Section 3. Recursive Backtracking


Section materials curated by Zheng Lian, drawing upon materials from previous quarters.

This week’s section exercises continue our exploration of recursion to tackle even more challenging and interesting problems. In particular, many of this week's section problems center around recursive backtracking, a very powerful and versatile problem solving technique.

Remember that every week we will also be releasing a Qt Creator project containing starter code and testing infrastructure for that week's section problems. When a problem name is followed by the name of a .cpp file, that means you can practice writing the code for that problem in the named file of the Qt Creator project. Here is the zip of the section starter code:

📦 Starter code

1) Recursion Mystery Part 2

Topics: recursive function calls, output tracing

void recursionMystery2(int x, int y) {
    if (y == 1) { 
        cout << x;
    } else {
        cout << (x * y) << ", ";
        recursionMystery2(x, y - 1);
        cout << ", " << (x * y);
    } 
}

For each call to the above recursive function, write the output that would be produced, as it would appear on the console.

Call                                   Output
        
recursionMystery2(4, 1);        ___________________________________
recursionMystery2(4, 2);        ___________________________________
recursionMystery2(8, 2);        ___________________________________
recursionMystery2(4, 3);        ___________________________________
recursionMystery2(3, 4);        ___________________________________
4
8, 4, 8
16, 8, 16
12, 8, 4, 8, 12
12, 9, 6, 3, 6, 9, 12

2) Recursion Tracing

Topics: Recursion, strings, recursion tracing

Below is a recursive function to reverse a string.

string reverseOf(string s) {
    if (s.empty()) {
        return "";
    } else {
        return reverseOf(s.substr(1)) + s[0];
    }
}

Trace through the execution of reverseOf("stop") along the lines of what we did in lecture, showing recursive call information for each call that’s made and how the final value gets computed.

Our initial call to reverseOf("stop") fires off a call to reverseOf("top"). This call fires off a call to reverseOf("op"). This in turn calls reverseOf("p"). This in turn calls reverseOf(""). This triggers the base case and returns the empty string. (Notice that the reverse of the empty string "" is indeed the empty string ""). We now append p to return "p". We now append o to return "po". We append t to return "pot". And finally we append s to return "pots" back to whoever called us. Yay!


3) Win some, lose sum (sum.cpp)

Topics: recursive backtracking

Write a recursive function named canMakeSum that takes a reference to a Vector<int> and an int target value and returns true if it is possible to have some selection of values from the Vector that sum to the target value. In particular, you should be implementing a function with the following declaration

bool canMakeSum(Vector<int>& values, int target)

For example, let's say that we executed the following code

Vector<int> nums = {1,1,2,3,5};
canMakeSum(nums, 9)

We should expect that the call to canMakeSum should return true. Given the values specified in nums, we can select 1, 3, and 5 such that 5 + 3 + 1 = 9.

However, let's say that we executed the following code instead

Vector<int> nums = {1,4,5,6};
canMakeSum(nums, 8);

We should expect that the call to canMakeSum in this case should return false, since there is no possible combination of values from the vector that sum up to the target value of 8.

bool canMakeSumHelper(Vector<int>& v, int target, int cur, int index) {
	if (index >= v.size()) {
		return cur == target;
	}
	return canMakeSumHelper(v, target, cur + v[index], index + 1) ||
		   canMakeSumHelper(v, target, cur, index + 1);
}

bool canMakeSum(Vector<int>& v, int target) {
	return canMakeSumHelper(v, target, 0, 0);
}

4) Weights and Balances (weights.cpp)

Topics: recursive backtracking

I am the only child of parents who weighed,
measured, and priced everything; for whom
what could not be weighed, measured, and
priced had no existence.
—Charles Dickens, Little Dorrit, 1857

In Dickens’s time, merchants measured many commodities using weights and a two-pan balance – a practice that continues in many parts of the world today. If you are using a limited set of weights, however, you can only measure certain quantities accurately.

For example, suppose that you have only two weights: a 1-ounce weight and a 3-ounce weight. With these you can easily measure out 4 ounces, as shown below:

two-pan balance with unmarked weight on the left side and two weights on the right side, marked 1 and 3. The balance is level.

It’s more interesting to discover that you can also measure out 2 ounces by shifting the 1-ounce weight to the other side, as follows below:

two-pan balance with weight marked 1 and unmarked weight on the left side and one weight on the right side, marked 3. The balance is level.

Write a recursive function

bool isMeasurable(int target, Vector<int>& weights)

that determines whether it is possible to measure out the desired target amount with a given set of weights, which is stored in the vector weights.

As an example, the function call

Vector<int> weights = {1, 3};
isMeasurable(2, weights);

should return true because it is possible to measure out two ounces using the sample weight set as illustrated in the preceding diagram. On the other hand, calling

Vector<int> weights = {1, 3};
isMeasurable(5, weights);

should return false because it is impossible to use the 1- and 3-ounce weights to add up to 5 ounces. However, the call

Vector<int> weights = {1, 3, 7};
isMeasurable(6, weights);

should return true: you can measure the six-ounce weight by placing it and the one-ounce weight on one side of the scale and the seven-ounce weight on the other.

Here’s a function question to ponder: let’s say that you get to choose n weights. Which ones would you pick to give yourself the best range of weights that you’d be capable of measuring?

bool isMeasurable(int target, Vector<int>& weights) {
	if (weights.isEmpty()) {
		return target == 0; // base case; no weights left to place
	} else {
		int last = weights[weights.size() - 1]; //using last index is fastest
		weights.remove(weights.size() - 1);

		// "choose and explore" all of the three possibilities
		bool result = isMeasurable(target + last, weights) ||
			   isMeasurable(target - last, weights) ||
			   isMeasurable(target, weights);
		// un-choose
		weights.add(last);
		return result;
 	}
}

5) Change We Can Believe In (change.cpp)

Topic: Recursive Backtracking

In the US, as is the case in most countries, the best way to give change for any total is to use a greedy strategy – find the highest-denomination coin that’s less than the total amount, give one of those coins, and repeat. For example, to pay someone 97¢ in the US in cash, the best strategy would be to

  • give a half dollar (50¢ given, 47¢ remain), then
  • give a quarter (75¢ given, 22¢ remain), then
  • give a dime (85¢ given, 12¢ remain), then
  • give a dime (95¢ given, 2¢ remain), then
  • give a penny (96¢ given, 1¢ remain), then
  • give another penny (97¢ given, 0¢ remain).

This uses six total coins, and there’s no way to use fewer coins to achieve the same total.

However, it’s possible to come up with coin systems where this greedy strategy doesn’t always use the fewest number of coins. For example, in the tiny country of Recursia, the residents have decided to use the denominations 1¢, 12¢, 14¢, and 63¢, for some strange reason. So suppose you need to give back 24¢ in change. The best way to do this would be to give back two 12¢ coins. However, with the greedy strategy of always picking the highest-denomination coin that’s less than the total, you’d pick a 14¢ coin and ten 1¢ coins for a total of eleven coins. That’s pretty bad!

Your task is to write a function

int fewestCoinsFor(int cents, Set<int>& coins)

that takes as input a number of cents and a Set indicating the different denominations of coins used in a country, then returns the minimum number of coins required to make change for that total. In the case of US coins, this should always return the same number as the greedy approach, but in general it might return a lot fewer!

You can assume that the set of coins always contains a 1¢ coin, so you never need to worry about the case where it’s simply not possible to make change for some total. You can also assume that there are no coins worth exactly 0¢ or a negative number of cents. Finally, you can assume that the number of cents to make change for is nonnegative.

Makes cents? I certainly hope so.

The idea behind this solution is the following: if we need to make change for zero cents, the only (and, therefore, best!) option is to use 0 coins. Otherwise, we need to give back at least one coin. What’s the first coin we should hand back? We don’t know which one it is, but we can say that it’s got to be one of the coins from our options and that that coin can’t be worth more than the total. So we’ll try each of those options in turn, see which one ends up requiring the fewest coins for the remainder, then go with that choice. The code for this is really elegant and is shown here:

/**
 * Given a collection of denominations and an amount to give in change, returns
 * the minimum number of coins required to make change for it.
 *
 * @param cents How many cents we need to give back.
 * @param coins The set of coins we can use.
 * @return The minimum number of coins needed to make change.
 */
int fewestCoinsFor(int cents, Set<int>& coins) {
	/* Base case: You need no coins to give change for no cents. */
	if (cents == 0) {
		return 0;
	}
	/* Recursive case: try each possible coin that doesn’t exceed the total as
	* as our first coin.
	*/
	else {
		int bestSoFar = cents + 1; // Can never need this many coins;
		for (int coin: coins) {
			/* If this coin doesn’t exceed the total, try using it. */
			if (coin <= cents) {
				bestSoFar = min(bestSoFar, 
					fewestCoinsFor(cents - coin, coins));
			}
		}
		return bestSoFar + 1; // For the coin we just used.
 	}
}

6) Longest Common Subsequence (lcs.cpp)

Topic: Recursive Backtracking

Write a recursive function named longestCommonSubsequence that returns the longest common subsequence of two strings passed as arguments. Some example function calls and return values are shown below.

Recall that if a string is a subsequence of another, each of its letters occurs in the longer string in the same order, but not necessarily consecutively.

Hint: In the recursive case, compare the first character of each string. What one recursive call can you make if they are the same? What two recursive calls do you try if they are different?

longestCommonSubsequence("cs106a", "cs106b") --> "cs106" 
longestCommonSubsequence("nick", "julie") --> "i" 
longestCommonSubsequence("karel", "c++") --> "" 
longestCommonSubsequence("she sells", "seashells") --> "sesells"
string longestCommonSubsequence(string s1, string s2) {
	if (s1.length() == 0 || s2.length() == 0) {
		return "";
 	} else if (s1[0] == s2[0]) {
 		return s1[0] + longestCommonSubsequence(s1.substr(1), 
 							s2.substr(1));
 	} else {
 		string choice1 = longestCommonSubsequence(s1, s2.substr(1));
		string choice2 = longestCommonSubsequence(s1.substr(1), s2);
		if (choice1.length() >= choice2.length()) {
			return choice1;
		} else {
			return choice2;
		}
	}
}

7) Cracking Passwords(crack.cpp)

Topic: Recursive Backtracking

Write a function crack that takes in the maximum length a site allows for a user's password and tries to find the password into an account by using recursive backtracking to attempt all possible passwords up to that length (inclusive). Assume you have access to the function bool login(string password) that returns true if a password is correct. You can also assume that the passwords are entirely alphabetic and case-sensitive. You should return the correct password you find, or the empty string if you cannot find the password. You should return the empty string if the maximum length passed is 0 or throw an integer exception if the length is negative.

Security note: The ease with which computers can brute-force passwords is the reason why login systems usually permit only a certain number of login attempts at a time before timing out. It’s also why long passwords that contain a variety of different characters are better! Try experimenting with how long it takes to crack longer and more complex passwords. See the comic here for more information: https://xkcd.com/936/

string crackHelper(string soFar, int maxLength) {
	if (login(soFar)) {
		return soFar;
	}
	if (soFar.size() == maxLength) {
		return "";
	}
	for (char c = 'a'; c <= 'z'; c++) {
		string password = crackHelper (soFar + c, maxLength);
		if (password != "") {
			return password;
		}
 		// Also check uppercase
 		char upperC = toupper(c);
 		password = crackHelper (soFar + upperC, maxLength);
		if (password != "") {
 			return password;
 		}
 	}
	return "";
}

string crack(int maxLength) {
	if (maxLength < 0) {
 		throw maxLength;
	}
 	return crackHelper("", maxLength);
}