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## Bisection Search

If the cards in the deck are not in order, there is no way to search that is faster than the linear search. We have to look at every card, since otherwise there is no way to be certain the card we want is not there.

But when you look for a word in a dictionary, you don't search linearly through every word. The reason is that the words are in alphabetical order. As a result, you probably use an algorithm that is similar to a bisection search:

1. Start in the middle somewhere.
2. Choose a word on the page and compare it to the word you are looking for.
3. If you found the word you are looking for, stop.
4. If the word you are looking for comes after the word on the page, flip to somewhere later in the dictionary and go to step 2.
5. If the word you are looking for comes before the word on the page, flip to somewhere earlier in the dictionary and go to step 2.

If you ever get to the point where there are two adjacent words on the page and your word comes between them, you can conclude that your word is not in the dictionary. The only alternative is that your word has been misfiled somewhere, but that contradicts our assumption that the words are in alphabetical order.

In the case of a deck of cards, if we know that the cards are in order, we can write a version of find that is much faster. The best way to write a bisection search is with a recursive function. That's because bisection is naturally recursive.

The trick is to write a function called findBisect that takes two indices as parameters, low and high, indicating the segment of the vector that should be searched (including both low and high).

1. To search the vector, choose an index between low and high, and call it mid. Compare the card at mid to the card you are looking for.
2. If you found it, stop.
3. If the card at mid is higher than your card, search in the range from low to mid-1.
4. If the card at mid is lower than your card, search in the range from mid+1 to high.

Steps 3 and 4 look suspiciously like recursive invocations. Here's what this all looks like translated into C++:

int findBisect (const Card& card, const pvector<Card>& deck,
int low, int high) {
int mid = (high + low) / 2;

// if we found the card, return its index
if (equals (deck[mid], card)) return mid;

// otherwise, compare the card to the middle card
if (deck[mid].isGreater (card)) {
// search the first half of the deck
return findBisect (card, deck, low, mid-1);
} else {
// search the second half of the deck
return findBisect (card, deck, mid+1, high);
}
}

Although this code contains the kernel of a bisection search, it is still missing a piece. As it is currently written, if the card is not in the deck, it will recurse forever. We need a way to detect this condition and deal with it properly (by returning -1).

The easiest way to tell that your card is not in the deck is if there are no cards in the deck, which is the case if high is less than low. Well, there are still cards in the deck, of course, but what I mean is that there are no cards in the segment of the deck indicated by low and high.

With that line added, the function works correctly:

int findBisect (const Card& card, const pvector<Card>& deck,
int low, int high) {

cout << low << ", " << high << endl;

if (high < low) return -1;

int mid = (high + low) / 2;

if (equals (deck[mid], card)) return mid;

if (deck[mid].isGreater (card)) {
return findBisect (card, deck, low, mid-1);
} else {
return findBisect (card, deck, mid+1, high);
}
}

I added an output statement at the beginning so I could watch the sequence of recursive calls and convince myself that it would eventually reach the base case. I tried out the following code:

cout << findBisect (deck, deck[23], 0, 51));

And got the following output:

0, 51
0, 24
13, 24
19, 24
22, 24
I found the card at index = 23

Then I made up a card that is not in the deck (the 15 of Diamonds), and tried to find it. I got the following:

0, 51
0, 24
13, 24
13, 17
13, 14
13, 12
I found the card at index = -1

These tests don't prove that this program is correct. In fact, no amount of testing can prove that a program is correct. On the other hand, by looking at a few cases and examining the code, you might be able to convince yourself.

The number of recursive calls is fairly small, typically 6 or 7. That means we only had to call equals and isGreater 6 or 7 times, compared to up to 52 times if we did a linear search. In general, bisection is much faster than a linear search, especially for large vectors.

Two common errors in recursive programs are forgetting to include a base case and writing the recursive call so that the base case is never reached. Either error will cause an infinite recursion, in which case C++ will (eventually) generate a run-time error.

Last Update: 2005-11-21