Mastering Dynamic Programming Techniques for Trees

Dynamic Programming (II) Tim Au Yeung

Tree DP • Dynamic Programming on Tree • “always” from leaves to root • Children node pass information to parent, obtain solution from root • Unrooted tree • Choose any node to be the root • Rooted tree: • Use specific root

Tree DP • Build tree • DFS/BFS • Set base case on leaves • Backtrack and update the optimal value for each node • Obtain solution at root

Tree DP - Balancing Act • http://poj.org/problem?id=1655 • Given a tree with N nodes • Define Balance[i] as the max size of subtrees after removing node i • Output: x • where Balance[x] is min

Tree DP - Balancing Act • PreProcess • DFS and DP: get num[i], the number of nodes in subtree rooted at i • Calculate Balance[i]

Tree DP - Apple Tree • http://poj.org/problem?id=2486 • Given undirected tree with N nodes (1..N) • Wi for node i (1<=i<=N) • Gain Wi score when you FIRST visit node i • Start from node 1 • At most K step • Output: Max score • N<=100; K<=200;

Tree DP - Apple Tree • DFS from node 1 • dp[i][j][0]: • max score obtained in subtree rooted at node i • starting at node i • walk at most j step • back to node i • dp[i][j][1]: • max score obtained in subtree rooted at node i • starting at node i • walk at most j step • May NOT back to node i • Ans = dp[1][K][1]

Tree DP - Rebuilding Roads • http://poj.org/problem?id=1947 • Given unrooted tree with N nodes • Output min number of edges whose destruction would isolate a subtree with exactly P nodes • 1 <= N <= 150; 1 <= P <= N;

Tree DP - Rebuilding Roads • dp[i][j]: min number of edges destruction to isolate a subtree rooted at i and with j nodes • Consider child x • If reserve x, • dp[i][j] = min(dp[i][j-k]+dp[x][k]) 0 <= k <= j • Else • dp[i][j] = dp[i][j] + 1 • Ans: min{dp[i][P]}

Advanced DP - Constants in the language of Shakespeare • http://codeforces.com/problemset/problem/132/D • You are given an integer n. You have to represent it as n = a1 + a2 + ... + am, where each of ai is a non-negative power of 2, possibly multiplied by -1. • Find a representation which minimizes the value of m. • Input: positive integer n, written as its binary notation. The length of the notation is at most 106.

Advanced DP - Lucky Country • http://codeforces.com/problemset/problem/95/E • Positive integers are lucky if their it doesn't contain digits other than 4 and 7. • Each island belongs to exactly one region, there is a path between any two islands located in the same region; there is no path between any two islands from different regions. • A region is lucky if the amount of islands in it is a lucky number. • Find the minimum number of roads needed to build to create a lucky region. • n: number of islands; m: the number of roads • 1 ≤ n, m ≤ 105

Advanced DP - Hyper String • http://codeforces.com/problemset/problem/176/D • A Hyper String is made by concatenation of some base strings. Suppose you are given a list of base strings b1, b2, ..., bn. Now the Hyper String made from indices list i1, i2, ..., im is concatenation of base strings bi1, bi2, ..., bim. • Compute the length of the longest common sub-sequence of a Hyper String t with a string s • 1 ≤ n ≤ 2000 • 1 ≤ m ≤ 2000

Problem List • Tree DP • NOI 2003 逃学的小孩 • NOI 2002 贪吃的九头龙 • Ural 1031 1039 1056 1073 1078 • POJ 1947 1155 1655 3107 2486 • http://codeforces.com/problemset/tags/dp

Mastering Dynamic Programming Techniques for Trees