Word Wrap

Given a sequence of n words and the limit of the length of each line M, let the cost of putting k words with total m characters in a line be (M - m - k + 1)^3, design an O(n lg n) time algorithm to find the optimal way to break the sequence of words into lines with minimum cost.


The solution is based on dynamic programming. Let p[i] be the total length for the first i words. Let c[i][j] be the cost of putting the i-th word to j-th word in the same line. By using the prefix sum of the length, each element of c[i][j] can be computed in constant time. Let F[i] be the optimal objective value for the subproblem containing the first i words. We get a recurrence relation F[j] = \min_{1 \leq i < j} F[i] + c[i][j]. By using the recurrence relation, we get an O(n^2) time algorithm.

In order to speed up, note that c[i][j] satisfies quadrangle inequality, that is c[i][k] + c[j][l] ≤ c[i][l] + c[j][k] for all i ≤ j ≤ k ≤ l. Furthermore, let d[i][j] = F[i] + c[i][j], the problem becomes finding the minimum for each column of matrix d. Note that the matrix d also satisfies quadrangle inequality. We will use a queue = (i_1, h_i), …, (i_k, h_k) to maintain the candidates, where i_1 \leq i_2 \dots \leq i_k and (i_1, h_1) is in the front of the queue. When processing to the j-th column, one element (i_r, h_r) in the queue means that in the submatrix d[1..j-2][j..n], row i_r is the column minimum for columns h_r to h_{r+1} - 1. Initially, the queue only contains (1, 1). When consider the column j, if d[j-1][n] is smaller then d[i_k][n], then we need to dequeue until the d[i_r][h_r] < d[j-1][h_r], where i_r is the rear of the queue. Then, we find the smallest h > h_r, such that d[i_r][h] < d[j-1][h] and enqueue (j-1, h+1). Finding such h needs O(lg n) time by using binary search. Since each index can be enqueue and dequeue once, the total complexity is O(n lg n).

Note: This problem can be solved in linear time by using slope optimization.


解法是基於動態規劃。令p[i]為第一個字到第i個字的總長度。令c[i][j]為把第i個字到第j個字放到一行的費用。利用陣列p,每一個c[i][j]都可以在常數時間內計算出。令F[i]為前i個字的最佳解之花費,我們可得一遞迴關係式F[j] = \min_{1 \leq i < j} F[i] + c[i][j]。 利用此遞迴關係,我們可得到O(n^2)時間的演算法。

有一個方法可以加速,因為c[i][j]滿足四邊形不等式,意即c[i][k] + c[j][l] ≤ c[i][l] + c[j][k]對所有的 i ≤ j ≤ k ≤ l。令d[i][j] = F[i] + c[i][j],問題就變成是在矩陣d中找出每列的最小值。 此時矩陣d同樣也滿足四邊形不等式。我們可以利用一個queue (i_1, h_i), …, (i_k, h_k)來維護可能的最佳解範圍,其中i_1 \leq i_2 \dots \leq i_k且(i_1, h_1)是queue的頭。當處理到第j列時,queue中的元素(i_r, h_r)表示在子矩陣d[1..j-2][j..n]中,第i_r行是第h_r列到第h_{r+1} - 1列的最小值。一開始queue中只包含(1, 1)。當處理到第j列時,如果d[j-1][n]比d[i_k][n]小,那麼我們必須把元素從queue中移除直到d[i_r][h_r] < d[j-1][h_r],i_r是queue的尾。然後我們找出最小的h > h_r,使得d[i_r][h] < d[j-1][h],然後把(j-1, h+1) enqueue。利用二分搜尋可以在O(lg n)的時間內找到此h。因為每一個索引只能被enqueue和dequeue一次,所以總時間複雜度為O(n lg n)。

Reference: Zvi Galil and Kunsoo Park, “Dynamic programming with convexity, concavity and sparsity,” Theoretical Computer Science Volume 92, Issue 1, 6 January 1992, Pages 49–76.



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