Change problem - Revision history

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‎Greedy algorithm

Brian at 18:00, 10 March 2012

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Brian: Created page with "The ''change-making problem'', often simply known as '''change''', occurs in two distinct but related flavours. Given a set of natural numbers $D_1, D_2, ..., D_n$ (th..."

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Created page with "The ''change-making problem'', often simply known as '''change''', occurs in two distinct but related flavours. Given a set of natural numbers <math>D_1, D_2, ..., D_n</math> (th..."

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The ''change-making problem'', often simply known as '''change''', occurs in two distinct but related flavours. Given a set of natural numbers <math>D_1, D_2, ..., D_n</math> (the ''denominations'') and a natural number <math>T</math> (the target amount for which to make change):
* ''Optimization problem'': Make change for the target amount using as few total coins as possible. Formally, find nonnegative integers <math>k_1, k_2, ..., k_n</math> that minimize <math>k_1 + k_2 + ... + k_n</math> subject to <math>k_1 D_1 + k_2 D_2 + ... + k_n D_n = T</math>, or determine that this is impossible.
* ''Counting problem'': Find the total number of different ways to make change for the target amount. Formally, count the number of tuples of nonnegative integers <math>(k_1, k_2, ..., k_n)</math> that satisfy <math>k_1 D_1 + k_2 D_2 + ... + k_n D_n = T</math>.
The former, the optimization problem, should be very familiar. It is a problem that cashiers solve when (for example) you hand them a five-dollar bill for a purchase of $3.44, and they have to find a way to give you back $1.56 in ''change'', preferably with as few coins as possible.<ref>The Canadians have a $2.00 coin, a $1.00 coin, a $0.25 coin, a $0.10 coin, a $0.05 coin, and a $0.01 coin. The solution to this problem in Canadian currency is one $1.00 coin, two $0.25 coins, one $0.05 coin, and one $0.01 coin.</ref> It is a well-studied problem that is extensively treated in the literature. The counting problem is somewhat more exotic; it arises mostly as a mathematical curiosity, often in the form of the puzzle: "How many ways are there to make change for a dollar?"<ref>The denominations in circulation in the United States are the same as those in Canada, except for the absence of a $2.00 coin and the presence of a $0.50 coin. In United States currency, the answer is 293 if the trivial solution consisting of a single one-dollar coin is counted, or 292 if it is not.</ref> It is also not often encountered in the literature, but it is discussed here because it occasionally appears in algorithmic programming competitions.

==Discussion of complexity==
The corresponding decision problem, which simply asks us to determine whether or not making change is ''possible'' with the given denominations (which it might not be, if we are missing the denomination 1) is known to be [[NP-complete]].<ref>G. S. Lueker. (1975). ''Two NP-complete problems in nonnegative integer programming.'' Technical Report 178, Computer Science Laboratory, Princeton University''</ref> It follows that the optimization and counting problems are both [[NP-hard]] (''e.g.'', because the result of 0 for the counting problem answers the decision problem in the negative, and any nonzero value answers it in the affirmative).

However, as we shall see, a simple <math>O(nT)</math> solution exists for both versions of the problem. Why then are these problems not in P? The answer is that the ''size'' of the input required to represent the number <math>T</math> is actually the ''length'' of the number <math>T</math>, which is <math>\Theta(\log T)</math> when <math>T</math> is expressed in binary (or decimal, or whatever). Thus, the time and space required by the algorithm is actually <math>O(n 2^{\lg T})</math>, that is, exponential in the size of the input. (This simplified analysis does not take into account the sizes of the denominations, but captures the essence of the argument.) This algorithm is then said to be ''pseudo-polynomial''. No true polynomial-time algorithm is known (and, indeed, none will be found unless it turns out that P = NP).

==Greedy algorithm==
Many real-world currency systems admit a [[greedy solution]] to the optimization version of the change problem. This algorithm is as follows: repeatedly choose the largest denomination that is less than or equal to the target amount, and ''use it'', that is, subtract it from the target amount, and then repeat this procedure on the reduced value, until the target amount decreases to zero. For example, with Canadian currency, we can greedily make change for $0.63 as follows: the largest denomination that fits into $0.63 is $0.25, so we subtract that (and thus resolve to use a $0.25 coin); we are left with $0.38, and take out another $0.25, so we subtract that again to obtain $0.13 (so that we have used two $0.25 coins so far); now the largest denomination that fits is $0.10, so we subtract that out, leaving us with $0.03; and then we subtract three $0.01 coins, leaving us with $0.00, at which point the algorithm terminates; so we have used six coins (two $0.25 coins, one $0.10 coin, and three $0.01 coins).

It turns out that the greedy algorithm ''always'' gives the correct result for both Canadian and United States currencies (the proof is left as an exercise for the reader). There are various other real-world currency systems for which this is also true. However, there are simple examples of sets of denominations for which the greedy algorithm does ''not'' give a correct solution. For example, with the set of denominations <math>{1, 3, 4}</math>, the greedy algorithm will change 6 as 4+1+1, using three coins, whereas the correct minimal solution is obviously 3+3. There are also cases in which the greedy algorithm will fail to make change at all (consider what happens if we try to change 7 using the denominations <math>{3, 4}</math>). This usually does not occur in real-world systems because they tend to have denominations that are quite a bit more "spaced out".

Obviously, there is no greedy solution to the counting problem.

==Dynamic programming solution==
The optimization problem exhibits optimal substructure, in the sense that if we remove any coin of value <math>D_i</math> from the optimal means of changing <math>T</math>, then the set of coins remaining is an optimal means of changing <math>T - D_i</math>. This is because if this were ''not'' so; that is, there existed a means of changing <math>T - D_i</math> that used fewer coins than what we obtained by removing the coin <math>D_i</math> from our supposed optimal change for <math>T</math>, then we could just add the coin <math>D_i</math> back in and get change for the original amount <math>T</math> in fewer coins, a contradiction. Therefore, if we let <math>f(x)</math> denote the minimal number of coins required to change amount <math>x</math>, then we can write <math>f(x) = 1 + \min_i f(x - D_i)</math>; we consider all possible minimal solutions to <math>x</math> minus one coin, and take the best one and add that coin back in to get minimal change for <math>x</math>. The base case is <math>f(0) = 0</math>; obviously, 0 coins are required to make change for 0. See [[Dynamic_programming#Optimization_example:_Change_problem|the DP article]] for details and an implementation.

TODO: counting problem

==Notes and References==
<references/>

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 The former, the optimization problem, should be very familiar. It is a problem that cashiers solve when (for example) you hand them a five-dollar bill for a purchase of $3.44, and they have to find a way to give you back $1.56 in ''change'', preferably with as few coins as possible.<ref>The Canadians have a $2.00 coin, a $1.00 coin, a $0.25 coin, a $0.10 coin, a $0.05 coin, and a $0.01 coin. The solution to this problem in Canadian currency is one $1.00 coin, two $0.25 coins, one $0.05 coin, and one $0.01 coin.</ref> It is a well-studied problem that is extensively treated in the literature. The counting problem is somewhat more exotic; it arises mostly as a mathematical curiosity, often in the form of the puzzle: "How many ways are there to make change for a dollar?"<ref>The denominations in circulation in the United States are the same as those in Canada, except for the absence of a $2.00 coin and the presence of a $0.50 coin. In United States currency, the answer is 293 if the trivial solution consisting of a single one-dollar coin is counted, or 292 if it is not.</ref> It is also not often encountered in the literature, but it is discussed here because it occasionally appears in algorithmic programming competitions.
-The men's prostate is central to the a part of a male's the reproductive system. It secretes fluids that aid in the transportation and activation of sperm. The prostate gland is situated just while watching rectum, below the bladder and around the urethra. When there is prostate problem, it will always be really miserable and inconvenient to the patient as his urinary system is directly affected.
+==Discussion of complexity==
+The corresponding decision problem, which simply asks us to determine whether or not making change is ''possible'' with the given denominations (which it might not be, if we are missing the denomination 1) is known to be [[NP-complete]].<ref>G. S. Lueker. (1975). ''Two NP-complete problems in nonnegative integer programming.'' Technical Report 178, Computer Science Laboratory, Princeton University. (The authors were not able to obtain a copy of this paper, but in the literature it is invariably cited to back up the claim that change is NP-complete.)</ref> It follows that the optimization and counting problems are both [[NP-hard]] (''e.g.'', because the result of 0 for the counting problem answers the decision problem in the negative, and any nonzero value answers it in the affirmative).
-The common prostate health conditions are prostate infection, enlarged prostate and prostate type of cancer.
+However, as we shall see, a simple <math>O(nT)</math> solution exists for both versions of the problem. Why then are these problems not in P? The answer is that the ''size'' of the input required to represent the number <math>T</math> is actually the ''length'' of the number <math>T</math>, which is <math>\Theta(\log T)</math> when <math>T</math> is expressed in binary (or decimal, or whatever). Thus, the time and space required by the algorithm is actually <math>O(n 2^{\lg T})</math>, that is, exponential in the size of the input. (This simplified analysis does not take into account the sizes of the denominations, but captures the essence of the argument.) This algorithm is then said to be ''pseudo-polynomial''. No true polynomial-time algorithm is known (and, indeed, none will be found unless it turns out that P = NP).
-−
-−
-Prostate infection, often known as prostatitis, is the most common prostate-related overuse injury in men younger than 55 years old. Infections in the prostate gland are classified into four types - acute bacterial prostatitis, chronic bacterial prostatitis, chronic abacterial prostatitis and prosttodynia.
-−
-Acute bacterial prostatitis may be the least common of all kinds of prostate infection. It is a result of bacteria based in the large intestines or urinary tract. Patients can experience fever, chills, body aches, back pains and urination problems. This condition is treated by utilizing antibiotics or non-steroid anti-inflammatory drugs (NSAIDs) to alleviate the swelling.
-−
-Chronic bacterial prostatitis is a condition of the particular defect inside the gland and also the persistence presence of bacteria inside urinary tract. It can be brought on by trauma to the urinary tract or by infections via other parts in the body. A patient may experience testicular pain, lower back pains and urination problems. Although it is uncommon, it could be treated by removal in the prostate defect as well as the use antibiotics and NSAIDs to deal with the inflammation.
-−
-Non-bacterial prostatitis is the reason approximately 90% of most prostatitis cases; however, researchers have not yet to create the cause of these conditions. Some researchers feel that chronic non-bacterial prostatitis occur as a result of unknown infectious agents while other think that intensive exercise and heavy lifting may cause these infections.
-−
-Maintaining a Healthy Prostate
-−
-To prevent prostate diseases, an appropriate diet is important. These are some with the things you can do and also hardwearing . prostate healthy.
-−
-. Drink sufficient water. Proper hydration is necessary for general health and it'll also keep your urinary track clean.
-−
-. Some studies declare that a few ejaculations weekly will help to prevent prostate cancer.
-−
-. Eat beef sparingly. It has been shown that consuming a lot more than four meals of beef per week will raise the chance of prostate diseases and cancer.
-−
-. Maintain a proper diet with cereals, vegetable and fruits to make sure sufficient intake of nutrients required for prostate health.
-−
-The most significant measure to look at to make certain a healthy prostate is usually to select regular prostate health screening. If you are forty yrs . old and above, you should select prostate examination at least per year.
 ==Greedy algorithm==

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 Many real-world currency systems admit a [[greedy solution]] to the optimization version of the change problem. This algorithm is as follows: repeatedly choose the largest denomination that is less than or equal to the target amount, and ''use it'', that is, subtract it from the target amount, and then repeat this procedure on the reduced value, until the target amount decreases to zero. For example, with Canadian currency, we can greedily make change for $0.63 as follows: the largest denomination that fits into $0.63 is $0.25, so we subtract that (and thus resolve to use a $0.25 coin); we are left with $0.38, and take out another $0.25, so we subtract that again to obtain $0.13 (so that we have used two $0.25 coins so far); now the largest denomination that fits is $0.10, so we subtract that out, leaving us with $0.03; and then we subtract three $0.01 coins, leaving us with $0.00, at which point the algorithm terminates; so we have used six coins (two $0.25 coins, one $0.10 coin, and three $0.01 coins).
-It turns out that the greedy algorithm ''always'' gives the correct result for both Canadian and United States currencies (the proof is left as an exercise for the reader). There are various other real-world currency systems for which this is also true. However, there are simple examples of sets of denominations for which the greedy algorithm does ''not'' give a correct solution. For example, with the set of denominations <math>{1, 3, 4}</math>, the greedy algorithm will change 6 as 4+1+1, using three coins, whereas the correct minimal solution is obviously 3+3. There are also cases in which the greedy algorithm will fail to make change at all (consider what happens if we try to change 7 using the denominations <math>{3, 4}</math>). This usually does not occur in real-world systems because they tend to have denominations that are quite a bit more "spaced out".
+It turns out that the greedy algorithm ''always'' gives the correct result for both Canadian and United States currencies (the proof is left as an exercise for the reader). There are various other real-world currency systems for which this is also true. However, there are simple examples of sets of denominations for which the greedy algorithm does ''not'' give a correct solution. For example, with the set of denominations <math>{1, 3, 4}</math>, the greedy algorithm will change 6 as 4+1+1, using three coins, whereas the correct minimal solution is obviously 3+3. There are also cases in which the greedy algorithm will fail to make change at all (consider what happens if we try to change 6 using the denominations <math>{3, 4}</math>). This usually does not occur in real-world systems because they tend to have denominations that are quite a bit more "spaced out".
 Obviously, there is no greedy solution to the counting problem.

@@ Line 5: / Line 5: @@
 ==Discussion of complexity==
-The corresponding decision problem, which simply asks us to determine whether or not making change is ''possible'' with the given denominations (which it might not be, if we are missing the denomination 1) is known to be [[NP-complete]].<ref>G. S. Lueker. (1975). ''Two NP-complete problems in nonnegative integer programming.'' Technical Report 178, Computer Science Laboratory, Princeton University''</ref> It follows that the optimization and counting problems are both [[NP-hard]] (''e.g.'', because the result of 0 for the counting problem answers the decision problem in the negative, and any nonzero value answers it in the affirmative).
+The corresponding decision problem, which simply asks us to determine whether or not making change is ''possible'' with the given denominations (which it might not be, if we are missing the denomination 1) is known to be [[NP-complete]].<ref>G. S. Lueker. (1975). ''Two NP-complete problems in nonnegative integer programming.'' Technical Report 178, Computer Science Laboratory, Princeton University. (The authors were not able to obtain a copy of this paper, but in the literature it is invariably cited to back up the claim that change is NP-complete.)</ref> It follows that the optimization and counting problems are both [[NP-hard]] (''e.g.'', because the result of 0 for the counting problem answers the decision problem in the negative, and any nonzero value answers it in the affirmative).
 However, as we shall see, a simple <math>O(nT)</math> solution exists for both versions of the problem. Why then are these problems not in P? The answer is that the ''size'' of the input required to represent the number <math>T</math> is actually the ''length'' of the number <math>T</math>, which is <math>\Theta(\log T)</math> when <math>T</math> is expressed in binary (or decimal, or whatever). Thus, the time and space required by the algorithm is actually <math>O(n 2^{\lg T})</math>, that is, exponential in the size of the input. (This simplified analysis does not take into account the sizes of the denominations, but captures the essence of the argument.) This algorithm is then said to be ''pseudo-polynomial''. No true polynomial-time algorithm is known (and, indeed, none will be found unless it turns out that P = NP).
@@ Line 19: / Line 19: @@
 The optimization problem exhibits optimal substructure, in the sense that if we remove any coin of value <math>D_i</math> from the optimal means of changing <math>T</math>, then the set of coins remaining is an optimal means of changing <math>T - D_i</math>. This is because if this were ''not'' so; that is, there existed a means of changing <math>T - D_i</math> that used fewer coins than what we obtained by removing the coin <math>D_i</math> from our supposed optimal change for <math>T</math>, then we could just add the coin <math>D_i</math> back in and get change for the original amount <math>T</math> in fewer coins, a contradiction. Therefore, if we let <math>f(x)</math> denote the minimal number of coins required to change amount <math>x</math>, then we can write <math>f(x) = 1 + \min_i f(x - D_i)</math>; we consider all possible minimal solutions to <math>x</math> minus one coin, and take the best one and add that coin back in to get minimal change for <math>x</math>. The base case is <math>f(0) = 0</math>; obviously, 0 coins are required to make change for 0. See [[Dynamic_programming#Optimization_example:_Change_problem|the DP article]] for details and an implementation.
-TODO: counting problem
+The counting problem is more subtle. We cannot approach it in quite the same way as we approach the optimization problem, because the counting problem does not exhibit disjoint substructure when it is "sliced" this way. For example, if the denominations are 2 and 3, and the target amount is 5, then we might try to conclude that the number of ways of changing 5 is the number of ways of changing 2 plus the number of ways of changing 3, because we can either add a coin of value 2 to any way of changing 3 or add a coin of value 3 to any way of changing 2. Alas, this gives the incorrect answer that there are 2 ways of changing 5, whereas in actual fact we double-counted the solution 2+3 and there is, in fact, only one way to change 5.
 ==Notes and References==
 <references/>