Problem

The thief has found himself a new place for his thievery again. There is only one entrance to this area, called the "root." Besides the root, each house has one and only one parent house. After a tour, the smart thief realized that "all houses in this place forms a binary tree". It will automatically contact the police if two directly-linked houses were broken into on the same night.

Determine the maximum amount of money the thief can rob tonight without alerting the police.

Example 1:

     3
    / \
   2   3
    \   \ 
     3   1

Maximum amount of money the thief can rob = 3 + 3 + 1 = 7

Example 2:

     3
    / \
   4   5
  / \   \ 
 1   3   1

Maximum amount of money the thief can rob = 4 + 5 = 9

错误思路

• 一开始单纯地以为是一层一层，所以用了bfs。实际上左右子树节点可以不在同一层

完美思路

Step I -- Think naively

At first glance, the problem exhibits the feature of "optimal substructure": if we want to rob maximum amount of money from current binary tree (rooted atroot), we surely hope that we can do the same to its left and right subtrees.

So going along this line, let's define the functionrob(root)which will return the maximum amount of money that we can rob for the binary tree rooted atroot; the key now is to construct the solution to the original problem from solutions to its subproblems, i.e., how to getrob(root)fromrob(root.left), rob(root.right), ...etc.

Apparently the analyses above suggest a recursive solution. And for recursion, it's always worthwhile figuring out the following two properties:

1. Termination condition: when do we know the answer torob(root)without any calculation? Of course when the tree is empty -- we've got nothing to rob so the amount of money is zero.

2. Recurrence relation: i.e., how to getrob(root)fromrob(root.left), rob(root.right), ...etc. From the point of view of the tree root, there are only two scenarios at the end:rootis robbed or is not. If it is, due to the constraint that "we cannot rob any two directly-linked houses", the next level of subtrees that are available would be the four "grandchild-subtrees" (root.left.left, root.left.right, root.right.left, root.right.right). However ifrootis not robbed, the next level of available subtrees would just be the two "child-subtrees" (root.left, root.right). We only need to choose the scenario which yields the larger amount of money.

Here is the program for the ideas above:

public int rob(TreeNode root) {
    if (root == null) return 0;

    int val = 0;

    if (root.left != null) {
        val += rob(root.left.left) + rob(root.left.right);
    }

    if (root.right != null) {
        val += rob(root.right.left) + rob(root.right.right);
    }

    return Math.max(val + root.val, rob(root.left) + rob(root.right));
}

However the solution runs very slowly (1186 ms) and barely got accepted (the time complexity turns out to be exponential, see my comments below).

Step II -- Think one step further

In stepI, we only considered the aspect of "optimal substructure", but think little about the possibilities of overlapping of the subproblems. For example, to obtainrob(root), we needrob(root.left), rob(root.right), rob(root.left.left), rob(root.left.right), rob(root.right.left), rob(root.right.right); but to getrob(root.left), we also needrob(root.left.left), rob(root.left.right), similarly forrob(root.right). The naive solution above computed these subproblems repeatedly, which resulted in bad time performance. Now if you recall the two conditions for dynamic programming: "optimal substructure" + "overlapping of subproblems", we actually have a DP problem. A naive way to implement DP here is to use a hash map to record the results for visited subtrees.

And here is the improved solution:

public int rob(TreeNode root) {
    return robSub(root, new HashMap
<
>
());
}

private int robSub(TreeNode root, Map
<
TreeNode, Integer
>
 map) {
    if (root == null) return 0;
    if (map.containsKey(root)) return map.get(root);

    int val = 0;

    if (root.left != null) {
        val += robSub(root.left.left, map) + robSub(root.left.right, map);
    }

    if (root.right != null) {
        val += robSub(root.right.left, map) + robSub(root.right.right, map);
    }

    val = Math.max(val + root.val, robSub(root.left, map) + robSub(root.right, map));
    map.put(root, val);

    return val;
}

The runtime is sharply reduced to9 ms, at the expense ofO(n)space cost (nis the total number of nodes; stack cost for recursion is not counted).

Step III -- Think one step back

In stepI, we defined our problem asrob(root), which will yield the maximum amount of money that can be robbed of the binary tree rooted atroot. This leads to the DP problem summarized in stepII.

Now let's take one step back and ask why we have overlapping subproblems. If you trace all the way back to the beginning, you'll find the answer lies in the way how we have definedrob(root). As I mentioned, for each tree root, there are two scenarios: it is robbed or is not.rob(root)does not distinguish between these two cases, so "information is lost as the recursion goes deeper and deeper", which results in repeated subproblems.

If we were able to maintain the information about the two scenarios for each tree root, let's see how it plays out. Redefinerob(root)as a new function which will return an array of two elements, the first element of which denotes the maximum amount of money that can be robbed ifrootisnot robbed, while the second element signifies the maximum amount of money robbed if it is robbed.

Let's relaterob(root)torob(root.left)androb(root.right)..., etc. For the 1st element ofrob(root), we only need to sum up the larger elements ofrob(root.left)androb(root.right), respectively, sincerootis not robbed and we are free to rob its left and right subtrees. For the 2nd element ofrob(root), however, we only need to add up the 1st elements ofrob(root.left)androb(root.right), respectively, plus the value robbed fromrootitself, since in this case it's guaranteed that we cannot rob the nodes ofroot.leftandroot.right.

As you can see, by keeping track of the information of both scenarios, we decoupled the subproblems and the solution essentially boiled down to a greedy one. Here is the program:

public int rob(TreeNode root) {
    int[] res = robSub(root);
    return Math.max(res[0], res[1]);
}

private int[] robSub(TreeNode root) {
    if (root == null) return new int[2];

    int[] left = robSub(root.left);
    int[] right = robSub(root.right);
    int[] res = new int[2];

    res[0] = Math.max(left[0], left[1]) + Math.max(right[0], right[1]);
    res[1] = root.val + left[0] + right[0];

    return res;
}