JavaScript is disabled on your browser. Enable JavaScript to use this site. Learn more

6.006 21. Dynamic Programming III : Text justification, Knapsack

Add to Favourites

Post to:

MIT OpenCourseWare http://ocw.mit.edu6.006Introduction to AlgorithmsSpring 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. �� • �� • �� • �� • �� • �� ×� �� = �� ⇒ � �� • �� • blahblahblahblahblahb l a hreallylongwordreallylongwordblahblahvs.....�� • �� (i, j) �� [i : j] �� > �� (�� )3 �� min ��• �� min �� [i :] = � �� = Θ(n) �� n = � �� ⇒ �� i : j = ⇒ � �� = n − i = O(n) �� • ��[i] = min��(i, j) � ��[j] �� j �� (i +1,n + 1)� • ��[n]= φ= �� = O(n)⇒ �� = O(n2) �� = ��[φ] �� ..ABC(AB)C costs (n2)A(BC) costs (n)�� Evaluation of an Expression �� (��)(��)······↑k−1 ↑k = � �� = O(n)⇒ �� A[i : j]= � �� Θ(n2)⇒ �� • ��[i, j] = min(�� [i, k]+ ��[k, j]+ �� (A[i] ··· A[k − 1]) �� (A[k] A[j − 1]) �� k �� (i +1,j))··· • ��[i, i +1] = φ= �� = O(n)⇒ �� = O(n3)�� = DP [0,n]�� S �� • �� i �� si � �� vi • �� ≤ S �� i = � �� 2⇒ �� • si ≤ S?!)DP [i] = max(DP [i + 1],vi + DP [i + 1] �� i �� • �� i�� X= � �� = O(nS) �⇒ �� • DP [i, X] = max(DP [i +1,X],vi + DP [i +1,X − si] �� si ≤ X) • DP [n, X]= φ= �� = O(1)⇒ �� = O(nS) �� = DP [φ, S] �� = �� ⇒ �� • �� =< S,s0, ··· ,sn−1,v0, ··· ,vn−1 > • �� O(lg S + lg s0 + ··· ) ≈ O(n lg ...) • �� O(nS) �� • O(nS) �� S �� • �� n �� w• �� x �� • �� • �� • �� w �� h• �� i�� i = � �� = O(w)⇒ �� DP [i]= DP [i + 1] �� i�� h0,h1, ,hw−1··· = � �� = O(nhw)⇒· �� DP [i, h] = max(DP [i, m] �� m �� i �� h)= �� = O(w)⇒ �� = O(nwhw) �� = DP [φ, φ] ��

Description
In this lecture notes we are going to continue with Dynamic Programming . The topics covered in this section are 1.Text Justification
2. Parenthesization, 3.Knapsack, 4.Pseudopolynomial Time and 5. Tetris Training. Dynamic programming is all about subproblems & guessing.

Instructors: Prof. Erik Demaine, Prof. Ronald Rivest, Prof. Srinivas Devadas, MIT Course Number: 6.006 Level: Undergraduate, 6.006 21. Dynamic programming III: text justification, parenthesization, knapsack, pseudopolynomial time, Tetris training ,Introduction to Algorithms, Electrical Engineering and Computer Science, Engineering, Massachusetts Institute of Technology: MIT Open Course Ware, http://ocw.mit.edu (02-11-2011). License: Creative Commons BY-NC-SA: http://ocw.mit.edu/terms/#cc.

Comments