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Back to Recursion, Revisited. On to Sorting Algorithms.
Held Wednesday, February 24
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/** * Compute the minimum number of stamps to make up some postage. * <br>Precondition: The postage is nonnegative. * <br>Precondition: Stamp prices are positive. * <br>Precondition: The stamp prices are sorted in increasing order. * <br>Precondition: There is a penny stamp (so that we can make * any price) * <br>Postcondition: Computes the minimum number of stamps. */ public int minimumStamps(int postage, int[] stampPrices) { /* Strategy: this algorithm uses a greedy technique. We * repeatedly buy the largest possible stamp. */ // The number of stamps needed. int count = 0; // The index of the most expensive stamp to consider. int stamp = stampPrices.length - 1; // Repeatedly choose a price and remove copies until the // postage reaches 0. while (postage > 0) { // Find the most expensive stamp we still need. while (stampPrices[stamp] > postage) --stamp; // Buy a stamp at the price. postage -= stampPrices[stamp]; count +- 1; } // while (postage > 0) // That's it, we're done. return count; } // minimumStamps(int, int[])
for each stamp price, x compute the minumum number of stamps for postage-x add 1 choose the minimum of all those prices
/** * Compute the minimum number of stamps required for some postage. * All values are expressed in terms of cents. * Pre: postage >= 0. * Pre: stamp prices are positive. * Pre: fewer than Integer.MAX_VALUE stamps are required. * Post: returns the minimum number of stamps required for that * postage (but does not print out which stamps). * Post: throws an exception if it's not possible to make that * postage with the given stamp prices. */ public int minStamps(int postage, int[] stampPrices) throws Exception { // Our guess. Initialized to some outlandish value so that // we can tell if we don't have a solution. int guess = Integer.MAX_VALUE; // The next guess. Used as we step through potential prices. int nextGuess; // Base case: no stamps are required for no postage. if (postage == 0) return 0; // Recursive case: try each price until we find the smallest. else { for(int i = 0; i < stampPrices.length; ++i) { // If the current stamp is cheap enough, try to // improve our guess. if (stampPrices[i] <= postage) { try { // Make a new guess nextGuess = 1 + minStamps(postage-stampPrices[i], stampPrices); if (nextGuess < guess) guess = nextGuess; } // try // No number of stamps could be used for revised postage. // Don't update our guess. catch (Exception e) { } } // if the price was cheap enough. } // for // We've now computed the minimum number of stamps. Ensure // that it's a reasonable value. if (guess != Integer.MAX_VALUE) return guess; // Otherwise, there was no way to sum stamp prices to the given // postage, so throw an exception else throw new Exception("No combination of stamps for " + postage); } // Recursive case } // minStamps(int,int[])
% ji StampComputer 7 2
% ji StampComputer 8 1 3 5
% ji StampComputer 8 5 3 1
% ji StampComputer 9 1 3 5
% ji StampComputer 27 10 9 1
MIN_STAMPS
.
int[] MIN_STAMPS
MIN_STAMPS = new int[postage+1]; MIN_STAMPS[0] = 0; for (int i = 1; i <= postage; ++i) { MIN_STAMPS[i] = Integer.MAX_VALUE; } // for
if (MIN_STAMPS[postage] != Integer.MAX_VALUE) return MIN_STAMPS[postage];
MIN_STAMPS[postage] = guess;
NewStampComputer.java
% ji NewStampComputer 7 2
% ji NewStampComputer 8 1 3 5
% ji NewStampComputer 8 5 3 1
% ji NewStampComputer 9 1 3 5
% ji NewStampComputer 27 10 9 1
History
Back to Recursion, Revisited. On to Sorting Algorithms.
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