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No of ways of coloring a tree black such that root is colored black initially and a node can only be colored black if its parent node is black?
By kritipandey, history, 3 years ago, In English

Appeared in Lowes test

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3 years ago, # |
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It can be solved by a dynamic approach. In order to get an answer for vertex, we can calculate answers for all children and then multiply them.

$$$dp[v]$$$ — number of ways color black the subtree of $$$v$$$.

$$$dp[leaf] = 1$$$, for all leafs

$$$dp[v] = \prod (dp[child] + 1)$$$

You can calculate it by simple dfs. Answer will be in $$$dp[root]$$$

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    3 years ago, # ^ |
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    isn't it $$$dp[v] = 1 + \prod dp[child]$$$?

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      3 years ago, # ^ |
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      $$$v$$$ has to be black, and a child can contain no black node at all, this is why there are $$$(dp[child] + 1)$$$ variations from each child. Am I wrong?

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        3 years ago, # ^ |
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        root
       /   \
      a     d
     / \
    c   b

        for this the answer should be 10 ig.

        when a is black, we can give c,b pair 4(2*2) different combinations. when a is white, c and b must be white . so here 4+1=5. 5 when d is white and 5 when d is black. so 5+5=10. as root can't be white ans will be 10

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          3 years ago, # ^ |
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          And by the algorithm above you get exactly this answer.

          $$$dp[c] = dp[b] = dp[d] = 1$$$

          $$$dp[a] = (1 + 1) * (1 + 1) = 4$$$

          $$$dp[root] = (4 + 1) * (1 + 1) = 10$$$

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    3 years ago, # ^ |
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    I think we need to color the whole tree with black color, and what matters is the ordering of coloring.. So, this dp approach won't work .. As it does not take into account the relative ordering.

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      3 years ago, # ^ |
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      Can you please show a case where the above solution won't work?

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      3 years ago, # ^ |
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      I think in this problem you don't need to consider an order of coloring, as it is not states so. Also, the author mentioned in comments that the answer for a rooted tree with two children is $$$4$$$ that proves order not matter.

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    3 years ago, # ^ |
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    oops, thought the root could be colored white

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3 years ago, # |
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https://atcoder.jp/contests/dp/tasks/dp_v Harder version of the same problem.

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3 years ago, # |
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I added Walmart to my boycott list ;)

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    3 years ago, # ^ |
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    You might want to add Lowe's as well . This problem appeared in my set in Lowe's Hiring contest Round 1 .

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3 years ago, # |
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This was the same question which was asked earlier in the LOWE's exam also. I solved it using DFS.

DFS
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