1 2 3 2 an = 2 (2an-3 1) 2 1 = 2 an-3 2 2 1 an = 2 (an-i) 2
i i -1
2
i -2
2
0
Set i = n, yielding: n n -1 n -2 0 n n -1 n -2 0 an = 2 (a0) 2 2 2 = 2 2 2 2 n n an = 2 (2 1) an = 1 Note that while the recurrence relation looks deceptively similar to the recurrence relation in the Tower of Hanoi problem, the solution to the recurrence relation here gives a vastly different result. c) Recurrence relation: a0 = 5 an = nan-1 Solution: an = n(n-1)an-2 an = n(n-1)(n-2)an-3 an = n(n-1)(n-2)(n-i+1)an-i Set i = n, yielding: an = n(n-1)(n-2)(1) a0 = a0 n! an = 5n!
Analyzing Recursive Programs
Solving recurrence relations is the key to analyzing recursive programs. We need a formula that represents the number of recursive calls and the work done in each call as the program converges to the base case. We will then use this formula to determine an upper bound on the program's worst-case performance. In terms of the analysis of algorithms, a recurrence relation has can be viewed as follows: A recurrence relation expresses the running time of a recursive algorithm. This includes how many recursive calls are generated at each level of the recursion, how much of the problem is solved by each recursive call, and how much work is done at each level.
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Say we have some unknown running time for function F that is defined in terms of F's parameter n (since this parameter controls the number of recursive calls). We call this TF(n). The value of TF(n) is established by an inductive definition based on the size of argument n. There are two cases, examined below: 1. n is sufficiently small that no recursive calls are made. This case corresponds to a base case in an inductive definition of TF(n). 2. n is sufficiently large that recursive calls made. are However, we assume that whatever recursive calls F makes, they will be made with smaller parameter values. This case corresponds to an inductive step in the definition of TF(n). The recurrence relation for TF(n) is derived by examining the code of F and defining the running times for the two cases stated above. We then solve the recurrence relation to derive a big-Oh expression for F.
Example 7
Consider the following function to compute factorials:
int factorial(int n) { if (n <= 1) return(1); else return(n * factorial(n-1)); }
1) 2) 3)
We will use T(n) to represent the unknown running time of this function with n as the size of the parameter. For the base case of the inductive definition of T(n), we will take n to equal 1, since no recursive calls are made in this case. When n = 1, we execute only lines 1 and 2, each of which has a constant running time (i.e., O(1)) which we will just designate as some time value a. When n > 1, the condition of line 1 is false so we execute lines 1 and 3. Line 1 takes some constant time value b and line 3 takes T(n-1) since we are making a recursive call. If the running time of factorial with parameter value n is T(n), then the running time for a call to factorial with parameter value n 1 will be T(n-1). This gives the following recurrence relation: base case: T(1) = a induction: T(n) = b + T(n-1), for n > 1
10 Now we must solve this recurrence relation to come up with a closed form formula. We can try just enumerating a few to see if there is a pattern: T(1) = a T(2) = b + T(1) = b + a T(3) = b + T(2) = b + (b + a) = a + 2b T(4) = b + T(3) = b + (a + 2b) = a + 3b T(n) = a + (n-1)b for all n >= 1 To be complete, we need to do an inductive proof to show that the running time formula T(n) = a + (n-1)b is true for all n >= 1. Proof: Let P(n) denote that the running time of factorial is T(n) = a + (n-1)b. And, recall that the recurrence relation definition give us: T(1) = a, and T(n) = b + T(n-1) for n > 1. Base case: prove that P(1) is true T(1) = a + ((1 1) * b) = a + (0 * b) = a. This equation holds since the base case of our inductive definition states that T(1) = a. Inductive case: inductive hypothesis is to assume P(k): T(k) = a + (k-1)b, and show P(k+1): T(k+1) = a + kb. We know from the recurrence relation definition that T(k+1) = b + T(k). We use the inductive hypothesis to substitute for T(k). This gives us: T(k+1) = b + T(k) = b + a + (k-1)b = b + a + kb - b = a + kb P(k+1) holds when P(k) holds, and P(1) holds, therefore P(n) is true for all n >= 1. Now that we know the closed form formula for this recurrence relation, we use this formula to determine the complexity. T(n) = a + (n-1)b = a + bn - b has a complexity of O(n). This makes sense: all it says is that to compute n!, we make n calls to factorial, each of which requires O(1) time.
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Example 8
Consider the following recursive selection sort algorithm:
void SelectionSort(int A[], int i, int n) {
1) 2) 3) 4) 5) 6) 7) 8) 9)
}
if (i < n) { int small = i; for (int j = i + 1, j <= n, j++) { if (A[j] < A[small]) small = j; } int temp = A[small]; A[small] = A[i]; A[i] = temp; SelectionSort(A, i + 1, n); }
Parameter n indicates the size of the array, parameter i tells us how much of the array is left to sort, specifically A[i...n]. So, a call to SelectionSort(A,1,n) will sort the entire array through recursive calls. We now develop a recurrence relation. Note that the size of the array to be sorted on each recursive call is equal to n i + 1. We will denote this value as m, the size of the array during any particular recursive call. There is one base case (m = 1). In this case, only line 1 is executed, taking some constant amount of time which we will call a. Note that m = 0 is not the base case because the recursion converges down to a list with one element; when i = n, m = 1. The inductive case is for m > 1: this is when recursive calls are made. Lines 2, 6, 7, and 8 each take a constant amount of time. The for loop of lines 3, 4 and 5 will execute m times (where m = n i + 1). So a recursive call to this function will be dominated by the time it takes to execute the for loop m times, which we shall designate O(m). The time for the recursive call of line 9 is T(m-1). So, the inductive definition of recursive SelectionSort is: T(1) = a T(m) = T(m-1) + O(m) To solve this recurrence relation, we first get rid of the big-Oh expression by substituting the definition of big-Oh: (f(n) = O(g(n)) if f(n) <= C * g(n)), so we can substitute C*m for O(m): T(1) = a T(m) = T(m-1) + C*m
12 Now, we can either try repeated substitutions or just enumerate a few cases to look for a pattern. Lets try repeated substitution: T(m) = T(m-1) + C*m = T(m-2) + 2Cm - C = T(m-3) + 3Cm - 3C = T(m-4) + 4Cm - 6C = T(m-5) + 5Cm - 10C ... = T(m-j) + jCm - (j(j-1)/2)C because T(m-1) = T(m-2) + C(m-1) because T(m-2) = T(m-3) + C(m-2) because T(m-3) = T(m-4) + C(m-3) because T(m-4) = T(m-5) + C(m-4)
To get a closed form formula we let j = m 1. We do this because our base case is T(1). If we were to continue the repeated substitution down to the last possible substitution, we want to stop at T(1) because T(0) is really not the base case that the recursion converges on. (Note that we have to do an inductive proof later to allow us to do this substitution, but for now): T(m) = T(1) + (m-1)Cm - ((m-1)(m-2)/2)C = a + m2C - Cm - (m2C - 3Cm + 2C)/2 = a + (2m2C - 2Cm - m2C + 3Cm - 2C)/2 = a + (m2C + Cm - 2C)/2 So, finally we have a closed form formula T(m) = a + (m2C + Cm - 2C)/2. The complexity of this formula is O(m2), which is the same complexity as iterative selection sort, so doing it recursively did not save us any time. The above recurrence relation generally occurs with any recursive subprogram that has a single loop of some form, and then does a recursive call (usually where there is tail recursion involved).
Example 9
MergeSort is an efficient sorting algorithm that uses a "divide and conquer" strategy. That is, the problem of sorting a list is reduced to the problem of sorting two smaller lists, and then merging those smaller sorted lists. Suppose that we have a sorted array A with r elements and another sorted array B with s elements. Merging is an operation that combines the elements of A with the elements of B into one large sorted array C that has (r + s) elements. MergeSort is based on this merging algorithm. What happens is we set up a recursive function that divides a list into two halves. We then sort the two halves and merge them together. The question is how do you sort the two halves? We sort them by using MergeSort recursively. The pseudocode for MergeSort is given below:
MergeSort(L) 1) if (length of L > 1) { 2) Split list into first half and second half 3) MergeSort(first half) 4) MergeSort(second half) 5) Merge first half and second half into sorted list }
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Note: we will also assume that we have Split and Merge functions that each run in O(n). The recursive calls continue dividing the list into halves until each half contains only one element; obviously a list of one element is sorted. The algorithm then merges these smaller halves into larger sorted halves until we have one large sorted list. For example, if the list contains the following integers: 7 9 2 4 3 6 1 8
The first thing that happens is we split the list into 2 halves (well refer to this call as MergeSort #0 = original call): 7 9 4 2 3 6 1 8
Then we call Mergesort recursively on the first half which again splits the list into two halves: (MergeSort #1): 7 9 4 2
We call Mergesort recursively again on the first half, which splits the list into two parts again: (MergeSort #2) 7 9
When we call MergeSort the next time, the if condition is false and we return up a level of recursion to MergeSort #2. Now we execute the next line in MergeSort #2: MergeSort(second half) which executes on the list containing just 9. Again the if condition is false, and we return back up to MergeSort #2. We execute the next line, which merges the sorted lists containing 7 and 9, producing: 7 9
We have completed MergeSort #2. Now we return up to recursive call MergeSort #1 and find we are on the list containing 4 and 2. We execute the next line in MergeSort #1, calling MergeSort(second half) on the list containing 4 and 2, and end up with separate lists of 4 and 2: 4 2
We end up merging these two sorted lists into one list containing 2 and 4 just as we did with the 7 and 9, producing: 2 4
Once that work is complete, we return to MergeSort #1 and execute the merge on the two lists each with two elements. This produces the sorted list: 2 4 7 9
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We return to recursive call MergeSort #0 (the original call) and execute the next line: MergeSort(second half). This sends us through another cycle of recursion just like what we did for the first part of the list. When we finish, we have a second sorted list: 1 3 6 8
The last line of MergeSort #0 merges these two sorted lists of four elements into one sorted list: 1 2 3 4 6 7 8 9
To analyze the complexity of MergeSort, we need to define a recurrence relation. The base case is when we have a list of 1 element and only line 1 of the function is executed. Thus, the base case is constant time, O(1). If the test of line 1 fails, we must execute lines 2-5. The time spent in this function when the length of the list > 1 is the sum of the following: 1) O(1) for the test on line 1 2) O(n) for the split function call on line 2 3) T(n/2) for recursive call on line 3 4) T(n/2) for recursive call on line 4 5) O(n) for the merge function call on line 5 If we drop the O(1)'s and apply the summation rule, we get 2T(n/2) + O(n). If we substitute constants in place of the big-Oh notation we obtain: T(1) = a T(n) = 2T(n/2) + bn To solve this recurrence relation, we will enumerate a few values. We will stick to n's that are powers of two so things divide evenly: T(2) = 2T(1) + 2b = 2a + 2b T(4) = 2T(2) + 4b = 2(2a + 2b) + 4b = 4a + 8b T(8) = 2T(4) + 8b = 2(4a + 8b) + 8b = 8a + 24b T(16) = 2T(8) + 16b = 2(8a + 24b) + 16b = 16a + 64b There is obviously a pattern but it is not as easy to represent as the others have been. Note the following relationships: value of n: coefficient of b: ratio: 2 2 1 4 8 2 8 24 3 16 64 4
So, it appears that the coefficient of b is n times another factor that grows by 1 each time n doubles. The ratio is log2 n because log2 2 = 1, log2 4 = 2, log2 8 = 3, etc. Our "guess" for the solution of this recurrence relation is T(n) = an + bn log2 n.
15 We could have used repeated substitution in which case we would have the following formula: T(n) = 2i T(n/2i) + ibn Now, if we let i = log2 n, we end up with: n*T(1) + bn log2 n = an + bn log2 n (because 2log2n = n). Of course, we would have to do an inductive proof to show that this formula does indeed hold for all n >= 1, but we leave the proof as an exercise for the reader. Finally, note that the complexity of an + bn log2 n is O(n log n), which is considerably faster than Selection Sort. You will often see this recurrence relation in "divide and conquer" sorts.
A Theorem in the Home or Office is a Good Thing
Fortunately, there is a handy theorem that is often useful in analyzing the complexity of general divide and conquer algorithms. Theorem 1 (Master Theorem): Let f be an increasing function that satisfies the recurrence relation: f(n) = a f(n/b) + cnd whenever n = bk, where k is a positive integer, a >= 1, b is an integer greater than 1, c is a positive real number, and d is a non-negative real number. Then: f(n) = O(nd) O(nd log n) O(n
logb a
if a < bd if a = bd if a > bd
)
Example 10
Problem: Use Theorem 1 above to find the big-Oh running time of MergeSort (from Example 9). Solution: In Example 9, we are given the recurrence relation for MergeSort as: T(n) = 2T(n/2) + xn where we have simply replaced the positive constant b in the original recurrence relation with the constant x, so that we do not confuse variable names below. In order to apply Theorem 1, we must show that this recurrence relation fits the functional form for f(n) in the theorem. We choose: a = 2, b = 2, c = x, and d = 1. With these constants, the function f(n) in the theorem becomes: f(n) = 2 f(n/2) + xn1
16 Note that f(n) now matches the function T(n) given earlier, making Theorem 1 applicable to the recurrence relation T(n). Given our choice of constants, we know that a = bd, since 2 = 21. Thus, Theorem 1 tell us the complexity of our recurrence relation is O(nd log n). Since we chose d = 1, we obtain the final complexity result O(n log n), which is the same complexity value we obtained via repeated substitution.
Example 11
Problem: Use repeated substitution to find the time complexity of the function recurse. Verify your result using induction.
/* Assume only non-negative even values of n are passed in */ void recurse(int n) { int total = 0;
1) 2) 3) 4)
}
if (n == 0) return 1; for (int i = 0; i < 4; i++) { total += recurse(n-2); } return total;
Solution: When n = 0 (the base case), this function only executes line 1, doing O(1) work. We will note this constant amount of work as a. When n >= 2 (the recursive case), this function does O(1) work in executing lines 1, 2, and 4, which we will denote by b, and also makes four recursive calls with parameter values n-2 in the body loop (line 3). We use these two points to perform the recurrence relation analysis below. From the base case, we have: From the recursive case, we have: T0 = a Tn = 4Tn-2 + b
We apply repeated substitution: Tn = 4Tn-2 + b 2 Tn = 4(4Tn-4 + b) + b = 4 Tn-4 + 4b + b 2 3 2 Tn = 4 (4Tn-6 + b) + 4b + b = 4 Tn-6 + 4 b + 4b + b Tn = 4 Tn-2i + 4
k 2k i i- 1
b+4
i -2
b++4 b
0
Note that 4 = 2 , yielding: 2i 2(i-1) 2(i-2) 0 Tn = 2 Tn-2i + 2 b+2 b++2 b Set i = n/2, yielding: n n -2 n -4 0 n n -2 n -4 0 Tn = 2 T0 + 2 b + 2 b + + 2 b = 2 a + (2 + 2 + + 2 ) b
17
So, we need to compute the closed form for (2
n -2
+2
n -4
+ + 2 ) = (2 1)/3.
0
n
Here is a brief digression on how we compute this closed form formula: n -2 n -4 0 Let x = (2 + 2 + + 2 ) 2 2 n -2 n -4 0 n n -2 2 So, 2 x = 4x = 2 (2 + 2 + + 2 ) = (2 + 2 + + 2 ) 2 x x = 3x = (2 + 2 n 0 n 3x = 2 2 = 2 1 x = (2 1)/3 = (2
n n -2 2 n n -2
+ + 2 ) (2
0
2
n -2
+2
n -4
++2 )
0
+2
n -4
++2 )
Substituting this closed form for the sum back in to the equation for Tn yields: n n -2 n -4 0 Tn = 2 a + (2 + 2 + + 2 ) b n n Tn = 2 a + ((2 1)/3) b Now, we verify this result using induction. Let P(n) denote Tn = 2 a + ((2 1)/3) b. Base case: Show P(0). 0 0 T0 = 2 a + ((2 1)/3) b = a + ((1 1)/3) b = a. This is the initial condition. Inductive case: Assume inductive hypothesis P(n): Tn = 2 a + ((2 1)/3) b Show P(n+2): Tn+2 = 2
n+2 n n n n
a + ((2
n+2
1)/3) b
Note: only consider even values of n
Tn+2 = 4Tn + b n n Tn+2 = 4(2 a + ((2 1)/3) b) + b 2n n Tn+2 = 2 (2 a + ((2 1)/3) b) + b n+2 n+2 2 Tn+2 = 2 a + ((2 2 )/3) b + b n+2 n+2 Tn+2 = 2 a + ((2 3 1)/3) b + b n+2 n+2 Tn+2 = 2 a + ((2 1)/3) b b + b n+2 n+2 Tn+2 = 2 a + ((2 1)/3) b QED.
by definition of the recurrence substituting Tn with the inductive hypothesis 2 algebra: 4 = 2 2 algebra: multiply in the 2 2 algebra: -2 = -3 - 1 algebra: (-3/3)b = -b this is the desired result
Finally, we calculate the big-Oh complexity for the function, as follows: n n From the recurrence relation Tn = 2 a + ((2 1)/3) b, we see that the algorithm has time n n n n n complexity: O(2 a) + O(2 b) = O(2 ) + O(2 ) = O(2 ).
Bibliography
A. Aho, J. D. Ullman, Foundations of Computer Science, New York: W.H. Freeman, 1992. T. Cormen, C. Leiserson, R. Rivest, Introduction to Algorithms, New York: McGraw-Hill, 1991. K. Rosen, Discrete Mathematics and its Applications (6th Ed.), New York: McGraw-Hill, 2007.